Banks v. United States

OPINION

1

HEWITT, Chief Judge.

This is an action for just compensation filed by owners of property along the eastern shore of Lake Michigan. Plaintiffs’ properties are located along an area of the shoreline that erodes naturally, but allege that the government’s construction and maintenance of a pair of jetties effected a taking by speeding the erosion of their properties.

This Opinion addresses a jurisdictional issue that arose after the second trial held by the court to address the merits of plaintiffs’ claims. The court holds that, because plaintiffs’ claims accrued earlier than 1952, plaintiffs filed this action outside of the limitations period. The court therefore dismisses plaintiffs’ claims for lack of jurisdiction.

“ ‘Without jurisdiction the court cannot proceed at all in any cause. Jurisdiction *120 is power to declare the law, and when it ceases to exist, the only function remaining to the court is that of announcing the fact and dismissing the cause.’” Steel Co. v. Citizens for a Better Env’t, 523 U.S. 83, 94 , 118 S.Ct. 1003 , 140 L.Ed.2d 210 (1998) (quoting Ex parte McCardle, 74 U.S. (7 Wall) 506, 514, 19 L.Ed. 264 (1868)). For purposes of judicial efficiency, if the reviewing court in any appeal should disagree with the court’s view of its jurisdiction, and to avoid the possibility of a trial opinion being drafted months or years after the trial, and the possibility of a repetitive trial, the court also presents here its findings from the trial. These findings are presented in the alternative and, in the absence of jurisdiction, do not entitle plaintiffs to just compensation in the amounts determined by the court.

Before the court are Plaintiffs’ Post-Trial Brief (Pis.’ Br.), Docket Number (Dkt. No.) 495, filed June 21, 2011; United States’ Post-Trial Memorandum (Def.’s Br.), Dkt. No. 496, filed June 21, 2011; Plaintiffs’ Response to the United States’ Post-Trial Brief (Pis.’ Resp.), Dkt. No. 497, filed July 12, 2011; and United States’ Post-Trial Response Memorandum (Def.’s Resp.), Dkt. No. 498, filed July 12, 2011.

Also before the court is the following briefing on the topic of jurisdiction, filed pursuant to the court’s August 9, 2011 Order, Dkt. No. 499: Plaintiffs’ Post-Trial Brief on Jurisdiction (Pis.’ Jur. Br.), Dkt. No. 501, filed September 7, 2011; United States’ Supplemental PosL-Trial Memorandum (Def.’s Jur. Br.), Dkt. No. 502, filed September 7, 2011; United States’ Supplemental Post-Trial Response Memorandum (Def.’s Jur. Resp.), Dkt. No. 503, filed September 21, 2011; and Plaintiffs’ Post-Trial Reply Brief on Jurisdiction (Pis.’ Jur. Resp.), Dkt. No. 504, filed September 21, 2011.

Aso before the court is the transcript of the trial of damages (Tr.), held from April 18 through April 21, 2011 and from April 25 through April 28, 2011.

I. Procedural Background and the Law of the Case

“Plaintiffs 2 are the owners of property along approximately four and a half miles of the eastern shore of Lake Michigan south of St. Joseph Harbor.” Sept. 28, 2007 Op., Banks v. United States (Liability Op.), 78 Fed.Cl. 603, 604 (2007). Beginning in the 1830s, the United States government, acting through the United States A’my Corps of Engineers (defendant or the Corps) re-constructed the mouth of the St. Joseph River and began constructing harbor jetties that jutted into Lake Michigan in order to accommodate commercial shipping vessels exiting the St. Joseph River into Lake Michigan. Id. Over time, the Corps lengthened the jetties and then encased them in steel. Id. Plaintiffs claim that the Corps’ construction and maintenance of the jetties caused erosion of their shoreline property. Id. Specifically, *121 plaintiffs allege that the encasement of the jetties in “sand-tight” steel sheet piling during the period from 1950 to 1989 interrupted the natural littoral 3 drift of sand to their properties, resulting in erosion. See Aug. 9, 2011 Order, Banks v. United States (Order to Brief Jurisdiction), 99 Fed.Cl. 622, 624 (2011).

Several factors obscure the effect of the jetties on plaintiffs’ properties. Plaintiffs’ properties are located along a shoreline that is eroding naturally, see Tr. 2594:24-25 (Nairn); Tr. 710:22-23 (Mackey), making it necessary to distinguish the baseline of natural erosion from erosion caused by defendant. The comparatively slow process of long-term erosion is also masked by far larger swings in the width of the beaches next to plaintiffs’ properties caused by cross-shore sand transport, a cyclical process by which sand is moved offshore during times of high lake levels and returned to the shore during times of low lake levels. See Tr. 2593:3-19, 2594:2-2595:2 (Nairn) (“So, I mean, you’ve got swings of hundreds of feet related to the cross or reversible process and then you’ve got a very [s]low retreat, we believe to be around .62 [feet] per year on the south end, going on in the background of all those very large swings back and forth.”); cf Tr. 1624:9-12 (Shabica) (stating that “predicting lake levels is like predicting the weather, but if we look in the past, high lake levels from one high lake level to the next have ranged anywhere between 11 years and 22 years”). Furthermore, the composition of a shoreline, a characteristic that impacts how the shoreline erodes and how it reacts to efforts to mitigate erosion, can be hidden by surface sediments, and may be complex and difficult to characterize into one of the two categories—sandy and cohesive—used by coastal engineers and geologists. See infra Part III.B.

Defendant filed a motion to dismiss on the ground that plaintiffs’ claims are time-barred. July 31, 2001 Op. and Order, Banks v. United States (Accrual Op. I), 49 Fed.Cl. 806, 809 (2001), rev’d, 314 F.3d 1304 (Fed.Cir.2003) (Accrual Op. II). The court granted defendant’s motion to dismiss, finding that plaintiffs’ claims accrued no later than 1989, more than six years before plaintiffs filed suit. See id. at 825. For reasons discussed below, see infra Part III.A.1, the United States Court of Appeals for the Federal Circuit (Federal Circuit) reversed and remanded for further proceedings, Accrual Op. II, 314 F.3d at 1310.

On remand, this ease has been bifurcated for trial of liability and damages. In 2007 the court held a one-week trial of liability and issued the court’s Liability Opinion. 4 See generally Liability Op., 78 Fed.Cl. 603 . The Liability Opinion summarizes the procedural history of the first eight years of this case, makes extensive findings of fact and describes many of the scientific concepts at issue in this case. See id. passim. In order to minimize repetition, this Opinion assumes substantial familiarity with the Liability Opinion.

After the court issued its Liability Opinion, the court directed the parties to “file a *122 joint status report or, if they cannot agree, separate status reports, proposing additional proceedings necessary to resolve the remaining issues in this matter.” Dec. 14, 2007 Order, Dkt. No. 250, at 1-2. Defendant filed a status report, stating that the parties had met in person “in an effort to discuss issues and share thoughts on how to calculate damages.” Def.’s Status Report, Dkt. No. 252, at 2. Defendant described several factual and legal issues that remained outstanding in regard to the measure of damages owed by defendant. See id. at 2-5. Defendant stated that, while defendant believed that the court had ruled on the issue of shoreline composition, plaintiffs believed that they would be able to introduce additional evidence on the topic. See id. at 4. Defendant noted that the “array of factors” that must be considered by the court in order to calculate damages is complicated by the fact that “plaintiffs present a variety of different circumstances: for example, some plaintiffs claim to have lost all of their property, some plaintiffs have sold their property since the onset of litigation, and some also claim loss and damage to structures and loss of rental income.” Id. at 2. Defendant suggested that appraisals of plaintiffs’ properties would be necessary to establish the amount of damage caused by the government’s actions. See id. at 2-3. Defendant proposed a schedule for briefing on the remaining legal issues, limited additional discovery and a trial of damages. See id. at 4-5.

Plaintiffs submitted a status report that proposed no further proceedings or discovery. Pis.’ Status Report, Dkt. No 251, passim. Plaintiffs’ status report proposed that the court order defendant to pay plaintiffs $124,918,219, an amount that plaintiffs argued was equal to the cost of the shore protection installed by plaintiffs, the loss of real property and structures to erosion, the loss of rent, the loss of beach access, “prospective damages” and interest. Id. at 1-3, 5. Plaintiffs did not attach adequate documentation or detailed calculations to then-status report to support the amount of their alleged damages. See id. passim. Plaintiffs stated that “[pjlaintiffs have reviewed numerous available appraisals and find them to be unhelpful in that they use comparables, 1. e., other sold properties, which likewise lack unhindered (i.e., no need for shore protection) lake access.” Id. at 6. Without explanation, plaintiffs attached to their status report a newspaper article titled “River dredging faces obstacles”; a list—largely typed but partly written by hand—of the plaintiffs and the sums alleged to be due to each; a copy of two filings in this ease related to interrogatories; and a list—largely written by hand but typed in part—entitled “Properties Sold or Destroyed[,] Reasonably Foreseeable Loss.” Id. at Exs. 1-4.

Plaintiffs then filed Plaintiffs’ Motion for Leave to Supplement Status Report, Dkt. No. 253, attached to which was a status report with the subtitle “taking is spoliation,” id. at 1. In their status report, plaintiffs stated that “defendanVspoliator, who has been found liable for taking private property landward of the high water mark (OHWM), now wants to foist upon the judicial process a tediously cumbersome effort to measure that very property which it has destroyed or disposed of by its own continuing activity.” Id. at 1-2. Plaintiffs argued that “[t]he best evidence of sand loss to each plaintiff is based on his or her own recollection, photographs, appraisals, etc. listed in plaintiffs’ answers to defendant’s 2nd set of interrogatories.” Id. at 2 (capitalization omitted). Plaintiffs further argued that “[a]n appropriate sanction for the defendant/spoliator would be for the [cjourt to accept the calculations which have been available to defendant since July 16, 2001—over six years.” Id. at 2. Plaintiffs argued that “defendant/spoliator has acted intentionally, in an anti-constitutional, anti-environmental, anti-due process, and unnecessary manner.” Id. at 3.

The court conducted two telephonic status conferences with the parties, see Jan. 22, 2008 Order, Dkt. No. 258, at 1-2, and scheduled briefing on the issue of whether plaintiffs would be permitted to present additional evidence of shoreline composition at the trial of damages, see Feb. 19, 2008 Order, Dkt. No. 260, passim. Plaintiffs filed a memorandum that the court treated as a motion for reconsideration on the issue of shoreline composition. Oct. 15, 2008 Op. and Order, Banks v. United States (Order Granting Re *123 cons.), 84 Fed.Cl. 288, 290 (2008). Plaintiffs argued that they should be permitted at the trial of damages to present additional evidence that the shoreline in plaintiffs’ zone 5 is cohesive. 6 Id. The court concluded that “[e]ontrary to plaintiffs’ contention, the court explicitly and conclusively determined the issue of the nearshore lakebed composition in plaintiffs’ zone.” Id. The court stated that “[plaintiffs did not carry their burden of proof regarding the composition of the near-shore lakebed during the liability trial.” Id. at 293 . 7 The court further stated that “[plaintiffs’ arguments fail to demonstrate any of the three circumstances which would support reconsideration: the occurrence of an intervening change in the controlling law, the availability of previously unavailable evidence, or the necessity of allowing the motion to prevent manifest injustice.” Id. at 292 (citing Matthews v. United States, 73 Fed.Cl. 524, 526 (2006)). However, notwithstanding that “[pjlaintiffs had the opportunity during trial to present evidence regarding the composition of the nearshore lakebed of plaintiffs’ zone,” id. at 297, the court held that “in order to avoid possible inefficiency and delay in resolving plaintiffs’ claims, the court will accept additional evidence regarding the composition of the nearshore lakebed in plaintiffs’ zone,” id. at 298; see also infra Parts III.B.2-3 (discussing additional evidence of shoreline composition). The court directed the parties to suggest a schedule for further proceedings. See Order Granting Recons., 84 Fed.Cl. at 298 .

After receiving separate status reports from the parties, the court set a schedule providing for discovery and the resolution of remaining legal issues. Nov. 12, 2008 Order, Dkt. No. 282, at 2-4; see also Aug. 11, 2009 Op. and Order, Banks v. United States (Law of Damages Op.), 88 Fed.Cl. 665, 669-70 (2009) (listing the motions and memoranda filed by the parties). The court then issued its Law of Damages Opinion, in which it stated that “[t]he court’s legal rulings contained within this Opinion and Order clarify the law that will govern the damages phase of the case.” 88 Fed.Cl. at 669 n. 1.

The damages available to plaintiffs are calculated in accordance with the court’s finding in the Liability Opinion that defendant is liable for the portion of 30% of the total erosion damages to each plaintiffs property not mitigated by defendant. 78 Fed.Cl. at 656-57 . Plaintiffs’ damages are to be divided into three time periods. See id. During each time period, defendant is liable for any portion of 30% the erosion damage to plaintiffs’ properties not mitigated by defendant. See id. During the first time period, which runs from 1950 to 1970, defendant undertook no mitigation activity and is liable for damages for 30% of the erosion of each plaintiffs property. 8 See id. at 656 . During the see *124 ond time period, which begins in 1970—when the Corps began its efforts to mitigate erosion caused by the jetties—defendant is liable “for any portion of 30% of each plaintiffs total erosion above the high water mark ... that was not effectively mitigated by the Corps’ nourishment.” See id. During the third time period, from January 2000 forward, defendant is liable for damages for any portion of 30% “of all reasonably foreseeable future loss” not mitigated by defendant. Id.

Plaintiffs who have installed shore protection may recover the government’s share of the cost to install and maintain the shore protection, Law of Damages Op., 88 Fed.Cl. at 685 & n. 16, that is, 30% of any costs incurred from 1950-1970, and a portion of the costs incurred from and after 1970 equal to the portion of the erosion caused by the jetties and not mitigated by defendant, cf. Liability Op., 78 Fed.Cl. at 656 .

Regarding defendant’s liability for “all reasonably foreseeable future loss,” id., the court stated that if further erosion of plaintiffs’ properties is “‘in fact preventable by prudent measures, the cost of that prevention is a proper basis for determining the damage,’ ” Law of Damages Op., 88 Fed.Cl. at 683 (quoting United States v. Dickinson, 331 U.S. 745, 751 , 67 S.Ct. 1382 , 91 L.Ed. 1789 (1947)). Relevant to whether the installation and maintenance of shore protection to prevent future erosion is a “prudent measure” is whether the shore protection costs more than the reasonably foreseeable erosion damage that would occur without it. See id: (“ ‘If revetments are less expensive than the value of the land forecast to be lost, then the [gjovernment may discharge its liability by bearing the cost of bank protection.’ ”) (quoting Boling v. United States (Boling I), 41 Fed.Cl. 674, 694 (1998), vacated on other grounds (Boling II), 220 F.3d 1365 (Fed.Cir.2000)).

“Although the general rule in takings cases is to award plaintiff the form of just compensation that will be least expensive for the government,” the court, in its Law of Damages Opinion, found this general rule “inapplicable in the particular circumstances of the present ease,” in which the government is not the sole source of erosion to plaintiffs’ properties. Id. The court stated that “[pjlaintiffs may not be financially able to pay 70% of the cost to construct shore protection measures out of their own pockets, resulting in continued erosion of their property.” Id. The court held that “as an alternative to seeking 30% of the cost of shore protection measures, plaintiffs may instead seek 30% of the cost of ‘reasonably foreseeable future loss’ to their property, although this amount may result in greater cost to the government.” Id. at 684.

The court held a trial of damages from April 18 through April 21, 2011 and from April 25 through April 28, 2011 in Niles, Michigan. 9 See Tr. passim. 10

*126 After the court received the parties’ post-trial briefing, the court directed the parties to file additional briefing addressing whether, in light of certain evidence presented at the trial of liability and certain findings of fact made in the court’s Liability Opinion, the court has jurisdiction to hear plaintiffs’ claims. See Order to Brief Jurisdiction, 99 Fed.Cl. at 625-26 .

II. Legal Standards

A. Jurisdiction

Subject matter jurisdiction is a threshold matter that a court must deter *127 mine at the outset of a case. See Steel Co., 523 U.S. at 94-95 , 118 S.Ct. 1003 ; PODS, Inc. v. Porta Stor, Inc., 484 F.3d 1359, 1365 (Fed.Cir.2007). “Plaintiffs] bear[ ] the burden of showing jurisdiction by a preponderance of the evidence.” Taylor v. United States, 303 F.3d 1357, 1359 (Fed.Cir.2002) (citing Thomson v. Gaskill, 315 U.S. 442, 446 , 62 S.Ct. 673 , 86 L.Ed. 951 (1942)). The court must consider jurisdictional issues at any point in a case that they arise. Steel Co., 523 U.S. at 93 , 118 S.Ct. 1003 . The court is obligated to raise the issue of its own jurisdiction sua sponte “if a question thereto exists.” Liberty Mut. Ins. Co. v. Wetzel, 424 U.S. 737, 740 , 96 S.Ct. 1202 , 47 L.Ed.2d 435 (1976) (citing Mansfield, C. & L.M. Ry. Co. v. Swan, 111 U.S. 379 , 4 S.Ct. 510 , 28 L.Ed. 462 (1884)).

The court must accept as true all undisputed allegations of fact made by the nonmoving party and draw all reasonable inferences from those facts in the non-moving party’s favor. See Henke v. United States, 60 F.3d 795, 797 (Fed.Cir.1995) (citing Scheuer v. Rhodes, 416 U.S. 232, 236-37 , 94 S.Ct. 1683 , 40 L.Ed.2d 90 (1974), abrogated on other grounds by Harlow v. Fitzgerald, 457 U.S. 800 , 102 S.Ct. 2727 , 73 L.Ed.2d 396 (1982)). However, “[w]hen a party challenges the jurisdictional facts alleged in the complaint, the court may consider relevant evidence outside the pleadings to resolve the factual dispute.” Arakaki v. United States, 62 Fed.Cl. 244, 247 (2004) (citing Reynolds v. Army & Air Force Exch. Serv., 846 F.2d 746, 747 (Fed.Cir.1988) and Indium Corp. of Am. v. Semi-Alloys, Inc., 781 F.2d 879, 884 (Fed.Cir.1985)); 2 James Wm. Moore et al., Moore’s Federal Practice § 12.30[3] (3d ed. 2004) (“[UJnlike a Rule 12(b)(6) dismissal, the court need not confine its evaluation to the face of the pleadings____”). If a court determines that it does not have jurisdiction, it must dismiss the claim. See Rules of the United States Court of Federal Claims (RCFC) 12(h)(3).

“Every claim of which the United States Court of Federal Claims [Court of Federal Claims] has jurisdiction shall be barred unless the petition thereon is filed within six year’s after such claim first accrues.” 28 U.S.C. § 2501 (2006). “It is well established that statutes of limitations for causes of action against the United States, being conditions on the waiver of sovereign immunity, are jurisdictional in nature.” Martinez v. United States, 333 F.3d 1295, 1316 (Fed.Cir.2003) (en bane) (citing, inter alia, Block v. North Dakota, 461 U.S. 273, 287 , 103 S.Ct. 1811 , 75 L.Ed.2d 840 (1983)). Because the statute of limitations in this court is jurisdictional, id., plaintiffs have the burden of showing by a preponderance of the evidence that their claims were timely filed, see Taylor, 303 F.3d at 1359 .

B. Accrual of Takings Claims

A takings claim must be filed “within six years after such claim first accrues.” See 28 U.S.C. § 2501 . Pursuant to the “stabilization doctrine,” “The accrual of a takings claim where the government leaves the taking of property to a gradual physical process occurs when the situation has ‘stabilized.’ ” Accrual Op. II, 314 F.3d at 1308 (quoting Boling II, 220 F.3d at 1370 ). The stabilization doctrine was first stated in a case involving a taking of property by a gradual process of flooding, as follows:

[A]s there is nothing in reason, so there is nothing in legal doctrine, to preclude the law from meeting such a process by postponing suit until the situation becomes stabilized. An owner of land flooded by the Government would not unnaturally postpone bringing a suit against the Government for the flooding until the consequences of inundation have so manifested themselves that a final account may be struck.

Dickinson, 331 U.S. at 749 , 67 S.Ct. 1382 . Ten years after Dickinson was decided, the United States Supreme Court (Supreme Court) clarified its holding, stating: “The expressly limited holding in Dickinson was that the statute of limitations did not bar an action under the Tucker Act for a taking by flooding when it was uncertain at what stage in the flooding operation the land had become appropriated to public use.” United States v. Dow, 357 U.S. 17, 27 , 78 S.Ct. 1039 , 2 L.Ed.2d 1109 (1958).

*128 Courts have recognized that interpreting the holding of Dickinson broadly would put it in “unending conflict with the statute of limitations.” (Gustine Land & Cattle Co. v. United States, 174 Ct.Cl. 556, 656 , 1966 WL 8856 (1966)); see also Del. State Coll. v. Ricks, 449 U.S. 250, 258 , 101 S.Ct. 498 , 66 L.Ed.2d 431 (1980); Boling II, 220 F.3d at 1371 (stating the holding of Ricks to be that “the proper focus in a claim accrual analysis ‘is upon the time of the [defendant’s] acts, not upon the time at which the consequences of the acts become most painful’ ” (internal quotation marks omitted) (alteration in original)); Columbia Basin Orchard v. United States, 116 Ct.Cl. 348, 357 , 88 F.Supp. 738, 739 (1950) (“[W]e do not think the Supreme Court, in the Dickinson case, meant to hold that plaintiff was entitled to wait until any possibility of further damage had been removed.”).

Stabilization of a claim for a taking by erosion occurs “when the erosion ha[s] substantially encroached the parcels at issue and the damages [are] reasonably foreseeable.” Boling II, 220 F.3d at 1373 . “ ‘[Stabilization occurs when it becomes clear that the gradual process set into motion by the government has effected a permanent taking, not when the process has ceased or when the entire extent of the damage is determined.’ ” Accrual Op. II, 314 F.3d at 1308 (quoting Boling II, 220 F.3d at 1370-71 ). The extent of the taldng must be “reasonably foreseeable,” Boling II, 220 F.3d at 1371 , but the damage need not be “complete and fully calculable before the cause of action accrues,” Fallini v. United States, 56 F.3d 1378, 1382 (Fed.Cir.1995).

C. Damages in Partial Takings Cases

The Fifth Amendment of the United States Constitution provides that private property shall not “be taken for public use, without just compensation.” U.S. Const, amend. V. Erosion of property due to government action is one type of physical injury that rises to the level of a taking. See, e.g., Boling II, 220 F.3d at 1370 . “When the government fails properly to compensate private property owners for a taking, this court has jurisdiction to enforce the owners’ right to just compensation.” Liability Op., 78 Fed.Cl. at 614 (citing 28 U.S.C. § 1491 ; Applegate v. United States, 25 F.3d 1579, 1581 (Fed.Cir.1994)). Such a claim is often referred to as a “inverse condemnation claim,” see, e.g., Owen v. United States, 851 F.2d 1404, 1418 (Fed.Cir.1988) (en banc), or as a “taking claim,” see, e.g., id. at 1406.

“If only a portion of a single tract is taken the owner’s compensation for that taking includes any element of value arising out of the relation of the part taken to the entire tract. Such damage is often, though somewhat loosely, spoken of as severance damage.” United States v. Miller, 317 U.S. 369, 376 , 63 S.Ct. 276 , 87 L.Ed. 336 (1943) (footnote omitted); see also Hendler v. United States, 175 F.3d 1374, 1383 (Fed.Cir.1999) (“In cases of a partial physical taking, as that here, just compensation under the takings clause of the Constitution includes ‘not only the market value of that part of the tract appropriated, but the damage to the remainder resulting from that taking, embracing ... injury due to the use to which the part appropriated is to be devoted.’ ” (alteration in original) (quoting United States v. Grizzard, 219 U.S. 180, 183 , 31 S.Ct. 162 , 55 L.Ed. 165 (1911))).

Plaintiffs carry the burden of proving “that a taking has occurred justifying the payment of just compensation.” Loesch v. United States, 227 Ct.Cl. 34, 44 , 645 F.2d 905, 914 (1981). Plaintiffs also carry the burden of proving the amount of just compensation to which they are entitled for severance damage. Ga.-Pac. Corp. v. United States, 226 Ct.Cl. 95, 107 , 640 F.2d 328, 336-37 (1980); Miller v. United States, 223 Ct.Cl. 352, 383-84 , 620 F.2d 812, 828-29 (1980).

Plaintiffs’ properties, being adjacent to the navigable waters of Lake Michigan, are subject to the navigational servitude held by the federal government, which servitude extends to the ordinary high water mark. See United States v. Cherokee Nation of Okla. (Cherokee), 480 U.S. 700, 704 , 107 S.Ct. 1487 , 94 L.Ed.2d 704 (1987); United States v. Chicago, M., St. P. & P.R. Co., 312 U.S. 592, 595-97 , 61 S.Ct. 772 , 85 L.Ed. 1064 (1941); Owen, 851 F.2d at 1408-10 . When the gov *129 ernment properly exercises its right to improve navigation in a manner that affects property within the boundaries of this servitude, “ ‘the damage sustained does not result from taking property from riparian owners within the meaning of the Fifth Amendment but from the lawful exercise of a power to which the interests of riparian owners have always been subject.’ ” Cherokee, 480 U.S. at 704 , 107 S.Ct. 1487 (quoting United States v. Rands, 389 U.S. 121, 123 , 88 S.Ct. 265 , 19 L.Ed.2d 329 (1967)).

Accordingly, plaintiffs are not entitled to just compensation for materials removed from the littoral zone, see Owen, 851 F.2d at 1413 , or eroded below and within the ordinary high water mark, see id. at 1412 (holding that the federal navigational servitude does not extend “to land located above or outside the bed of the stream as delineated by the ... high-water mark at the time of construction”); see generally May 26, 2006 Op., Banks v. United States (OHWM Op.), 71 Fed.Cl. 501 (2006) (discussing the definition of the term “ordinary high water mark”); June 23, 2005 Op. and Order, Banks v. United States, 68 Fed.Cl. 524 (2005) (discussing the federal navigational servitude and the date on which the ordinary high water mark is to be measured).

D. The Law of the Case Doctrine and the Mandate Rule

The law of the case doctrine “posits that when a court decides upon a rule of law, that decision should continue to govern the same issues in subsequent stages in the same case. This rule of practice promotes the finality and efficiency of the judicial process by protecting against the agitation of settled issues.” Christianson v. Colt Indus. Operating Corp., 486 U.S. 800, 815-816 , 108 S.Ct. 2166 , 100 L.Ed.2d 811 (1988) (citations and internal quotation marks omitted); see also Gould, Inc. v. United States, 67 F.3d 925, 930 (Fed.Cir.1995) (“The law of the case is a judicially created doctrine, the purposes of which are to prevent the relitigation of issues that have been decided and to ensure that trial courts follow the decisions of appellate courts.” (internal quotation marks omitted)). The law of the case doctrine is applicable both to issues decided explicitly and to issues decided “by necessary implication.” Laitram Corp. v. NEC Corp., 115 F.3d 947, 951 (Fed.Cir.1997).

The Federal Circuit has explained, regarding its adherence to prior appellate decisions:

[T]he law of the case doctrine is a policy not a command even respecting a prior appellate decision in the ease, and should be applied “as a matter of sound judicial practice, under which a court generally adheres to a decision in a prior appeal in the case unless one of three ‘exceptional circumstances’ exists: the evidence on a subsequent trial was substantially different, controlling authority has since made a contrary decision of the law applicable to such issues, or the decision was clearly erroneous and would work a manifest injustice.”

Mendenhall v. Barber-Greene Co., 26 F.3d 1573, 1582 (Fed.Cir.1994) (quoting Cent. Soya Co. v. Geo. A. Hormel & Co., 723 F.2d 1573, 1580 (Fed.Cir.1983)).

“ ‘[T]he law of the case doctrine is not applicable to issues neither presented nor decided in a former proceeding in the case.’ ” Halpern v. Principi, 384 F.3d 1297, 1301 (Fed.Cir.2004) (quoting Stearns v. Beckman Instruments, Inc., 737 F.2d 1565, 1568 (Fed.Cir.1984)).

“The mandate rule requires that the [trial] court follow an appellate decree as the law of the case.” Cardiac Pacemakers, Inc. v. St. Jude Med., Inc., 576 F.3d 1348, 1356 (Fed.Cir.2009) (citing Sibbald v. United States, 37 U.S. (12 Pet.) 488, 492 , 9 L.Ed. 1167 (1838)); accord In re Sanford Fork & Tool Co., 160 U.S. 247, 255 , 16 S.Ct. 291 , 40 L.Ed. 414 (1895). “The mandate rule provides that ‘issues actually decided [on appeal]—those within the scope of the judgment appealed from, minus those explicitly reserved or remanded by the court—are foreclosed from further consideration.’ ” Amado v. Microsoft Corp., 517 F.3d 1353, 1360 (Fed. Cir.2008) (quoting Engel Indus., Inc. v. Lockformer Co., 166 F.3d 1379, 1383 (Fed.Cir.1999)). “Unless remanded by [the appellate] court, all issues within the scope of the appealed judgment are deemed incorpo *130 rated within the mandate and thus are precluded from further adjudication.” Engel Indus., 166 F.3d at 1383 .

Much like the law of the case doctrine, which “is not applicable to issues neither presented nor decided in a former proceeding,” Halpern, 384 F.3d at 1301 , the mandate rule provides that the trial court “may act on matters left open by the mandate,” Laitram, 115 F.3d at 951 (internal quotation marks omitted); see, e.g., Exxon Chem. Patents, Inc. v. Lubrizol Corp. (Exxon), 137 F.3d 1475, 1484 (Fed.Cir.1998) (holding that the scope of the judgment appealed from, limited to literal infringement, did not preclude adjudication of infringement by the doctrine of equivalents). “Interpretation of an appellate mandate entails more than examining the language of the court’s judgment in a vacuum.” Exxon, 137 F.3d at 1484 . “The scope of the issues presented ... on appeal must be measured by the scope of the judgment appealed from, not by the arguments advanced by the appellant.” Engel Indus., 166 F.3d at 1382 (citations omitted). The Federal Circuit has cautioned that, “both the letter and the spirit of the mandate must be considered.” Id. at 1383 .

III. Discussion

A. Jurisdiction

In their original complaint, 11 plaintiffs alleged “that defendant effected a gradual taking of their shorefront property through the construction and maintenance of ‘a series of 15 jetties along 200 miles of the eastern coast of Lake Michigan for over 100 years.’ ” Accrual Op. I, 49 Fed.Cl. at 810 (quoting Compl. ¶ 26, Banks v. United States, No. 99-445 L (Fed.Cl. filed July 9, 1999), Dkt. No. 1). Plaintiffs narrowed their claims in the individual complaints they filed after the court denied class certification, focusing on the effect of the jetties at St. Joseph Harbor. Compl. ¶¶ 4-7. The jetties at St. Joseph Harbor were originally built in the 1830s, and were lengthened several times, reaching their current length in 1903. Liability Op., 78 Fed.Cl. at 604 (citations omitted).

Although the jetties had been in place at their current length for nearly a century—a period of time longer than the six-year limitations period applicable takings claims, see 28 U.S.C. § 2501—plaintiffs’ complaints alleged that the accelerated erosion of their properties was caused not by construction of the jetties, but by a maintenance activity that took place more recently. Specifically, plaintiffs alleged that the jetties “continued to exist without harmful interference to the natural littoral flow of sand and river sediment until the [Corps] gradually installed sand-tight steel sheet piling during the period of 1950 to 1989,” which installation has “alter[ed] the supply of sand to the lake bed and subaerial visible beach in front of the plaintiffs’ property.” Compl. ¶¶ 6-7. After plaintiffs filed their individual complaints, defendant filed a motion to dismiss on jurisdictional grounds, arguing that “the takings causes of action accrued, at the latest, in 1989,” when the government completed its encasement of the jetties in the steel sheet piling. Accrual Op. I, 49 Fed.Cl. at 811 .

The briefing filed by defendant in support of its motion to dismiss and the opinions filed by the court and by the Federal Circuit make *131 it clear that defendant, the court and the Federal Circuit presumed the truth of plaintiffs’ allegation that encasing the jetties in steel sheet piling made them impermeable to sand, interfering with the littoral flow of sand and damaging plaintiffs’ properties. Defendant stated in its motion that it “[ajccepted as true for the sake of defendant’s motion” that the steel sheet piling “prevents the drift of sand from passing through the jetties and proceeding south along the eastern shore toward plaintiffs’ lands.” Mot. to Dismiss at 6, Banks v. United States, 49 Fed.Cl. 806 (2001), Dkt. No. 64 (Mot. to Dismiss).

In its opinion addressing defendant’s motion to dismiss, the court noted that “[wjhen considering a motion to dismiss, the court must presume that well pleaded factual allegations in the complaint are true.” Accrual Op. I, 49 Fed.Cl. at 808 (citing, inter alia, Miree v. DeKalb Cnty., 433 U.S. 25 , 27 n. 2, 97 S.Ct. 2490 , 53 L.Ed.2d 557 (1977)). Quoting the allegations in the Banks plaintiffs’ complaint, the court stated, “Plaintiffs claim that the jetties did not cause ‘harmful interference to the natural littoral flow of sand and river sediment until the Corps gradually installed sand-tight steel sheet piling during the period of 1950 to 1989.’ ” Id, at 808 (quoting Compl. ¶ 6); see also id. at 824 (“Plaintiffs allege in their complaints, which the court construes in favor of the complainants, see Scheuer, 416 U.S. at 236 , 94 S.Ct. 1683 , that the Corps completed the installation of ‘sand-tight steel sheet piling’ in 1989.” (quoting Compl. ¶ 6)).

Applying Applegate , the court considered and rejected plaintiffs’ argument that, after the installation of the steel sheet piling was completed in 1989, the government delayed the accrual of plaintiffs’ claims by promising to mitigate the erosion caused by the jetties. Accrual Op. I, 49 Fed.CI. at 812-13, 822-23. In Applegate , the Federal Circuit held that the plaintiffs’ claims for a taking by erosion did not accrue while promises by the government to mitigate the damage made the plaintiffs “justifiably uncertain” whether a permanent taking had occurred. Applegate, 25 F.3d at 1583-84 ; see also infra Part III.A.1 (applying Applegate to the facts of this case).

Examining the evidence cited by the parties, the court found that “the evidence here does not show that the Corps’ sand transfer program constituted a promise on which plaintiffs could rely to postpone the filing of their suits, as contemplated by the Applegate case.” Accrual Op. I, 49 Fed.CI. at 823. The court therefore found that “the time for filing suit expired in 1995,” six years after installation of the steel sheet piling was completed. Id, at 825. Because plaintiffs did not file suit until 1999, the court granted defendant’s motion to dismiss. Id. at 825-26.

The Federal Circuit reversed, finding that after the steel sheet piling was installed, mitigation efforts that the government had begun in 1970 created uncertainty as to whether any erosion damage caused by the jetties was “permanent and irreversible.” Accrual Op. II, 314 F.3d at 1309-10. The Federal Circuit found that plaintiffs’ claims accrued with the publication of three Corps reports that concluded that the mitigation efforts were ineffective. See id. On remand, the court determined that, because the last of these reports was published no earlier than January of 2000, plaintiffs’ claims accrued in January of 2000. May 3, 2007 Op., Banks v. United States, 76 Fed.Cl. 686, 696 (2007).

After conducting a trial of liability, the court found that, contrary to the allegations in plaintiffs’ complaints—allegations that the court had presumed to be true when considering defendant’s motion to dismiss, see Accrual Op. I, 49 Fed.Cl. at 808—the jetties were impermeable to sand before they were encased in steel sheet piling, Liability Op., 78 Fed.CI. at 636 (stating that “plaintiffs’ own expert witness testified that the piers were impermeable even prior to their encasement in steel”). The court further found that “plaintiffs have failed to prove that the piers were ever permeable.” Liability Op., 78 Fed.Cl. at 635 .

The implication of the court’s finding is that the installation of steel sheet piling had not altered the supply of littoral sand to plaintiffs’ properties as plaintiffs had alleged, see Compl. ¶¶ 6-7, thereby exacerbating the erosion of plaintiffs’ properties. The most recent government action that could have effected a taking by increasing the erosion of *132 plaintiffs’ properties was not the installation of steel sheet piling, a process that ended in 1989, but rather the extension of the jetties, a process that ended in 1903. See Liability Op., 78 Fed.Cl. at 604 .

At the time of the court’s Liability Opinion, neither the court nor the parties addressed the impact of the court’s finding on the timeliness of plaintiffs’ claims. See Liability Op., 78 Fed.Cl. passim. However, after the trial of damages, the court directed the pai’ties to file briefing on the issue. See Order to Brief Jurisdiction, 99 Fed.Cl. at 626 . The court now considers, in light of the extensive factual record developed at two trials, whether plaintiffs’ claims were timely filed, given that the jetties were impermeable to sand before they were encased in steel sheet piling. See Steel Co., 523 U.S. at 93 , 118 S.Ct. 1003 (noting that the court must consider jurisdictional issues at any point in a case that they arise).

To determine whether plaintiffs’ claims are timely, the court must address four questions. First, working backward from the date of filing, the court must determine the period of time during which plaintiffs’ claims could not have accrued, under Applegate , because the government’s promises and efforts to mitigate erosion caused by the jetties created justifiable uncertainty about the permanence of any erosion damage. Second, working forward from 1903, the year that the jetties reached their final length, the court must determine whether the situation stabilized sufficiently for plaintiffs’ claims to accrue and for the statute of limitations to run before the government’s promises and efforts to mitigate erosion made the permanence of any damage uncertain. Third, the court must consider plaintiffs’ argument that their claims did not accrue until certain adverse precedent was overruled by the Federal Circuit, sitting en banc. Finally, the court must consider plaintiffs’ argument that the law of the case doctrine prohibits the court from finding that it lacks jurisdiction over plaintiffs’ claims.

1. Justifiable Uncertainty Caused by the Corps’ Promises of Mitigation

The court first considers the period of time during which plaintiffs’ claims could not have acci’ued because of the government’s promises and efforts to mitigate erosion caused by the jetties. The Federal Circuit in Applegate “analyzed the stabilization doctrine set forth in Dickinson as it applied to situations in which the government was attempting to mitigate actions that would otherwise constitute a permanent taking.” Accrual Op. II, 314 F.3d at 1308. In Applegate , the government repeatedly promised to build a sand transfer plant to mitigate erosion of the plaintiffs’ properties caused by the government’s construction and maintenance of a deep water harbor. See Applegate, 25 F.3d at 1580 . The Federal Circuit detei’mined that “[t]he gi’adual character of the natural erosion process set in motion by the Cox’ps, compounded by the Government’s promises of a sand transfer plant, have indeed made acci’ual of the landowner’s claim uncertain.” Id. at 1582 .

When this coux’t addressed defendant’s motion to dismiss, the court found Applegate inapplicable, Accrual Op. I, 49 Fed.Cl. at 818-23 ; but the Federal Circuit disagreed and held that this court had “misread Apple-gate as x’equixing the presence of a legally binding promise or duty or a matter requiring a congx’essional appropi’iation,” Accrual Op. II, 314 F.3d at 1309. This coui’t noted that, according to the rate of erosion alleged in plaintiffs’ complaints to be caused by the jetties, plaintiffs lost almost eighty feet of shoreline to erosion between 1950 and 1989, an amount “sufficient to put plaintiffs on inquiry notice of their potential takings claims.” Accrual Op. I, 49 Fed.Cl. at 825 . The court gi’anted defendant’s motion, finding that “plaintiffs’ takings claims accnxed no later than 1989 and the time for filing suit expired in 1995.” Id. at 825-26 .

The Federal Circuit revei’sed, stating:

In Applegate , the mei’e promises of a sand ti’ansfer plant, held out by the Coi’ps and repeatedly I’enewed but never implemented, indicated that “the landownex’s did not know when or if their land would be permanently destroyed.” Hei’e, even gx-eater uncei’tainty was ex-eated by the Coi’ps’ mitigation plan. While the Coi’ps in Apple-gate made promises of a mitigating sand *133 transfer plant, the Corps in this case actually performed its mitigation activities for several years before the filing of this action.

Accrual Op. II, 314 F.3d at 1309-10 (quoting Applegate, 25 F.3d at 1582 ). The Federal Circuit noted that the government’s mitigation activities began in 1970, delaying accrual of plaintiffs’ claims until the publication of three reports by the Corps, which determined that the mitigation program had not been effective and “collectively indicated that erosion was permanent and irreversible.” Id. at 1310.

The Federal Circuit did not discuss whether the government’s promises to mitigate erosion created justifiable uncertainty before the Corps began mitigation efforts in 1970. See id. passim. However, beginning in 1958, “[t]he Corps released a series of reports ... over several decades describing the erosion caused south of St. Joseph Harbor by the jetties, outlining a plan to mitigate the erosion attributable to the jetties, and evaluating the effectiveness of the mitigation program that was eventually implemented.” Liability Op., 78 Fed.Cl. at 612 .

The first of these reports, a study released in 1958 (1958 Study), was published before mitigation efforts began. See id. at 604, 612 . Plaintiffs contend that the 1958 Study delayed the accrual of their claims, pursuant to Applegate , because the 1958 Study “promises that the government will attempt to mitigate the loss.” Pis.’ Jur. Br. 16-18. The 1958 Study is a “beach erosion control study,” transmitted by the Corps to the Speaker of the United States House of Representatives. Pis.’ Ex. (PX) 132 (1958 Study) v. It assessed erosion in a “study area” including plaintiffs’ zone that is “about 32 miles in length from the north city limit of Benton Harbor to the Michigan-Indiana State line.” Id. at 3. The study recommended building a sand berm to protect a portion of the shoreline north of plaintiffs’ zone. See id. at 24-25. It calculated the economic benefits of extending the sand berm south to the southern limit of the village of Shoreham, id. at 44, which is in plaintiffs’ zone. The 1958 Study determined, however, that “[t]his downcoast reach is entirely privately owned, would have no public benefits to make it eligible for Federal aid,” and would cost more to protect than the value of extending the sand berm. Id. The study noted, however, that even without including this area in the project, “this reach and adjacent shores to the south would receive substantial benefits of shore stabilization due to restoration of normal littoral drift.” Id.

Plaintiffs’ argument is that the 1958 Study created justifiable uncertainty about the permanence and extent of the taking of their property, when viewed in the light of Apple-gate. See Pis.’ Jur. Br. 16-18. The 1958 Study proposes a shore protection project that, while not intended to protect plaintiffs’ properties directly, was expected to benefit them by “restor[ing] normal littoral drift.” PX 132 (1958 Study) 44. The sand transfer plant promised by the Corps in Applegate , unlike the sand berm proposed here, was intended directly to address the erosion of the plaintiffs’ properties. See Applegate, 25 F.3d at 1580 . However, it was uncertainty about the permanence of the taking, rather than the government’s intentions in undertaking shore protection efforts, that delayed accrual of the plaintiffs’ claims in Applegate. See id. at 1582 (“With plans for a sand transfer plant pending, the landowners had no way to determine the extent, if any, of the permanent physical occupation.”); Boling II, 220 F.3d at 1372 (“[T]he critical element that delayed stabilization in Applegate [is] the justifiable uncertainty about the permanency of the taking.”).

Even accepting plaintiffs’ contention that the justifiable uncertainty created by the Corps began, not with the commencement of mitigation efforts in 1970, but with publication of the 1958 Study, 12 for plaintiffs’ claims to be timely, the situation must not have stabilized, Accrual Op. II, 314 F.3d at 1308, before 1952, six years before the possible *134 creation of justifiable uncertainty by the publication of the 1958 Study. 13

2. Claim Accrual and Stabilization: the Impermeability of the Jetties is the Law of the Case

“The accrual of a takings claim where the government leaves the taking of property to a gradual physical process occurs when the situation has ‘stabilized.’ ” Id. (quoting Boling II, 220 F.3d at 1370 ). Stabilization of claims for a taking by erosion occurs “when the erosion ha[s] substantially encroached the parcels at issue and the damages [are] reasonably foreseeable.” Boling II, 220 F.3d at 1373 . “ ‘ [Stabilization occurs when it becomes clear that the gradual process set into motion by the government has effected a permanent taking, not when the process has ceased or when the entire extent of the damage is determined.’ ” Accrual Op. II, 314 F.3d at 1308 (alteration in original) (quoting Boling II, 220 F.3d at 1370-71 ).

Defendant contends that plaintiffs’ claims stabilized with the Corps’ publication of the 1958 Study, which “acknowledged that the construction and maintenance of the St. Joseph Harbor was interfering with the littoral flow to properties south of the jetties and advised that the United States did not intend to take action to protect that shoreline.” Def.’s Jur. Br. 12.

The court, in its Liability Opinion, found the 1958 Study to be highly persuasive evidence that plaintiffs’ properties are, and were understood to be, within the zone of influence of the jetties. Liability Op., 78 Fed.Cl. at 621 (“Although not specifically pointed out by plaintiffs, the court views the most persuasive evidence to be contained in the 1958 Study, an admission by defendant.”). Although a number of Corps reports were introduced as exhibits and were before the court, id. at 612 , the court found the 1958 Study to be “the most persuasive evidence” of whether and for how long plaintiffs’ properties have been considered to be within the jetties’ zone of influence, id. at 621 . The court concluded that “plaintiffs’ properties have been considered in the zone of influence of the jetties at least as far back as the 1950s, even if the specific impact of the jetties was not quantified until later.” Id.

It is therefore the law of this case that plaintiffs’ properties were considered to be within the zone of influence of the jetties at least as far back as the 1950s. It is also the law of the case that the jetties have caused 30% of the erosion in plaintiffs’ zone since 1950. See id. at 654-57 . In its Liability Opinion, the court agreed with plaintiffs’ argument that “[u]ntil this litigation, the Army Corps had repeatedly endorsed the fact that the structures at St. Joseph had interrupted, dredged or diverted” a volume of sand equal to “30% of the total annual loss to the littoral zone,” id. at 636 (quoting Pis.’ Opening Post Trial Br., Dkt. No. 241, at 30-31), finding that the Corps had maintained, “as an admission, that the piers caused approximately 30% of the erosion to the south of the harbor,” id. The court therefore found, based on plaintiffs’ argument that the Corps had admitted the percentage of the erosion in plaintiffs’ zone caused by the jetties, 14 id., that the government is liable for the portion of 30% of the erosion in plaintiffs’ zone since 1950 not mitigated by the government, see id. at 654-57 .

Plaintiffs do not argue that any exception to the law of the case doctrine applies here. See Pis.’ Jur. Br. passim; Pis.’ Jur. Resp. passim. Plaintiffs do not argue that the jetties did not begin to impact their properties until later than 1950. See Pis.’ Jur. Br. passim; Pis.’ Jur. Resp. passim. Plaintiffs, in fact, argue that the jetties caused significantly more than 30% of the erosion in plain *135 tiffs’ zone since 1950. See infra. Part III.C.l (discussing plaintiffs’ arguments that the jetties have caused 60-70% of the erosion in plaintiffs’ zone since 1950).

It therefore remains the law of the case— based on an argument made by plaintiffs and found persuasive by the court, see Liability Op., 78 Fed.Cl. at 636—that the zone of influence of the jetties reached plaintiffs no later than 1950, causing 30% of the erosion of plaintiffs’ properties for no fewer than two years before 1952. Moreover, for the reasons stated below, it is the view of the court that, based upon the documentation of erosion in plaintiffs’ zone contained in the 1958 Study and upon the conclusions reached by defendant’s expert, Dr. Nairn, the jetties began to increase the erosion of plaintiffs’ properties before 1950 and that stabilization occurred significantly earlier than 1952, more than six years before plaintiffs argue, see Pis.’ Jur. Br. 16-18, that the government began to create justifiable uncertainty by publishing the 1958 Study. 15

Defendant is correct that the Corps acknowledged in the 1958 Study that the jetties had long caused erosion in plaintiffs’ zone. 16 The 1958 Study notes that “[t]he navigation channel at St. Joseph Harbor is dredged annually to maintain project depth, and for this reason little or no beach building material is believed to pass the harbor entrance and reach the downdrift shore.” PX 132 (1958 Study) 4; see also id. at 20-21 (describing the quantity of sand blocked by the jetties since 1907 and the amount of sediment dredged from the navigation channel).

The 1958 Study concluded that, if the shore protection recommended in the study *136 were constructed, an area continuing south “to the south limit of the village of Shore-ham” and the “adjacent shores to the south would receive substantial benefits of shore stabilization due to restoration of normal littoral drift.” PX 132 (1958 Study) 44. Implicit in the statement that the project would restore “normal littoral drift,” id., is the understanding that normal littoral drift had been disrupted. Plaintiffs acknowledge that this statement in the 1958 Study refers to a segment of shoreline that includes their properties. See Pis.’ Br. 18 (stating that “the 1958 [Study] indicates that the properties in the [plaintiffs’ zone would benefit from the mitigation program”).

The 1958 Study also documented the greater rate of erosion south of the jetties since their construction: 17

The shores for about a mile north of the harbor structures and 1,200 feet south thereof have been advancing lakeward since the entrance structures were built. South of the latter accreting area, however, erosion of the bluffs has been severe, causing the loss of, or necessitating the movement of, a number of valuable residences and threatening a railroad and public highway. Numerous protective structures have been erected but have exhibited only moderate effectiveness. 18

PX 132 (1958 Study) 3. The 1958 Study observed that, in the area up to 500 feet north of the jetties, “the shoreline has been accreting at a rate of approximately 4 feet per year since 1830,” id. at 10, a rate of growth that, in the court’s view, would have been apparent to landowners.

In addition to the disruption of normal littoral drift in plaintiffs’ zone and the increase in erosion, the 1958 Study also documented the proliferation of shore protection structures south of the jetties after their construction, a trend that is consistent with—although it does not independently establish—an increased rate of erosion south of the jetties after their construction and lengthening. See id. at 18, 20. Describing *137 the area northward from the jetties to the northernmost edge of Benton Harbor, the 1958 Study states:

Beach erosion in this reach of shoreline has been limited to the extreme northern end near the Benton Harbor city limits. No extensive beach protective structures have been built in this reach with protective efforts limited to the placement of an occasional steel barrel filled with sand. The remaining portion of this shoreline has been accreting.

Id. at 18. In contrast, in the area extending southward from the edge of the shore protection adjacent to-a highway to the southern edge of the village of Shoreham, shore protection was very common: “Approximately 22 individual structures built by property owners are in existence and range from typical groin construction to steel sheet-pile seawalls and revetments. The protection provided by a large portion of this work is limited, and erosion is continuing in those intermediate areas not protected.” Id. at 20. Further to the south, where the southernmost properties owned by plaintiffs are located, there were few shore protection structures, but the land was largely undeveloped, see id., making them less necessary.

The disruption of normal littoral drift and the increased erosion south of the jetties documented in the 1958 Study are consistent with the conclusion reached by Dr. Nairn in his 2006 expert report that the jetties have long contributed to erosion south of the harbor. See Def.’s Ex. (DX) 1 (Nairn Report) 4-151 to 4-152. Dr. Nairn, using a sediment budget 19 and several types of numerical modeling, quantified the impact of the jetties on erosion in the an area of shoreline beginning at the jetties and continuing for ten miles to the south. See infra Parts III.B-C (explaining and finding persuasive Dr. Nairn’s use of sediment budgets and numerical modeling to predict rates of erosion and the effectiveness of mitigation). Dr. Nairn determined that, during the period from 1836 to 1875, when the jetties were shorter, erosion in the study area continued to fall “within the range of pre-harbor erosion estimates.” 20 DX 1 (Nairn Report) 4-151. Between 1876 and 1903, as the Corps continued to lengthen the jetties, the rate of erosion in the study area did not increase and, in fact, decreased slightly from pre-harbor levels. See id. at 4-152.

The impact of the jetties began to be felt when they reached their final length. In terms of volume, Dr. Nairn calculates that between 1904 and 1969, the year before the Corps began its mitigation efforts, the jetties were responsible for 25% of all erosion in the study area. See id. at 4-158. This impact would have been greatest in the northern part of the study area and would not be the

*138 same for each property. 21 See id. at 4-159.

In their post-trial briefing addressing the measure of their damages, plaintiffs argue that Dr. Nairn’s approach significantly underestimates the impact of the jetties on their properties since 1950. 22 See infra Part III.C.l. Plaintiffs contend that:

the United States has (1) substantially undercounted the volume of sand being blocked by the jetties, as well as the jettiesf] percentage contribution to the erosion in the [plaintiffs’ zone; and (2) substantially overcounted the volume of sand block[ed] by the C & O and MDOT revetments, so that the jetties[’] percentage contribution to the erosion in the [plaintiffs’ zone is well in excess of 30% and is fairly calculated as being between 60% and 70%.

Pis.’ Br. 23; see also Tr. 19:7-24 (plaintiffs’ counsel) (stating, in his opening argument, that Dr. Mackey’s testimony about the flaws in Dr. Nairn’s methods will demonstrate that the jetties have caused 60 to 70 percent of the erosion in plaintiffs’ zone since 1950). As described below, the court finds plaintiffs’ criticisms of Dr. Nairn’s erosion analysis unpersuasive. See infra Parts III.B-C. However, the court finds plaintiffs’ argument that the jetties caused 60-70% of the erosion in plaintiffs’ zone after 1950 inconsistent—absent evidence cited by plaintiffs to explain the sudden change—with a finding that the jetties did not cause significant and noticeable erosion in plaintiffs’ zone prior to 1950.

Because of the interruption of littoral drift and the erosion documented in the 1958 Study and because of Dr. Nairn’s calculations of the jetties’ impact on erosion, which plaintiffs fail persuasively to contradict (suggesting instead that Dr. Nairn has underestimated the amount of erosion caused by the jetties), the court finds that the erosion caused by the jetties in plaintiffs’ zone was a longstanding problem by 1952, beginning as early as 1903. 23

*139 To determine the point at which plaintiffs’ claims stabilized, the court must determine the point at which this additional erosion “substantially encroached the parcels at issue and the damages were reasonably foreseeable.” Boling II, 220 F.3d at 1373 . The Federal Circuit has advised that, in making this determination, a court must “take into account the uncertainties of the terrain, the difficulty in determining the location of the government’s easement, and the irregular progress of erosion.” Id. Plaintiffs are correct that, to some extent, the exact rate of erosion—and therefore changes in the rate of erosion—would have been obscured by natural variations, such as changes in lake level, barometric pressure, and wind direction, as well as “the dynamic nature of the [plaintiffs’ shoreline.” Pis.’ Jur. Br. 13; see also supra Part I (describing variations in beach width caused by cross-shore sand transport and changes in lake levels). Plaintiffs contend that fluctuations in lake levels of up to 1.8 feet occur with annual frequency and that, as numerous plaintiffs testified at trial, beaches “ ‘come and go’ as lake levels changed and storms occurred.” Pis.’ Jur. Br. 13.

However, the erosion relevant to plaintiffs’ claims, that is, the erosion of property above the ordinary high water mark, can be gauged more clearly because the erosion of such property does not vary as widely as does the profile of beaches below the ordinary high water mark with changes in lake depth, barometric pressure, and wind direction. As described below in Part III.D.l, the court finds persuasive a delineation of the high water mark that lies, on properties characterized by bluffs, at the toe of the bluff, and, on properties characterized by dunes, at the edge of permanent vegetation. Although the evidence presented in this case establishes that beaches are somewhat ephemeral, see infra Part III.D.3.b.i, the ordinary high water mark is defined by these, more permanent, features, see infra Part III.D.l; Tr. 2343:18-22 (Nairn) (defining the ordinary high water mark for bluff properties as the toe of the bluff and distinguishing permanent vegetation from “ephemeral vegetation,” such as marine grass and dune grass); Tr. 2363:11-13 (Nairn) (stating that the ordinary high water mark, when defined by bluffs, normally moves only in one direction). Plaintiffs cite no evidence that bluffs and permanent vegetation “come and go” as rapidly as beaches do. See Pis.’ Jur. Br. passim; Pis.’ Jur. Resp. passim. The rate of erosion above the ordinary high water mark, therefore, can be determined with some confidence.

The location of the ordinary high water mark, because it is characterized by observable features, can be identified by landowners such as plaintiffs. See Loesch, 227 Ct.Cl. at 61 , 645 F.2d at 925 (stating that “the OHWM on a riverbank is a physical fact, subject to determination by inspection of the riverbank” (citation omitted) (citing Kelley’s Creek & Nw. R.R. Co. v. United States, 100 Ct.Cl. 396, 406 , 1943 WL 4239 (1943) (“The high watermark is not to be determined by arithmetical calculation; it is a physical fact to be determined by inspection of the river bank. It is the line where the water stands sufficiently long to destroy vegetation below it.”)). “It is not unreasonable to expect that plaintiffs, as riparian landowners, were familiar to some degree with their [shoreline]).” Id. at 61, 645 F.2d at 925 . It is therefore not unreasonable to expect that plaintiffs, as well as the prior owners of their properties, would have observed where the ordinary high water mark lay on their properties and would have observed its changing location over time.

Any uncertainty would have been further reduced by the significant span of time— forty-nine years—between the year 1903, when the jetties reached their final length, and 1952, the date after which plaintiffs’ claims must have accrued, under plaintiffs’ reading of Applegate, see supra Part III.A.1, for the uncertainty created by the 1958 Study to render plaintiffs’ claims timely. Although bluffs and permanent vegetation would not be expected to “come and go” as beaches do, erosion is often an irregular process, accelerating at times of high lake levels *140 and during storms. See infra Part III.D.4.b. The rate of erosion, and therefore the effect of the jetties, would have become clearer with the passage of time. 24

During the forty-nine years between 1903, when the jetties reached their final length, and 1952, the jetties were responsible for 25% of the material eroded from Dr. Nairn’s study area, the ten mile segment of shoreline south of the jetties. See DX 1 (Nairn Report) 4-158. Neither party cites—and the court does not find in the record—estimates of how many additional feet of erosion occurred in plaintiffs’ zone during this period as a result of the jetties. See supra note 19 (discussing Dr. Nairn’s analysis of bluff recession rates). However, a study published in 1976 determined the rate of bluff recession, averaged across all of Berrien County over 120 years to be .6 meters, or two feet, per year. See DX 40 (1997 Report) 3. Given this rate of erosion and the passage of forty-nine years between 1903 an 1952, the court concludes that the additional erosion caused by the jetties was not limited to “mere inches.” Boling II, 220 F.3d at 1372 .

The well-documented additional erosion caused by the lengthened jetties would have made it clear to a reasonable landowner well before 1952 that the government had effected a permanent taking. See PX 132 (1958 Study) 3-4, 10-12, 17-18, 20-21; DX 1 (Nairn Report) 4-152, 4-158. This is particularly true given the lower rates of erosion (and the accretion of sand) occurring further north since construction of the jetties, and given the diminishing impact of the jetties on properties further south from the jetties. At this time, the extent of the damage would have been reasonably foreseeable from the increased rate at which erosion had occurred for forty-nine years.

Plaintiffs assert that “[e]ven if, for the sake of argument, the 1958 [Study] placed [plaintiffs on inquiry notice that the harbor jetties caused some degree of erosion, they were clearly not on notice that the jetties had caused erosion to their specific properties, much less a permanent loss.” Pis.’ Jur. Br. 14-15. Plaintiffs are correct that the 1958 Study did not discuss their properties individually. See PX 132 (1958 Study) passim. Because the 1958 Study was not prepared for purposes of this litigation, it would not be expected to discuss plaintiffs’ properties individually, or to measure separately the amount of erosion caused by the jetties in plaintiffs’ zone.

However, “plaintiff[s] bear[ ] the burden of showing jurisdiction by a preponderance of the evidence.” Taylor, 303 F.3d at 1359 (citing Thomson, 315 U.S. at 446 , 62 S.Ct. 673 ). It is not enough for plaintiffs to argue in briefing that the general pattern of erosion that followed the lengthening of the jetties in 1903 might not have affected certain of their individual properties. Because plaintiffs have the burden of showing the court’s jurisdiction, they must cite evidence that erosion caused by the jetties had not reached their properties when the 1958 Study, PX 132 (1958 Study) 3-4, 10-12, 17-18, 20-21, and Dr. Nairn’s analysis, see DX 1 (Nairn Re *141 port) 4-152, 4-158, indicate that it reached the rest of the area. Apart from their interpretation of the 1958 Study, plaintiffs have cited no such evidence, see Pis.’ Jur. Br. passim; Pis.’ Jur. Resp. passim, and have failed to carry their burden of proof.

Plaintiffs argue that “the effect of the harbor jetties on the lakebed and adjoining shoreline to the south was a very gradual process that occurred over decades of time.” Pis.’ Jur. Br. 14. Citing a single page of a report published by the Corps in 1997, coauthored by defendant’s expert witness, Dr. Nairn (1997 Report), plaintiffs argue that as late as 1997 it was not understood that the harbor jetties caused increased erosion in plaintiffs’ zone. 25 Id. The implication of plaintiffs’ argument is that their claims stabilized no earlier than 1997 because it was not understood at that time 26 that the jetties were causing erosion in plaintiffs’ zone. 27

The only relevant sentence on the page of the 1997 Report cited by plaintiffs describes an area approximately 8.2 kilometers south of the jetties, which, “up until recently, may not have been significantly influenced by the harbor jetties.” DX 40 (1997 Report) 58. 28 The page of the 1997 Report cited by plaintiffs does not explain why the report’s authors believed that the area may not have been significantly influenced by the jetties or what they meant by “until recently.” See id. Neither does the referenced page of the 1997 Report explain the term *142 “significantly influenced.” 29 See id. Other portions of the 1997 Report suggest that its authors assumed that the area was not affected by the jetties. See, e.g., id. at 25, 27 (listing among the authors’ “assumptions about the nearshore conditions and profile evolution prior to the comparison of the data” that this area was “not influenced by reduced sediment supply from the north—[and is] representative of natural conditions or background erosion rate”); id. at 79 (assuming that, because the rate of erosion in this area was similar to the rate of erosion north of the jetties, this area is not affected by the jetties). Plaintiffs cite no page of the 1997 Report on which the authors state that they made a detailed study of the effect of the jetties on this portion of plaintiffs’ zone. See Pis.’ Jur. Br. passim; Pis.’ Jur. Resp. passim. Rather, the court has found that the 1997 Report was largely based on data collected north of plaintiffs’ zone. Liability Op., 78 Fed.Cl. at 624-25 .

The court does not view an assumption in the single sentence cited by plaintiffs of a report published in 1997 as establishing that it was not known that the jetties caused erosion in plaintiffs’ zone until at least 1997. Because there was evidence apparent to an ordinary landowner that the jetties accelerated erosion, plaintiffs were not entitled to postpone the filing of their claims until they received scientific confirmation or a technical explanation of the subsurface processes causing the harm. See Accrual Op. I, 49 Fed.Cl. at 820 (quoting Fallini, 56 F.3d at 1380 ).

The court therefore finds that, significantly earlier than 1952, the erosion caused by the jetties “had substantially encroached the parcels at issue and the damages were reasonably foreseeable.” Boling II, 220 F.3d at 1373 . Because the situation stabilized prior to 1952, the court finds that plaintiffs’ claims accrued prior to 1952, see id., and that, by waiting to file a complaint until 1999, plaintiffs failed to file their claims within the six-year limitations period set out in 28 U.S.C. § 2501 .

3. The Accrual Suspension Rule

In their opening brief on jurisdiction, plaintiffs argue that their claims “could not have accrued, as a matter of law, prior to the issuance of the Owen decision” because “[p]rior to Owen[,] it was settled federal law that a property owner could not recover for a taking of its fast land absent a physical invasion.” 30 Pis.’ Jur. Br. 21. According to *143 plaintiffs, “[p]rior to Owen [ ], federal eases uniformly held that erosion damage was not a taking where the damage from the federal project was caused, not by a rise in water levels, but as a consequence of some other interference with the natural water flow.” Id. Plaintiffs contend that “to argue that [p]laintiffs[’] takings claim was barred before they even had a recognized claim would be patently unfair.” Id. at 22.

Defendant, interpreting plaintiffs’ argument as an argument for the application of equitable tolling, correctly notes that equitable tolling is not available in actions brought under the Tucker Act. Def.’s Jur. Resp. 13-14 (citations omitted); see John R. Sand & Gravel Co. v. United States, 552 U.S. 130, 133-34 , 128 S.Ct. 750 , 169 L.Ed.2d 591 (2008); Young v. United States, 529 F.3d 1380, 1384 (Fed. Cir.2008) (stating the holding of John R. Sand & Gravel to be that “the statute of limitations applicable to Tucker Act claims, 28 U.S.C. § 2501 , is jurisdictional and not susceptible to equitable tolling”). Equitable tolling extends the statute of limitations ‘“where the claimant has actively pursued his judicial remedies by filing a defective pleading during the statutory period, or where the complainant has been induced or tricked by his adversary’s misconduct into allowing the filing deadline to pass.’ ” Young v. United States, 535 U.S. 43, 50 , 122 S.Ct. 1036 , 152 L.Ed.2d 79 (2002) (quoting Irwin v. Dep’t of Veterans Affairs, 498 U.S. 89, 96 , 111 S.Ct. 453 , 112 L.Ed.2d 435 (1990)).

Plaintiffs, in their jurisdictional response brief, do not argue for equitable tolling or repeat their previous argument that it would be “patently unfair” to find that the limitations period ended before the Federal Circuit’s decision in Oiven. See Pis.’ Resp. passim. Plaintiffs instead raise the new argument that their claims are timely under the “accrual suspension rule,” which, plaintiffs assert, directs that ‘“[n]o cause of action generally accrues until the plaintiff has a right to enforce his cause.’” Pis.’ Jur. Resp. 12 (alteration in original) (quoting United States v. One Red Chevrolet Impala Sedan (Red Chevrolet), 457 F.2d 1353 , 1358 (5th Cir.1972)).

Accrual suspension is an exception to the general rule “that a claim ‘first accrues’ when all the events have occurred which fix the alleged liability of the defendant and entitle the plaintiff to institute an action.” Hopland Band of Porno Indians v. United States, 855 F.2d 1573, 1577 (Fed.Cir.1988) (citing Japanese War Notes Claimants Ass’n of the Phil., Inc. v. United States, 178 Ct.Cl. 630, 632 , 373 F.2d 356, 358 (1967)). It directs “that the accrual of a claim against the United States is suspended, for purposes of 28 U.S.C. § 2501 , until the claimant knew or should have known that the claim existed.” Martinez, 333 F.3d at 1319 ; see also Ingrum v. United States, 560 F.3d 1311 , 1315 n. 1 (Fed.Cir.2009) (stating that it “is both more common and more precise” to describe the rule as suspending accrual while a claim is “concealed or inherently unknowable” (internal quotation marks omitted) (citing Martinez, 333 F.3d at 1319 )).

“The ‘accrual suspension’ rule is ‘strictly and narrowly applied: ... [The plaintiff] must either show that defendant has concealed its acts with the result that plaintiff was unaware of their existence or it must show that its injury was “inherently unknowable” at the accrual date.’” Martinez, 333 F.3d at 1319 (alteration in original) (quoting Welcker v. United States, 752 F.2d 1577, 1580 (Fed.Cir.1985)). “Mindful that the ‘accrual suspension rule’ is to be ‘strictly and narrowly applied,’ courts have concluded that a misunderstanding of the meaning of the law or one’s legal rights does not trigger this rule.” Petro-Hunt, L.L.C. v. United States, 90 Fed.Cl. 51, 62 (2009) (citation omit *144 ted); see also Catawba Indian Tribe v. United States, 982 F.2d 1564, 1572 (Fed.Cir.1993) (“[A]ll the relevant facts were known. It was the meaning of the law that was misunderstood.”). The accrual suspension rule “is based on a construction of the term ‘accrues’ in section 2501,” and “is distinct from the question whether equitable tolling is available under that statute, although the term ‘tolling’ is sometimes used in describing the rule.” Martinez, 333 F.3d at 1319 .

Because plaintiffs did not raise their accrual suspension argument in their opening brief, see Pis.’ Jur. Br. passim, defendant did not have an opportunity to respond to it, and the court finds the argument to be waived, see Cross Med. Prods., Inc. v. Medtronic Sofamor Danek, Inc., 424 F.3d 1293 , 1320 n. 3 (Fed.Cir.2005) (finding an issue not properly raised in an opening brief to be waived); Becton Dickinson & Co. v. C.R. Bard, Inc., 922 F.2d 792, 800 (Fed.Cir.1990) (“[W]e see no reason to depart from the sound practice that an issue not raised by an appellant in its opening brief ... is waived.”).

Even if the accrual suspension rule had been properly raised, it is inapplicable to plaintiffs’ claims. Plaintiffs’ argument is that, before Owen was decided in 1988, plaintiffs “had no cause of action” because “federal cases uniformly held that erosion damage was not a taking where the damage from the federal project was caused, not by a rise in water levels, but as a consequence of some other interference with the natural water flow.” 31 Pis.’ Jur. Br. 21. Plaintiffs are therefore arguing that the legal basis—rather than the factual basis—of their claims was “inherently unknowable,” Ingrum, 560 F.3d at 1315 n. 1 (citing Martinez, 333 F.3d at 1319 ), until Owen was decided.

The accrual suspension rule is generally applied in situations where the factual basis, rather than the legal basis, of a plaintiffs claim is concealed or inherently unknowable. 32 See, e.g., Young, 529 F.3d at 1385 (finding that the trial court had properly declined to suspend accrual because “[i]t is a plaintiffs knowledge of the facts of the claim that determines the accrual date”) (citing, inter alia, United States v. Kubrick, 444 U.S. 111, 122 , 100 S.Ct. 352 , 62 L.Ed.2d 259 (1979)); Martinez, 333 F.3d at 1319 (finding that the accrual suspension rule was not applicable because “Mr. Martinez was not unaware of the existence of his injury and the *145 acts giving rise to his claim”); Welcker, 752 F.2d at 1580 (finding that the plaintiffs claims accrued in 1949 or in 1950 because “[c]learly, appellant was aware in 1949-1950 that he had been dismissed from the federal service on grounds that he himself considered to be wrong and improper”). Plaintiffs do not cite—and the court’s research has not found'—any decision in which the Court of Federal Claims or the Federal Circuit has found that adverse precedent tolled a plaintiffs claims under the Tucker Act, rendering them timely. 33 See Pis.’ Jur. Br. passim; Pis.’ Jur. Resp. passim.

The Federal Circuit has repeatedly stated that plaintiffs must file their claims within the statute of limitations, notwithstanding the presence of adverse precedent or the futility of filing. See, e.g., Aectra Ref. & Mktg., Inc. v. United States, 565 F.3d 1364, 1373-74 (Fed.Cir.2009) (finding, in the internal revenue context, that “futility does not excuse the failure to file a proper claim for limitations purposes”) (citing, inter alia, United States v. Clintwood Elkhorn Mining Co., 553 U.S. 1, 5-6 , 128 S.Ct. 1511 , 170 L.Ed.2d 392 (2008)); Frazer v. United States, 288 F.3d 1347, 1354-55 (Fed.Cir.2002) (finding that “it is, of course, irrelevant” to the timeliness of the plaintiffs’ claims whether similar’ claims had been unsuccessful in the past (citing Boling II, 220 F.3d at 1374 )); Boling II, 220 F.3d at 1374 (stating that the difficulty of pursuing a takings claim due to adverse precedent “does not justify tolling the statute of limitations”); Welcker, 752 F.2d at 1583 (rejecting the plaintiffs argument that any attempt to pursue his claims would have been “an exercise in futility” because of the “tenor of the times” and noting that “the statute of limitations is not tolled by litigative timidity”); see also Venture Coal Sales Co. v. United States (Venture Coal), 57 Fed.Cl. 52, 54-55 (2003) (finding that accrual of plaintiffs’ claims for a tax refund was not suspended until a decision finding the tax unconstitutional), aff'd, 370 F.3d 1102 (Fed.Cir.2004).

Plaintiffs rely upon two related cases not binding on this court for the proposition that, although the factual circumstances of a plaintiffs claim are not concealed or inherently unknowable, “ ‘[n]o cause of action generally accrues until the plaintiff has a right to enforce his cause.’ ” Pis.’ Jur. Resp. 12 (alteration in original) (quoting Red Chevrolet, 457 F.2d at 1358); id. at 13 (citing Neely v. United States, 546 F.2d 1059, 1068 (3d Cir.1976)). The plaintiffs in Red Chevrolet sued for the return of property forfeited to the United States government on the ground that it “had been used in conducting a gambling business without payment of the taxes required” by the Internal Revenue Code. Red Chevrolet, 457 F.2d at 1355. The plaintiffs in Neely sued for annulment of their convictions under the federal wagering statutes and the return of the fines they had paid. Neely, 546 F.2d at 1061 . In both cases, the courts determined that the plaintiffs’ claims were timely, notwithstanding that they were filed more than six years after the fines and forfeitures that gave rise to them. 34 See Red Chevrolet, 457 F.2d at 1358; Neely, 546 F.2d at 1068 .

*146 However, both Red Chevrolet and Neely “involve[ed] claims that were filed to obtain the benefit of a new constitutional rule that the Supreme Court expressly held to have retroactive application.” 35 Venture Coal, 57 Fed.Cl. at 55 (citing United States v. United States Coin & Currency (U.S. Coin), 401 U.S. 715, 723-24 , 91 S.Ct. 1041 , 28 L.Ed.2d 434 (1971)). The statutory provisions at issue in Red Chevrolet and Neely “commanded that gamblers submit special registration statements and tax returns that contained information which could well incriminate them in many circumstances.” U.S. Coin, 401 U.S. at 717 , 91 S.Ct. 1041 . In a pair of cases decided on the same day, the Supreme Court held that, “[b]eeause the risk of self-incrimination was substantial, ... a Fifth Amendment privilege could be raised as a defense to a criminal prosecution charging failure to file the required forms.” Id. (construing Marchetti v. United States, 390 U.S. 39 , 88 S.Ct. 697 , 19 L.Ed.2d 889 (1968) and Grosso v. United States, 390 U.S. 62 , 88 S.Ct. 709 , 19 L.Ed.2d 906 (1968)). The Supreme Court determined that the rule announced in Marchetti and Grosso applied retroactively, stating that, because “the conduct being penalized is constitutionally immune from punishment, ... [n]o circumstances call more for the invocation of a rule of complete retroactivity.” Id. at 724, 91 S.Ct. 1041 .

The courts in Red Chevrolet and Neely both noted the Supreme Court’s holding in U.S. Coin that the rule protecting the plaintiffs from self incrimination should be applied retroactively. Red Chevrolet, 457 F.2d at 1355; Neely, 546 F.2d at 1061 . In contrast, the Federal Circuit did not hold in Owen that its ruling was to be applied retroactively. See Owen, 851 F.2d passim. Plaintiffs cite no decision in which the holding of Owen was applied retroactively to extend the statute of limitations for a claim that was otherwise untimely. See Pis.’ Jur. Br. passim; Pis.’ Jur. Resp. passim.

To the contrary, the Federal Circuit has expressly determined that the overruling of advei’se precedent in Owen did not extend the limitations period within which erosion claims could timely be filed. The plaintiffs in Boling II argued that the “Ballam decision, which effectively barred the cause of action they intended to assert, and its subsequent reversal in Owen , form another basis for equitable tolling.” Boling II, 220 F.3d at 1374 . The Boling II plaintiffs further argued that “after the Ballam decision was rendered, any attempt to assert their rights to compensation under the Fifth Amendment would have been futile, because the identical claim had been previously rejected.” Id. The *147 Federal Circuit observed that “the plaintiffs point to no authority which would suggest that the presence of adverse precedent automatically leads to equitable tolling.” Id. The Federal Circuit concluded: “It is true that during the period between the decision in Ballam and its subsequent reversal in Owen , any claim by the plaintiffs for compensation would have been difficult. However, this difficulty does not justify tolling the statute of limitations.” Id.

Although the Federal Circuit used the term “equitable tolling” rather than “accrual suspension,” the Boling II plaintiffs’ argument is sufficiently analogous to the argument raised by plaintiffs in this case that the court views the Federal Circuit’s decision as binding upon the court. See Martinez, 333 F.3d at 1319 (stating that the accrual suspension rule is distinct from equitable tolling, “although the term ‘tolling’ is sometimes used in describing the rule.”). The court therefore holds that the accrual suspension rule did not delay the accrual of plaintiffs’ claims until the Owen decision was entered in 1988.

4. The Law of the Case and the Mandate Rule

Because the Federal Circuit has previously ruled on the timeliness of plaintiffs’ claims, the court directed the parties to address in their briefing whether the law of the case doctrine prohibits the court from holding, in light of its finding that the jetties were impermeable to sand before installation of steel sheet piling, that plaintiffs did not timely file their claims. See Order to Brief Jurisdiction, 99 Fed.Cl. at 626 .

Plaintiffs, in their briefing, argue that the law of the ease doctrine bars the court from considering whether plaintiffs’ claims are timely. Plaintiffs argue that, in its decision reversing the dismissal of plaintiffs’ claims on statute of limitations grounds, the Federal Circuit “held that [pjlaintiffs’ claims did not accrue until January 2000, and were not time barred.” Pis.’ Jur. Br. 5 (citing Accrual Op. II, 314 F.3d at 1310). Plaintiffs contend that “[sjpeeifically, the Federal Circuit held: ‘We are satisfied that the plaintiffs met their jurisdictional burden before the Court of Federal Claims.’ ” Id. (quoting Accrual Op. II, 314 F.3d at 1310). Plaintiffs therefore believe that, because they have met their jurisdictional burden, the court may not reexamine any aspect of the timeliness of plaintiffs’ claims.

Defendant argues that “[bjeeause the Federal Circuit’s decision ... incorporated [pjlaintiffs’ legal theory—a theory premised on facts substantially different from those facts subsequently proven at trial—the law of the ease doctrine does not preclude this [ejourt from dismissing [pjlaintiffs’ action at this time.” Def.’s Jur. Br. 12. Defendant further argues that “[pjlaintiffs initially alleged that the installation of steel sheet-piling was the government action that began the erosion process; but the evidence presented at trial established that the construction and maintenance of St. Joseph Harbor initiated the erosion process many years before the sheet-piling installation began.” Id. at 14. Defendant therefore contends that the Federal Circuit’s decision “is inapposite” to the question currently before the court. Id. at 15.

Defendant’s argument is best understood as two distinct arguments. First, defendant argues that the law of the case doctrine is inapplicable because the law of the case does not address the aspect of the court’s jurisdiction currently at issue, which was “ ‘neither presented nor decided in a former proceeding in the case.’” Def.’s Jur. Resp. 11 (quoting Halpern, 384 F.3d at 1301 ). Second, defendant argues that, even if the law of the case addresses the aspect of the court’s jurisdiction currently at issue, a change in circumstances—specifically, a change in the evidence available to the court—after the Federal Circuit’s decision warrants application of one of the exceptions to the law of the case doctrine. Id. at 10-12 (citation omitted). For the reasons stated below, the court agrees with defendant that the law of the case doctrine or, more specifically, the mandate rule, is inapplicable because the aspect of the court’s jurisdiction currently at issue was not decided by the Federal Circuit on appeal. Rather than addressing the aspect of the court’s jurisdiction currently at issue, the Federal Circuit pre *148 sumed the truth of the allegations in plaintiffs’ complaints related to steel sheet piling and addressed only the jurisdictional facts relevant to the judgment from which plaintiffs appealed.

“The mandate rule provides that ‘issues actually decided [on appeal]—those within the scope of the judgment appealed from, minus those explicitly reserved or remanded by the court—are foreclosed from further consideration.’ ” Amado, 517 F.3d at 1360 (alteration in original). “The scope of the issues presented ... on appeal must be measured by the scope of the judgment appealed from, not by the arguments advanced by the appellant.” Engel Indus., 166 F.3d at 1382 (citations omitted). The court must therefore determine whether the scope of the judgment that was appealed to the Federal Circuit encompassed the effect of the steel sheet piling on the erosion of plaintiffs’ property.

As discussed above in Parts III.A-A.1, the question addressed by the court in its order granting defendant’s motion to dismiss was whether plaintiffs’ claims accrued when the government finished the installation of steel sheet piling in 1989 or whether, under Apple-gate, the government’s promises to mitigate erosion damage created justifiable uncertainty as to the permanence of erosion caused by the jetties, delaying accrual of plaintiffs’ claims. The court stated that “[w]hen considering a motion to dismiss, the court must presume that well[-]pleaded factual allegations in the complaint are true.” Accrual Op. I, 49 Fed.Cl. at 808 .

Plaintiffs had alleged that the installation of steel sheet piling increased the rate of erosion of their properties. See id. at 810 (stating that plaintiffs allege “that it was the Corps’ installation of ‘sand-tight steel sheet piling during the period of 1950 to 1989’ that has ‘alter[ed] the supply of sand to the lake bed and subaerial visible beach in front of the plaintiffs’ property’ ” (alteration in original)); id. (“Referring to the installation of the steel sheet piling, plaintiffs state that as a ‘direct result of [defendant’s actions,’ they ‘have suffered a gradual and continued taking of their property’----”).

The court presumed this allegation to be true, analyzing when, after installation of the piling, plaintiffs’ claims for erosion damage caused by the steel-clad jetties accrued. See id. passim. Plaintiffs cite no portion of the court’s opinion in which the court examined, either expressly or by necessary implication, see Laitram, 115 F.3d at 951 , whether the steel sheet piling does, in fact, exacerbate erosion of plaintiffs’ properties, see Pis.’ Jur. Br. passim; Pis.’ Jur. Resp. passim.

Plaintiffs correctly note that the exhibits that the parties intended to present at trial were before the court when it ruled on defendant’s motion to dismiss. Pis.’ Jur. Br. 3; see also Accrual Op. I, 49 Fed.Cl. at 809 n. 4 (stating that the parties’ trial exhibits had been filed in accordance with the court’s pretrial order). Plaintiffs are also correct, see Pis.’ Jur. Resp. 5, that some of the evidence before the court suggested that the jetties had increased erosion in plaintiffs’ zone to some extent before the installation of steel sheet piling, see, e.g., Accrual Op. I, 49 Fed.Cl. at 820 -21 (quoting a newspaper interview in which a Corps official stated that “we have 80, 90, or 100 years of non-mitiga: tion to make up for”). Plaintiffs correctly note, see Pis.’ Jur. Resp. 2-3, that defendant argued in its motion to dismiss that the erosion caused by the jetties did not begin only with the installation of steel sheet piling, 36 see Mot. to Dismiss at 10 (stating that a nongovernmental report “indicate[d] that erosive processes preceded plaintiffs’ theory that such erosion only began after the commencement of sheet piling maintenance by the Corps in 1950”); id. at 8 (stating that many of the exhibits submitted by plaintiffs with an earlier filing “reiterate the conditions of erosion plaintiffs complain of as having *149 existed for decades or even more than a century”).

The court, however, reviewed the evidence submitted by the parties only to resolve the dispute upon which the court decided defendant’s motion: whether plaintiffs’ claims had accrued, at the latest, in 1989, when the Corps finished installation of steel sheet piling, or whether the government’s promises to mitigate erosion delayed accrual. The court stated that “[i]f the jurisdictional facts in the complaint are disputed, ... the court may consider relevant evidence beyond the pleadings to decide the jurisdictional question.” Accrual Op. I, 49 Fed.Cl. at 809 (emphasis added) (footnote omitted) (citing, inter alia, Land v. Dollar, 330 U.S. 731 , 735 n. 4, 67 S.Ct. 1009 , 91 L.Ed. 1209 (1947)). The court therefore considered the evidence submitted by the parties as it pertained to when, after installation of steel sheet piling, plaintiffs’ claims would have accrued. The court did not weigh the evidence of whether the jetties caused erosion in plaintiffs’ zone or whether the steel sheet piling, in fact, exacerbated the erosion caused by the jetties. See id. passim. The court made no findings of fact regarding whether the jetties caused erosion in plaintiffs’ zone or whether the steel sheet piling did, in fact, exacerbate erosion caused by the jetties. See id. passim. The court resolved these fact-intensive questions, with the assistance of extensive expert testimony and the parties’ post-trial briefing, only after the trial of liability. See Liability Op., 78 Fed.Cl. passim.

On appeal of the court’s dismissal of plaintiffs’ claims, the Federal Circuit addressed the proper application of accrual principles rather than the factual issues of whether the jetties actually caused erosion in plaintiffs’ zone or whether the steel sheet piling exacerbated this erosion. The Federal Circuit stated that “the question is whether the ‘predictability [and permanence] of the extent of damage to the [plaintiffs’] land’ was made justifiably uncertain by the Corps’ mitigation efforts.” Accrual Op. II, 314 F.3d at 1309 (alterations in original) (quoting Applegate, 25 F.3d at 1583 ). The Federal Circuit determined that “the Court of Federal Claims and defendant misread Applegate as requiring the presence of a legally binding promise or duty or a matter requiring a congressional appropriation.” Id. The Federal Circuit reversed the dismissal of plaintiffs’ claims “[b]ecause the Court of Federal Claims misapplied the standard for claim accrual under Applegate , and because plaintiffs remained uncertain as to the permanent nature of the taking until the Corps reported that the erosion was permanent and irreversible.” Id. at 1310. The Federal Circuit did not address whether the installation of steel sheet piling increased erosion in plaintiffs’ zone. See id. passim.

In addition to the text of the Federal Circuit's decision and the judgment from which plaintiffs appealed, the court is mindful of the Federal Circuit’s admonition that “both the letter and the spirit of the mandate must be considered.” Engel Indus., 166 F.3d at 1383 . Whether the jetties, with or without the steel sheet piling, cause erosion in plaintiffs’ zone goes to the merits of plaintiffs’ claims. As the Federal Circuit has explained, an appellate court reviewing the disposition of a motion to dismiss does not examine the ultimate merits of the case: “That is what the trial judge will do on remand. Appellate courts do not do that, even if we had a record on which to do it.” Henke, 60 F.3d at 801 n. 5; see also Cordis Corp. v. Boston Scientific Corp., 658 F.3d 1347, 1360 (Fed.Cir.2011) (“It would be illogical for this court to remand for findings on unresolved outcome determinative issues, while simultaneously foreclosing reconsideration of the outcome after the district court considered those issues for the first time.”). The court therefore concludes that the Federal Circuit did not intend to address the merits of whether the jetties cause erosion in plaintiffs’ zone before the issue was taken up, and before trial was held on the issue, by the trial court. 37

*150 A trial court may act on matters left open by the mandate. Laitram, 115 F.3d at 951 . The issue “actually decided on appeal,” Amado, 517 F.3d at 1360 (brackets omitted), was whether, under Applegate , plaintiffs’ claims could have accrued between 1989 and 1999 despite promises and efforts by the government during that period of time to mitigate erosion damage caused by the jetties. Since the decision of the Federal Circuit, the court has found, based on evidence presented at trial rather than allegations made in plaintiffs’ complaints, that the most recent government act that increased the rate of erosion of plaintiffs’ properties took place, not in 1989, but in 1903, more than fifty years before the government first proposed to mitigate erosion in plaintiffs’ zone. See supra Parts III.A.1-2. Because the Federal Circuit addressed only the delay of accrual resulting from promises and efforts to mitigate erosion, the court finds that whether plaintiffs’ claims accrued before the government made its first promises of mitigation is an issue left open by the mandate. The mandate rule therefore does not bar the court’s holding that plaintiffs’ claims were filed after the end of the limitations period.

B. Composition of the Shoreline

1. The Court’s Previous Findings on Shoreline Composition

In its Liability Opinion, the court addressed the evidence presented at trial as to the composition of the shoreline adjacent to plaintiffs’ properties. See Liability Op., 78 Fed.Cl. at 621-28 . “The composition of the lake bed is relevant because the composition affects erosion and mitigation processes. If the shore is composed of sand, the quantity of sand that is depleted is directly proportional to the quantity of sand that needs to be replaced.” Id. at 622 (footnote omitted). Therefore, “‘[F]or a predominantly sandy shore ... as long as the sand supply south of the harbor is restored to the pre-harbor levels, then we can assume directly that the erosion will remain the same as the preharbor levels, all other things aside----’’’Id. (alteration and omissions in original) (quoting transcript of the trial of liability (Liability Tr.) 1215:11-16, 1296:4-10 (Nairn)). As Dr. Nairn explained at the trial of liability, “ ‘it’s a simple sediment budget’ and a ‘simpler assessment.’ ” Id. (quoting Liability Tr. 1215:12,1215:8 (Nairn)).

Assessing the erosion of cohesive 38 shores is “much more complicated. We know the sand acts to abrade, sort of like sandpaper, the till. And it also acts to protect it----” Id. (omission in original). Once cohesive material is eroded, “it cannot reconstitute itself, and the cohesive form is lost forever.” Id. (citation and quotation marks omitted). However, it is possible for sand cover to increase to the point that a cohesive section of shoreline would be categorized as sandy. See PX 178 (Coastal Engineering Manual (CEM)) III—5—9 39 (“[Ojnly when the sand cover is sufficient to protect the cohesive substratum at all times will the shore revert to a sandy classification (i.e., truly a ‘thick pile of sand’).”); Tr. 512:11-513:15 (colloquy between Dr. Mackey and plaintiffs’ counsel) (quoting same with approval); cf. PX 178(CEM) III—5—13 (“Investigations of Great Lakes sites have shown that approximately 200 m 3/m of sand cover (measured from the top of the beach out to the 4-m contour) is required to halt the downcutting process____”). Short of providing such large quantities of sand, however, “it is unclear whether mitigation is ever possible. There *151 no longer is a direct correlation between replacing material and effective mitigation because nourishment can act as an abrasive agent that exacerbates erosion, and the erosion of the lake bottom is considered permanent.” Liability Op., 78 Fed.Cl. at 624 (citation omitted).

Determining shoreline composition can be difficult for two reasons. First, “cohesive shores are often difficult to identify owing to the presence of a sand beach at the shore,” PX 178(CEM) III—5—3, which can mask the underlying substrates, see generally infra Part III.B.2.b (discussing the CEM and its role in the parties’ arguments). Cohesive shores often are covered by a thin “veneer of sand and gravel” that the CEM states “is usually in the range of a few centimeters to 2 or 3 [meters].” 40 PX 178(CEM) III-5-3. Second, a shoreline may contain both sandy and cohesive material as a result of its geological history. See, e.g., DX 3 (Larson Report) 34, Fig. 9 (geological map and cross section of plaintiffs’ zone) (showing layers, pockets, and lenses of cohesive material embedded between layers of sand); Tr. 2035:3-2036:12 (colloquy between Dr. Mickelson and defendant’s counsel) (describing DX 293 (Mickelson Report) 8, Fig. 3 41 ) For instance, Dr. Mickelson and Dr. Larson described a thin layer, rich in organic material, that formed approximately 6,000 years ago, which is sandwiched between thick deposits of sand in the southern portion of plaintiffs’ zone. See infra Part III.B.3.a (describing the formation of the thin, organic-rich layer between thick layers of sand).

The composition of the shoreline is significant in this case because it indicates how the shoreline will erode, whether any erosion is permanent and whether it is possible to mitigate any ongoing erosion. Liability Op., 78 Fed.Cl. at 622 . After analyzing the evidence of shoreline composition presented at the trial of liability, the court concluded that “plaintiffs failed to prove by a preponderance of the credible evidence that plaintiffs’ properties are located on a cohesive lake bottom.” Id. at 628 . The court found persuasive the testimony of defendant’s expert witness in geology, Dr. Larson, see id. at 628 , who testified in detail about the geological history of the area and described the stratigraphy 42 he had prepared by interpreting well logs and engineering borings and by traveling every path through the area to observe exposures of sediment, 43 see id. at 625 . The *152 process of using well logs collected nearby to assemble a stratigraphy of the shoreline has been found persuasive in the past—by this court, by the Federal Circuit and by the United States Court of Claims—when direct evidence as to actual subsurface conditions is unavailable. 44 Order Granting Recons., 84 Fed.Cl. at 296-97 (discussing Renda Marine, Inc. v. United States, 509 F.3d 1372, 1378 (Fed.Cir.2007), aff'g Renda Marine, Inc. v. United States, 66 Fed.Cl. 639 (2005)); id. at 297 n. 6 (discussing Arundel Corp. v. United States, 207 Ct.Cl. 84, 98 , 515 F.2d 1116, 1124 (1975)).

Based on his stratigraphy and the geological history of the area, Dr. Larson concluded that “the shoreline for some of the northernmost plaintiffs’ properties is a cohesive shore that then transitions to a sandy shore for the remaining larger group of plaintiffs’ properties.” Liability Op., 78 Fed.Cl. at 625 (internal quotation marks omitted). He further testified that south of the harbor, the exposed layer of till gives way to an area that is “mainly raw sand.” Id. (quotation marks omitted). Dr. Nairn, defendant’s expert witness in coastal engineering, reviewed Dr. Larson’s stratigraphy and conducted measurements and numerical modeling, also concluding that “the properties belonging to a majority of the plaintiffs are located on a sandy, not a cohesive, shoreline.” Id.

At the trial of liability, plaintiffs presented “no expert evidence ... to counter defendant’s expert’s studies and explanations, and no expert review of Dr. Nairn’s—and particularly Dr. Larson’s—research conclusions regarding the lake bottom composition.” Id. at 628 . Plaintiffs’ expert, Dr. Chrzastowski, testified that he had not reviewed Dr. Nairn’s report, and plaintiffs provided no evidence that Dr. Chrzastowski had reviewed Dr. Larson’s report or testimony. Id. Dr. Chrzastowski instead “opined that the lake bottom was cohesive, basting] his evaluation on existing literature, aerial photography, historical maps, and ground photography,” documentary evidence that the court found had been “credibly refuted” by Dr. Larson. Id. Dr. Chrzastowski did not take any measurements in connection with his formulation of his expert opinion. See id.

The court noted that “[t]here is no dispute that, prior to this litigation, defendant consistently held the position that the shore in the area south of St. Joseph Harbor was cohesive.” Id. However, defendant had previously held this position as a result of studies conducted prior to this litigation that the court found less probative than the evidence presented at trial. Two of the earlier studies, which the court referred to in its Liability Opinion as the “1996 Report” and the “1997 Report,” had focused on a broader area of shoreline with little data from plaintiffs’ zone. See id. at 625, 627 . A study that the court referred to in its Liability Opinion as the “1992 Pilot Study” stated on its title page that it was a preliminary study, failed to discuss the geological history of the area *153 and, in Dr. Larson’s opinion, made “no geological sense.” See id. at 627 (quotation marks omitted).

The court also found the use of sidescan sonar in the 1992 Pilot Study problematic. 45 Sidescan sonar, the court stated, “scans only the surface of the lake bed and does not penetrate it.” Id. at 626 . Sidescan sonar “doesn’t [actually] tell you what’s there, it simply produces a picture of what the surface of the lake bed looks like, that is, it provides information on whether it is bumpy or flat.” Id. (alteration in original) (internal quotation marks omitted). The radar used to supplement sidescan sonar in the 1992 Pilot Study penetrates the ground but “can’t penetrate very far ... because the energy is consumed very rapidly.” Id. at 626 (internal quotation marks omitted). Dr. Larson testified that sidescan sonar and the other techniques used in the 1992 Pilot Study were “good science” but explained that it “should not stand alone; it has to be verified,” for example with well logs. Id. (citation and internal quotation marks omitted). Instead of using well logs to verify their conclusions about the composition of the lakebed, the authors of the 1992 Pilot Study performed “periodic sampling of the surface” of the lakebed by collecting sediment samples. Id. (citation and quotation marks omitted). The court found these superficial sediment samples less persuasive than “Dr. Larson’s extensive use of well logs, which obtains samples of materials in strata that are at the same depth as the subsurface of the adjacent shore.” Id. at 626-27 .

The court, having found Dr. Larson’s and Dr. Nairn’s testimony persuasive, stated that “[t]he inference of this conclusion is that for any given plaintiff it is more likely than not that his or her property is located in a sandy zone.” Id. However, “The evidence at trial did not permit the court to determine (by, for example, an overlay of plaintiffs’ property lines onto the lakebed composition data presented by defendant) exactly which of plaintiffs’ properties in the northernmost portion of plaintiffs’ zone were adjacent to the cohesive nearshore lakebed shown on defendant’s exhibits.” Order Granting Recons., 84 Fed.Cl. at 291 . The court stated that the issue of which specific properties are located along the small section of cohesive shoreline would be determined in future proceedings. Liability Op., 78 Fed.Cl. at 628.

2. Additional Evidence of Shoreline Composition Presented by Plaintiffs at the Trial of Damages

a. Observation of the Surface of the Lakebed by Sidescan Sonar, Underwater Video and Sediment Grab Sampling

At the trial of damages, plaintiffs presented additional evidence addressed to the composition of their section of shoreline. 46 However, plaintiffs did not provide an interpretation of the geological history of the area or present a stratigraphy of the area to challenge the geological history and stratigraphy developed by Dr. Larson, see Tr. passim, and found persuasive by the court at the trial of liability, see Liability Op., 78 Fed.Cl. at 628 . Plaintiffs’ expert witness on the topic of shoreline composition, Dr. Mackey, testified at the trial of damages that he had not reviewed the well logs that Dr. Larson used to develop his stratigraphy. See Tr. 710:5 (Mackey). Plaintiffs did not present evidence that the properties of individual plaintiffs had sandy or cohesive shorelines, see Tr. passim, but instead argued that the entirety of plaintiffs’ zone is located along a cohesive shoreline, see Pis.’ Br. 15.

Plaintiffs retained Dr. Mackey to perform a sidescan sonar survey of the nearshore lakebed adjacent to their properties, to map the distribution of surface sediments and to determine whether the shoreline along which their properties are located is cohesive. Tr. 488:7-13 (Mackey). To aid in the interpretation of the sidescan sonar data, Dr. Mackey recorded underwater video of plaintiffs’ zone. Tr. 507:17-24 (colloquy between Dr. Mackey and plaintiffs’ counsel). As Dr. Mackey ex *154 plains in his report, “In many respects, sidesean sonar data is similar to aerial photographs taken of the earth’s surface. The only difference is that the images are produced with sound instead of light.” PX 136 (Mackey Report) 3. A device called a towfish is pulled behind a boat, id. at 2, 3; the towfish emits acoustic pulses and records the acoustic energy reflected by the lakebed, id. at 2. To survey the nearshore lakebed adjacent to plaintiffs’ properties, Dr. Mackey piloted a boat along five survey lines parallel to the shore, see id. at 5-6, and used specialized software to merge the data from the five survey lines “into a seamless image of the lakebed,” id. at 7.

Sidescan sonar can be used to determine whether materials at the surface of the lake-bed are sandy or cohesive. See id. at 2 (“Generally, harder materials (bedrock, sand, metal) will give a stronger acoustic return than softer materials (silt, clay, or mud).”); DX 136 (McNineh Report) 2 (stating that “different surfaces often generate characteristic and distinguishable changes to the reflected sound”). Dr. Mackey explained that “[i]n general, harder materials ... will show up as ‘bright’ patterns on the sidescan sonar. Softer materials ... will show up as ‘dark’ patterns on the sidescan sonar.” PX 136 (Mackey Report) 10; see, e.g., id. at 11, Fig. 11.

Because sidescan sonar maps the surface of the lakebed, it is of limited value to researchers attempting to determine the composition of substrates located below the surface. Dr. Mackey testified that the sidescan sonar penetrated the lake bottom to a depth of “a couple of inches, at most.” Tr. 728:16 (Mackey). Under the correct conditions, sidescan sonar can be used to detect subsurface materials located very near the surface, but it does not indicate the depth at which the subsurface materials are located. 47 Tr. 727:9-11 (Mackey). Accordingly, Dr. Mackey agreed that the “mapped image usually does not provide a cross section of or insight into the depth of a substrate that appears on the sui’face of a lake bed.” Tr. 721:2-6 (colloquy between Dr. Mackey and defendant’s counsel).

In his trial testimony, Dr. Mackey agreed that he is aware of acoustic devices, including “sparkers,” “boomers” and “ehirper[s],” which “produce fairly high energy that allow[s] the sound of the side scan sonar to penetrate into the earth’s crust and then reflect off different areas of different density in the earth’s crust at varying depths.” Tr. 721:7-20 (colloquy between Dr. Mackey and defendant’s counsel). Dr. Mackey did not suggest to plaintiffs that they employ any of these devices, id. at 722:5-7, because he “didn’t think it was [his] place to suggest the use of those sort of tools because [he does not] own that type of equipment and [he] typically [does not] do that kind of work,” id. at 725:9-22. Dr. Mackey also testified that he was “not directly aware” that the question of what substrates lie below the sui’face of the lakebed and at what depth was an issue in this ease. Id. 722:8-13.

Interpreting the results of his sidescan sonar, Dr. Mackey concludes that approximately 5% of the lakebed surface consists of cohesive material, rather than boulder cobble, bedrock, gravel or sand. 48 See PX 136 *155 (Mackey Report) 13, Table 1. Dr. Mackey infers, without explanation, that an additional 5.7% of the lake bottom is covered by “[t]hin deposits of fine-med-sand overlying cohesive clay/silt deposits.” Id. at 13. Dr. Mackey states that his observations “fit the typical cohesive profile.” Id. at 15. 49 However, Dr. McNinch, reviewing Dr. Mackey’s results, argues persuasively that far less of the lake-bed surface is composed of cohesive materials. Dr. McNinch “found it very difficult to duplicate or even identify many of the same surface expressions of the cohesive clay substrate” identified by Dr. Mackey. DX 294 (McNinch Report) 4; but see PX 139 (Mackey Response to McNinch) 6 (stating that Dr. Mackey has more experience than Dr. McNinch in interpreting sidescan sonar data collected in the Great Lakes). Regarding Dr. Mackey’s inference that five percent of the lake bottom is composed of a thin layer of sand overlying cohesive material, Dr. McNinch stated that “[interpreting thickness based solely on the patchiness of exposed substrates ... is highly speculative” and noted that none of Dr. Mackey’s sedi-

ment samples revealed “a non-sandy, underlying layer.” 50 DX 294 (McNinch Report) 3.

Dr. McNinch also argues that, although “[t]he data acquisition and processing methodologies used by Dr. Mackey ... were quite good and met conventional standards,” id. at 2, Dr. Mackey failed properly to corroborate the results of his sidescan sonar, see Tr. 1799:25-1800:6 (colloquy between Dr. McNinch and defendant’s counsel); DX 294 (McNinch Report) 2-5 (describing several respects in which Dr. Mackey’s measurements could have been improved). Dr. Mackey acknowledges in his report that “it is always necessary (and appropriate) to validate the acoustic responses in the sidescan data either by sampling or direct observation.” 51 PX 136 (Mackey Report) 7.

One reason that it is necessary to validate sidescan sonar data with other measurements is that even sandy lakebeds contain a measure of finer material, which will often become separated from the sand and settle on the surface. Dr. McNinch explained that *156 “waves and currents typically winnow and sort unconsolidated sediment on the lakebed. These processes often result in what appears as a complex, patchwork pattern of different substrates at the surface but which, in fact, could simply originate from an underlying layer that is composed of a mixture of these sediment types.” DX 294 (McNinch Report) 3.

However, the sampling and direct observation undertaken by Dr. Mackey were quite limited and provide little corroboration of Dr. Mackey’s conclusion that his observations “fit the typical cohesive profile.” See PX 136 (Mackey Report) 15. Dr. Mackey recorded underwater videos and collected sediment samples from the surface of the lakebed. Id. at 6. After studying Dr. Mackey’s underwater videos, which total fifty-five minutes in length, Dr. McNinch determined that cohesive material was visible for thirty-seven seconds, or slightly more than 1% of the time. 52 Tr. 1835:3-7 (McNinch); DX 294 (McNinch Report) 5. Several of Dr. Mackey’s underwater videos document no cohesive sediment at all. DX 293 (Mickelson Report) 27. Dr. Mackey does not argue that Dr. McNinch has incorrectly measured the amount of cohesive material visible in the underwater videos. See PX 139 (Mackey Response to McNinch) 3-4.

None of the sediment grab samples 53 collected by Dr. Mackey verified the existence of exposed cohesive material. See PX 136 (Mackey Report) App. A (sample descriptions) (describing every sediment sample as consisting of sand or “[v]ery coarse sand/[g]ravel”); Tr. 1799:22-24 (McNinch) (“I believe from his table at the back of his report, all the samples from his petite ponar were sandy, sandy material”). 54 Most significantly, Dr. Mackey failed to verify his observations with subsurface investigation. See PX 136 (Mackey Report) passim (offering no discussion of subsurface investigation). Dri McNinch stated that, when studying lakebed composition, “[w]e often take cores, and we often take sub-bottom profiles 55 so that we know what the strat[um] is like just below.” Tr. 1798:13-15 (McNinch). Sub-bottom profiles also allow an observer to “see where the base of the sand is, and where other layers of different geology might be present.” 56 Tr. *157 1804:24-1805:5 (McNinch). Dr. Mackey did not ground truth his results with cores or sub-bottom profiles. 57 See PX 136 (Mackey Report) passim.

Dr. Mackey’s observations of the surface of the lakebed are simply insufficient to establish that the shoreline is cohesive. The court discussed the limited value of sidescan sonar coupled with surface sampling in its Liability Opinion—in which it found Dr. Larson’s stratigraphy more persuasive than the results of a 1992 study that relied on sidescan sonar and sampling of the lakebed surface, Liability Op., 78 Fed.Cl. at 626 -28— and again pointed out the limited value of sidescan sonar and other surface evidence in its Order Granting Reconsideration, 84 Fed. Cl. at 295 . Plaintiffs do not explain why they presented again at the trial of damages the results of sidescan sonar uncorroborated by subsurface investigation, such as cores or sub-bottom profiles. See Pis.’ Br. passim; Pis.’ Resp. passim; cf. Tr. 1798:13-15 (McNinch) (stating that, when studying lake-bed composition, “[w]e often take cores, and we often take sub-bottom profiles so that we know what the strat[um] is like just below”). Neither do plaintiffs attempt to distinguish the techniques used by Dr. Mackey from the techniques used in the 1992 study, presented at the trial of liability, and found unpersuasive by the court. See Pis.’ Br. passim; Pis.’ Resp. passim.

The court agrees with Dr. McNinch that “sediment thickness and the character of underlying substrates cannot be directly measured from sidescan sonar, underwater video, and surface sediment samples. No corroborating evidence, such as sediment cores or sub-bottom seismic 58 profiles, that explicitly measure the thickness and nature of buried substrates[,] are provided to support these interpretations.” DX 294 (McNinch Report) 1. The court therefore finds that Dr. Mackey’s observations of the lakebed surface, without supplementation by subsurface sampling or sub-bottom profiles, are insufficient to establish that plaintiffs’ zone is cohesive.

b. CEM Techniques to Identify Cohesive Shorelines

Plaintiffs offer two arguments that supplement Dr. Mackey’s observations of the surface of the lakebed with analysis drawn from the CEM. The CEM is an engineering manual drafted by the Corps to describe “the basic principles of coastal processes ... and [to provide] guidance on how to formulate and conduct studies in support of coastal flooding, shore protection, and navigation projects.” PX 316 (Guide for Preparation of the CEM) 5. Part III, Chapter 5 of the CEM, entitled “Erosion, Transport, and Deposition of Cohesive Sediments,” has been admitted into evidence as PX 178 and addresses, among other things, the identification of cohesive shores. PX 178(CEM) III—5—2 to III-5-8.

Plaintiffs’ first argument concerns six visual features that the CEM states may help in the identification of a cohesive shoreline. See Pis.’ Br. 9-12. Plaintiffs argue that four of the six visual features set out by the CEM are present in plaintiffs’ zone. Tr. 17:5-8 (plaintiffs’ counsel). Plaintiffs’ second argument purports to summarize with a two-part analysis the CEM’s discussion of cohesive shorelines and how they are identified. Pis.’ Br. 8. Because neither argument employs the *158 detailed analysis of shoreline composition described in the CEM, the court does not find either argument persuasive.

i. Six Visual Features that May Indicate a Cohesive Shore

Although the CEM notes that “cohesive shores are often difficult to identify owing to the presence of a sand beach at the shore,” the CEM states that “[t]here are at least six ways of visually identifying the presence of underlying consolidated cohesive sediment.” 59 PX 178(CEM) III—5—3. Drawing from Dr. Mackey’s initial expert report, 60 plaintiffs contend that “the six factors set forth in the CEM provide strong visual clues to the presence of a cohesive shore.” Pis.’ Br. 12; see also Tr. 17:5-8 (plaintiffs’ counsel) (“There are six ways, in the [CEM], to identify whether a [cohesive] 61 lakebed exists. We have four out of six.”); PX 136 (Mackey Report) 17-25 (discussing the presence of four of the six factors in plaintiffs’ zone).

The six visual features listed by the CEM are (1) “the presence of exposed cohesive sediment on the beach”; (2) pieces of clay or peat that have washed onto the beach; (3) “[s]prings or surface runoff across a beach”; (4) “[discoloration of water in the nearshore zone”; (5) “[permanent undulations in the shoreline platform”; and (6) exposed cohesive sediments in troughs between offshore bars. PX 178(CEM) III-5-3 to III-5-6. In addition to the six visual indicators, the CEM describes the “[u]se of more detailed subsurface investigations to confirm visual observations and provide more detailed information.” Id. at III—5—6. Plaintiffs claim that four of the six CEM factors are present on their properties: exposed cohesive material on the beach, pieces of cohesive material washed up on the beach, discoloration of the water in the nearshore zone and exposed cohesive sediments in the troughs between offshore bars. Pis.’ Br. 10-11; see also PX 136 (Mackey Report) 17-25 (discussing the presence of four of the six visual features in plaintiffs’ zone); PX 141 (Shires Report) 6-7 (discussing Dr. Mackey’s analysis of the six visual features).

The CEM contemplates that the six visual indicators would be used in conjunction with “more detailed subsurface investigations.” PX 178(CEM) III—5—6; see DX 155 (Nairn Composition Report) 17 (stating that the six visual factors are “clues”). The CEM describes a shoreline in Ghana where “initial *159 visual observations” were followed by “a series of subsurface investigations” including “augers, boreholes, vibraeores and sub-bottom profiling.” PX 178(CEM) III-5-6. A section of the CEM describes “a variety of techniques for characterizing the surface and subsurface conditions, with particular focus on the sand cover thickness,” including the use of steel probes, jet probes and sub-bottom profiling. Id. at III—5—20.

Plaintiffs’ expert witness, Mr. Shires, examined samples obtained by divers and determined them to be cohesive. Pis.’ Br. 12-13. These samples, however, are not “subsurface investigations” as claimed by plaintiffs. Id. at 12. The fact that the samples were collected below the surface of the water does not make them “subsurface investigations” as contemplated by the CEM. Compare PX 178(CEM) III—5—6 (describing the use of the visual features in conjunction with techniques that sampled material below the surface of the lakebed), with Pis.’ Br. 12-13 (characterizing as “subsurface investigations” Mr. Shires’ analysis of samples collected at “various depths (between 10 and 20 feet)” below the surface of the water).

Mr. Shires testified that the samples were taken from the surface of the lakebed. Tr. 789:24-790:1 (Shires) (“So these are right on the bottom, they’re exposed on the bottom.”). Mr. Shires’ expert report, the permits obtained before the samples were collected and the photographs taken by the divers similarly indicate that the samples were collected from the surface of the lakebed and that the divers were, in fact, prohibited from collecting samples more than four inches below the surface. See PX 141 (Shires Report) App. A., Attachment B (letter from the Corps) (authorizing collection of samples two inches thick), App. A, Attachment B (letter from Michigan Department of Natural Resources & Environment) (authorizing the collection of samples two to four inches thick), App. A, Attachment C (photographs taken by divers) (showing exposed samples of cohesive material lying on the surface of the lakebed). The letter from Prism Environmental Services (Prism), the company retained to collect the samples, states that “[t]he primary purpose of this investigation was to locate in-place exposures of cohesive bottom sediments.” Id. at App. A (letter from Prism) 1 (emphasis added).

The first and second CEM features relate to the presence of cohesive materials on the beach, either as a result of intact cohesive units becoming exposed or as a result of cohesive materials washing up on the beach. PX 178(CEM) III—5—3. Plaintiffs argue that both features are present on the beaches adjacent to their properties. Pis.’ Br. 10. However, the evidence cited by plaintiffs to document the existence of exposed, intact cohesive units on the beach is limited and unpersuasive. Plaintiffs cite the trial testimony and expert report of Dr. Mackey, who, they argue, observed “exposed cohesive material and eroding cohesive scarps in the lakebed adjacent to [plaintiffs’ properties.” Pis.’ Br. 10 (emphasis added) (citing Tr. 538:19-539:7 (colloquy between Dr. Mackey and plaintiffs’ counsel); PX 136 (Mackey Report) 20). But the CEM refers to exposed cohesive sediment “on the beach,” PX 178(CEM) III—5—3, not the lakebed. As Dr. Nairn explained, “A ‘beach’ is defined as the above water or ‘subaerial’ part of the near-shore profile.” DX 155 (Nairn Composition Report) 12; see also The American Heritage Dictionary of the English Language (American Heritage Dictionary) 155 (4th ed. 2000) (including in its definition of a beach: “[t]he zone above the water line at a shore of a body of water”); Coastal Engineering Manual, App. A (Glossary) A-6 (2003) (defining a beach as “[t]he zone of unconsolidated material that extends landward from the low water line”), available at http://140.194.76.129/ publications/eng-manuals/emlll0-2-1100/ AppA/a-a.pdf.

Plaintiffs also rely on the trial testimony of Mr. Shires, Pis.’ Br. 10, who identified cohesive materials in two pictures of bluffs above beaches in plaintiffs’ zone by observing that “there appear[s] to be some vertically standing materials in the beach environment just upslope of the beach,” Tr. 817:16-18 (Shires) (discussing PX 161 (unidentified photos of beaches) l). 62 Mr. Shires explained that “in *160 order for it to stand vertically!],] it’s got to have some consolidation, some consolidated cohesive properties to it.” Tr. 817:25-818:2 (Shires).

Mr. Shires did not testify, however, that he had collected or tested samples of the material in the photos or even examined the materials in person. See Tr. 816:4-818:6 (colloquy between Mr. Shires and plaintiffs’ counsel). The court finds Mr. Shires’ characterization of the exposed materials as cohesive, which is based on photographs, to be less reliable evidence of their composition than the more sophisticated laboratory tests Mr. Shires ordered performed on the samples collected from the lake bottom. See Tr. 796:19-810:19 (colloquy between Mr. Shires and plaintiffs’ counsel) (discussing the laboratory tests conducted on soil samples—including tests of shear strength, dry unit weight, moisture content, loss on ignition, plasticity and grain size). Furthermore, the materials Mr. Shires identified in photographs are not “on the beach” as contemplated by the OEM, see PX 178(CEM) III—5—3, but are bluffs, see PX 161 (unidentified photos of beaches) 1, located, in Mr. Shires’ words, “upslope of the beach,” Tr. 817:17-18 (Shires).

Nor are the pieces of sediment that wash up on beaches in the southern reach of plaintiffs’ properties, see Pis.’ Br. 10 (describing this sediment as “pieces of exposed cohesive clay and consolidated peat,” “chunks of grayish brown clay material” and “pieces of peat”), evidence of a cohesive shoreline. Defendant points out that the presence of rafts of organic-rich sediments washing onto the beach is consistent with defendant’s explanation of the geological history and the stratigraphy of the area. See Def.’s Br. 22-23. Dr. Miekelson and Dr. Larson testified that changing lake levels deposited a thin layer of organic-rich material in the southern reach of plaintiffs’ properties about seven feet below mean lake level approximately 6,600 years ago. See infra Part III.B.3.a (describing the formation of the thin, organic-rich layer); DX 293 (Miekelson Report) 3-6 (discussing the formation of a layer of “marshy soil” approximately 6,600 years ago that has become exposed in areas and creates “rafts of peaty sand that occasionally wash up on the beach”). 63 Dr. Larson states that radiocarbon testing of one of the rafts of material that washed up on shore determined its organic material to be approximately 6,980 years old. DX 3 (Larson Report) 17. The court is persuaded by Dr. Mickelson’s conclusion that “it all fits together that the cohesive material that’s showing up out here ... is this organic-rich sand and silt.” Tr. 2100:4-6 (Miekelson). The presence of rafts of “organic-rich sand and silt,” id., is not evidence that the shoreline as a whole is cohesive.

The two other visual features drawn from the OEM and relied upon by plaintiffs'— exposed sediment in the troughs between offshore bars and discoloration of water in the nearshore zone—are similarly unpersuasive as evidence of a cohesive shoreline. Plaintiffs argue that Dr. Mackey observed exposed cohesive sediment in the troughs between offshore sand bars. Pis.’ Br. 11 (citing Tr. 542:10-543:10 (colloquy between Dr. Mackey and plaintiffs’ counsel); PX 136 (Mackey Report) at 22-23). However, as Dr, MeNinch explained, “waves and currents typically winnow and sort unconsolidated sedi *161 ment on the lakebed. These processes often result in what appears as a complex, patchwork pattern of different substrates at the surface but which, in fact, could simply originate from an underlying layer that is composed of a mixture of these sediment types.” DX 294 (McNineh Report) 3. Plaintiffs have not rebutted Dr. McNineh’s account of the source and composition of the sediment and have not, therefore, earned their burden of proof.

Plaintiffs contend that “[ae]rial photographs taken of the Plaintiffs’ nearshore zone document discoloration in the nearshore zone.” Pis.’ Br. 10. However, Dr. Nairn determined that the plumes of discoloration actually travel downstream to plaintiffs’ properties from areas that defendant describes as cohesive. See DX 155 (Nairn Composition Report) 12-13; see also id. at 2-26 (describing a section of cohesive shoreline approximately 1.9 miles long and located south of the jetties). Dr. McNineh notes that “many sandy beaches worldwide exhibit ... plumes of discoloration in the surf.” DX 294 (McNineh Report) 7. Plaintiffs failed to rebut defendant’s evidence on the source of the discoloration and similarly failed to carry their burden of proof on this issue.

The court finds the explanation by defendant’s expert witnesses of the presence of the four visual features in plaintiffs’ zone to be both credible and persuasive. Because plaintiffs failed either to rebut defendant’s evidence or to supplement its evidence of the visual features on which they rely with “more detailed subsurface investigations” such as “augers, boreholes, vibraeores and sub-bottom profiling,” as contemplated by the CEM, PX 178(CEM) III—5—6, the court finds unpersuasive plaintiffs’ argument that the claimed existence of four of the six CEM visual features establishes that plaintiffs are located along a cohesive shoreline.

ii. Plaintiffs’ Two-Part “Test” of Composition

Plaintiffs’ second argument purports to summarize the CEM’s discussion of cohesive shorelines as a two-part “test”:

In short, if (1) cohesive materials are exposed on the lakebed or covered with less than 9 feet of sand and (2) those materials are subject to irreversible lakebed erosion, a shoreline is cohesive.

Pis.’ Br. 8. 64 For the following reasons, the court finds that plaintiffs’ proposed two-part “test” does not accurately represent the analysis recommended by the CEM to use in identifying cohesive shorelines. Nor do the facts adduced at trial support plaintiffs’ proposed analysis.

Nowhere does the CEM reduce the detailed analysis it describes to a two-part “test.” See PX 178(CEM) passim. Instead, the CEM recommends a detailed analysis. The CEM contains a non-exhaustive list of six visual factors to consider, see id. at III—5— 3 to III—5—6, and describes “a variety of techniques for characterizing the surface and subsurface conditions,” id. at III-5-20, which are to be used to verify the visual indicators, see id. at III—5—6 (stating that, at the shoreline used as an example, such techniques included “augers, boreholes, vibraeores and sub-bottom profiling”).

Not only has plaintiffs’ proposed two-part “test” oversimplified the detailed analysis *162 suggested by the CEM, but plaintiffs have also reduced the analysis described in the CEM to a “test” that is more easily satisfied. With regard to the first part of the “test,” that cohesive materials are exposed on the lakebed or covered with less than nine feet of sand, Pis.’ Br. 8, Dr. Mackey testified that even a very small amount of exposed cohesive material would be sufficient under plaintiffs’ analysis:

Q: [S]o the presence of those two substrates, that fulfills that part of the definition for you ... ?

A: Yeah.

Q: Sir, would that still be your opinion if the percentage of those two substrates were say 5 percent?

A: Yes.

Q: 1 percent?

A: Yes.

Tr. 715:17-716:10 (colloquy between Dr. Mackey and defendant’s counsel); but see DX 294 (McNinch Report) 3 (“[W]aves and currents typically winnow and sort unconsolidated sediment on the lakebed. These processes often result in what appears as a complex, patchwork pattern of different substrates at the surface but which, in fact, could simply originate from an underlying layer that is composed of a mixture of these sediment types.”).

To demonstrate the existence of cohesive materials, plaintiffs presented the testimony of Mr. Shires, who reviewed the results of laboratory tests run on samples taken from the surface of the lakebed. See PX 141 (Shires Report) 2. Rather than sampling at regular or pre-determined intervals, divers hired by plaintiffs searched the lakebed for “in-place exposures of cohesive bottom sediments” and provided samples to Mr. Shires. Id. at App. A (Letter from Prism) 1. Mr. Shires states that, although some of the samples contained as much as 87% sand, id. at 9, they could be considered cohesive, because they contain organic material that “provides binding properties (i.e., apparent cohesion) not accounted for by simply considering grain size,” id. at 5; see id. at 9. Mr. Shires opined that the samples he tested are of near-shore lake bottom that is composed of “glacial till that was consolidated in the geologic past by glaciers.” Id. at 7; see also id. at 1, 5.

The theory that the samples were taken from a thick layer of till formed and consolidated by glaciers, however, is inconsistent with the geological history of the area. More persuasive is the explanation that Dr. Shires’ samples were collected from outcroppings of a thin layer of organic-rich soil. See infra Part III.B.3.a (describing the formation of the thin, organic-rich layer). Plaintiffs acknowledge that radiocarbon dating indicates that a layer of “peat and logs” is present seven feet below the mean lake level in the southern portion of plaintiffs’ zone and contains organic material approximately 6,600 years old. Pis.’ Br. 13. As Dr. Mickelson explained, the organic materials were deposited too recently to be glacial till: “the glacier was gone by 10,000 years ago ... [T]hat layer is dated as just under 6,000 years old.... So, it’s much later, much younger.” Tr. 2096:10-21 (Mickelson); see DX 293 (Mickelson Report) 3-6 (discussing the formation of a layer of “marshy soil” approximately 6,600 years ago), 34-36 (discussing radiocarbon dating of the peat layer penetrated by engineering borings). The fact that “rafts of peaty sand,” DX 293 (Mickelson Report) 3, which have been determined by radiocarbon dating to be approximately 6,600 years old, Pis.’ Br. 13 (citing Tr. 2099:1-2101:4 (colloquy between Dr. Mickelson and plaintiffs’ counsel)), wash up on the beach indicates that this thin, peaty layer has become exposed in some areas of the nearshore adjacent to plaintiffs’ properties, see DX 293 (Mickelson Report) 3; cf. PX 136 (Mackey Report) (stating that pieces of cohesive material Dr. Mackey observed on the beach had been “eroded from exposed cohesive deposits on the lakebed located immediately adjacent to[,] and offshore from,” the areas where they were found); Tr. 532:1-533:3 (colloquy between Dr. Mackey and court) (describing a scarp of material he interpreted as cohesive, visible in his sidescan sonar survey).

In addition to the age of the materials, their content is also inconsistent with their being composed of glacial till. The existence

*163 of a large proportion of organic material in the samples analyzed by Mr. Shires is consistent with defendant’s explanation that the samples were collected from a thin layer, rich in peat and other organic material, formed after the retreat of the glaciers, see Def.’s Br. 22-23, and is inconsistent with Mr. Shires’ contention that the material was deposited by glaciers, see PX 141 (Shires Report) 1, 5, 7. Dr. Mickelson testified that glacial till “has a mix of grain sizes and typically isn’t organic. This has organic materials, fine sand grains that look like wind[-Jblown sand and no pebbles, no stones. They’re certainly not till.” Tr. 2096:3-9 (Mickelson); .see also DX 293 (Mickelson Report) 2 (stating that glacial till is “characterized by containing a wide range of grain sizes compared to sediment deposited in streams or lakes”). The court is persuaded by Dr. Mickelson’s report and related testimony that the samples analyzed by Mr. Shires are pieces of the thin layer rich in organic materials located between thick deposits of sand. See Tr. 2096:10-21, 2096:3-9 (Mickelson); DX 293 (Mickelson Report) 27, 33-34 (describing the formation of the thin, organic-rich layer); cf. DX 3 (Larson Report) 16 (describing same); see infra, Part III.B.3.a (describing how the thin, organic-rich layer formed and is eroding, forming rafts that wash up on the beach). 65

Plaintiffs do not reconcile their “test” for the presence of a consolidated shoreline— which is satisfied by even the very small amount of cohesive material on the surface of the lakebed found by plaintiffs’ experts, see Tr. 715:17-716:10 (colloquy between Dr. Mackey and defendant’s counsel)—-with the geological history of the area, which resulted in the incorporation of small amounts of cohesive material into otherwise sandy areas of shoreline, see infra Part III.B.3.a (describing the geological history and stratigraphy of plaintiffs’ zone).

The second part of plaintiffs’ “test” merely restates the fact—well established in this case—that the erosion of cohesive materials is irreversible. See Pis.’ Br. 8; Liability Op., 78 Fed.Cl. at 628 (finding that the erosion of cohesive material is permanent and irreversible). The CEM provides:

A shore is defined as consolidated cohesive when the erosion process is directly related to the irreversible removal of a cohesive sediment substratum____This differs fundamentally from sandy shores where erosion (or deposition) is directly related to the net loss (or gain) of noncohesive sediment from a given surface area. Erosion on a sandy shore is a potentially reversible process (i.e., due to natural processes), while erosion on a consolidated cohesive shore is irreversible.

PX 178(CEM) III—5—2 to III-5-3. It is the nature of cohesive materials that any erosion they undergo is irreversible. See id. The second part of plaintiffs’ “test” merely restates this fact.

The “test” plaintiffs propose reduces the analysis of whether a shoreline is cohesive to the narrow inquiry of (1) whether there are any “exposed cohesive materials on the lake-bed in [plaintiffs’ zone which [ (2) ] are subject to irreversible downeutting.” Pis.’ Resp. 18 n.6; see also Pis.’ Br. 8 (stating same). Because this approach is inconsistent with the thorough analysis of shoreline composition recommended by the CEM, see PX 178(CEM) III—5—3 to III—5—6 (describing six visual features of cohesive shorelines); id. at *164 III—5—20 (describing “a variety of techniques for characterizing the surface and subsurface conditions”), from which plaintiffs purport to draw their two-part “test,” see Pis.’ Br. 8 (citing, inter alia, PX 178(CEM)); Tr. 716:25-717:3 (colloquy between Dr. Mackey and defendant’s counsel) (agreeing that Dr. Mackey limited his analysis of shoreline composition to “what is set forth in Chapter 5 [of the OEM]”), the court declines to adopt plaintiffs’ approach.

3. Additional Evidence of Shoreline Composition Presented by Defendant at the Trial of Damages

In addition to the evidence of shoreline composition submitted by defendant at the trial of liability, which the court found persuasive, see Liability Op., 78 Fed.Cl. at 621-28 , defendant submitted additional evidence of shoreline composition at the trial of damages. Defendant states that assessing the composition of the shoreline “requires locating the shoreline at a point on a spectrum encompassing ‘two ends of a limiting system, one of which is almost all cohesive, and the other which is ... all sand.’ ” Def.’s Br. 5 (omission in original) (quoting Tr. 2781:1-3 (Nairn)). Defendant, therefore, “supplemented its understanding of shoreline composition with geological and physical evidence related to the shoreline’s material composition, as well as computer modeling and a sediment budget that assesses the shoreline’s behavior.” Id.

Specifically, defendant presented the testimony of Dr. Miekelson, a geologist, and Dr. Nairn, a coastal engineer. See supra note 9. Dr. Miekelson reviewed the stratigraphy and geological history presented by Dr. Larson at the trial of liability 66 and supplemented both with his own analysis of the history and grain size distribution of the northern and southern areas of plaintiffs’ zone. See Tr. 2011:7-2154:14 (testimony of Dr. Miekelson). Dr. Nairn used the stratigraphy developed by Dr. Miekelson and Dr. Larson to determine, on a property-by-property basis, whether each plaintiffs property has a sandy or a cohesive shoreline. See DX 155 (Nairn Composition Report) 27-32. Dr. Nairn also discussed the erosion and accretion of different areas in plaintiffs’ zone over time, concluding that most of plaintiffs’ properties were located in areas that behaved as a sandy shoreline would be expected to behave. See Tr. 2748:2-5 (Nairn) (stating that the shoreline in plaintiffs’ zone behaves as a sandy shoreline).

The record indicates that the shoreline in plaintiffs’ zone is neither pure sand nor pure cohesive material. Certain areas contain layers of both cohesive and sandy material. See infra Part III.B.3.a (describing the formation of a thin band of sediment rich in cohesive organic material between thick layers of sand); DX 293 (Miekelson Report) 2-3 (describing the complex stratigraphy formed in the northerly portion of plaintiffs’ properties by the retreat and advance of glaciers, as well as deposition and erosion by lake and stream environments). Certain sediments sampled in plaintiffs’ zone are difficult to classify as sandy or cohesive. See, e.g., PX 141 (Shires Report) 5 (discussing sediment samples with both high sand content and cohesive characteristics). Because the shoreline composition is more complex than pure sand or pure cohesive material, the court finds the approach used by defendant—an approach that employs several metrics to evaluate shoreline composition—to be useful in determining where on the spectrum of composition plaintiffs’ zone is located.

a. Additional Evidence Regarding Geological History and Stratigraphy

Dr. Miekelson provided additional detail to supplement the geological history and stratigraphy presented by Dr. Larson at the trial of liability. See generally DX 293 (Miekelson Report). Regarding the stratigraphy, Dr. Miekelson reviewed all of the well logs Dr. Larson had used to assemble his stratigraphy and concluded that “for the most part I agreed with the stratigraphy that he had put *165 together.” 67 Tr. 2027:5-9 (Mickelson). Dr. Miekelson states that, because of the dearth of subsurface samples, “the best information available about what is below the surface in the nearshore is extrapolation of well log data from onshore.” DX 293 (Miekelson Report) 20.

Based on the geological history and stratigraphy of plaintiffs’ zone, Dr. Mickelson concludes that there are “two distinctly different regions” in plaintiffs’ zone, which he “treat[s] ... separately.” Tr. 2027:12-15 (Mickelson). The dividing line between Dr. Mickelson’s regions, or “reaches,” is West Glenlord Road, which intersects Lake Michigan in approximately the same location as the Inner Lake Border moraine, a ridge built by glaciers. Tr. 2029:7-12 (Miekelson); see also DX 293 (Miekelson Report) 4, Fig. 1 (map of plaintiffs’ zone). Dr. Mickelson concludes that the compositions of the northern reach and the southern reach are very different: “The northern reach is made up of glacial till and sand and gravel. The southern reach is made up of sand dunes and some organic material.” Tr. 2030:19-22 (Mickelson). The northern reach is characterized by “bluff shoreline. And south of that[,] it’s dominated by dunes.” Tr. 2023:11-13 (Mickelson). The twelve northernmost properties are located in the northern reach, and the twenty-eight remaining properties are located in the southern reach. DX 293 (Mickelson Report) 2. 68

Dr. Mickelson explained that the differing composition of the northern and southern reaches results from their differing geological histories. 69 All of the sediment underlying plaintiffs’ properties was deposited since the last glaciation of the area—25,000 years ago. Id. “The glacier lobe scoured much of the Lake Michigan basin, enlarging it and removing most of the older glacial deposits down to the bedrock surface.” Id. Starting approximately 20,000 years ago, the glacier began to retreat, depositing glacial till, “which is sediment deposited directly by glacier ice and which is characterized by containing a wide range of grain sizes compared to sediment deposited in lakes or streams.” Id. Dr. Mickelson described how the retreat of the glaciers formed the layers that underlie the northern reach of plaintiffs’ properties:

The retreat of the glacier was interrupted by numerous re-advances of the glacier as [the] climate fluctuated. Each re-advance of the glacier deposited a layer of till with a distinct composition. The two till layers present along this shoreline are the younger Saugatuek till and the somewhat older Ganges till. Gravel, sand, silt, and clay were deposited in shallow lake and stream environments near the front of the retreating glacier. These deposits lie between the Saugatuek and Ganges till units as well as beneath the Ganges till where it is present.

Id. at 2-3. The resulting stratigraphy is complex, consisting of two layers of glacial till, with deposits of gravel, sand, silt and clay between the layers of till and-in certain loeations-beneath both layers of till. See DX 3 (Larson Report) 34, Fig. 9 (geological map and cross section of plaintiffs’ zone). Certain of the layers of sediment are located above the surface of the lake and therefore outside of the lakebed. See id.; DX 293 (Mickelson *166 Report) 11 (describing cohesive material located in onshore bluffs). Other layers of sediment are located below the depth to which the nearshore lakebed is eroding. See DX 155 (Nairn Composition Report) 30 (stating that, for this reason, Dr. Nairn would examine sediments to a depth of eight meters); DX 293 (Mickelson Report) 20, 34 (stating that, in characterizing the composition of plaintiffs’ shoreline, Dr. Mickelson would focus on sediments up to twenty feet below the surface of the lake).

At this point, the geological histories of the northern reach and the southern reach diverge because the southern reach underwent further change. When the glacier that formed the moraine retreated, lake levels rose and the edge of the lake reached the edge of the moraine, submerging the southern reach under water. 70 Tr. 2029:19-2030:7 (colloquy between Dr. Mickelson and defendant’s counsel). As a result, the surface was “cleaned off by waves” and covered in sand. Tr. 2029:25-2030:12 (colloquy between Dr. Mickelson and defendant’s counsel); Tr. 2032:3-4 (Mickelson) (stating that “sand would have been delivered to the area at the time the lake level was high”).

Approximately 9,000 years ago, the lake level dropped, creating a land surface with shallow ponded areas, on which organic material began to collect. Tr. 2032:4-10 (Mickelson). This organic material accumulated into a layer of peat and logs approximately ten centimeters thick and located approximately seven feet below the current surface of the lake. DX 3 (Larson Report) 16. This material is cohesive, but because of its thinness, constitutes a small fraction of the sediment in the shoreline. DX 293 (Mickelson Report) 36. Approximately 6,000 years ago, lake levels again rose, submerging this layer and depositing sand above it. See Tr. 2032:22-2033:8 (Mickelson). The thin layer of organic-rich materials was penetrated by two of the well logs, as well as the two engineering borings conducted by the United States Geological Service (USGS). See DX 293 (Mickelson Report) 3. 71

This thin, organic-rich layer has become exposed offshore in some areas in the southern reach of plaintiffs’ properties. See id. at 36. As it erodes, it creates rafts of organic-rich material that wash up on the beach. See id. at 3. Radiocarbon testing of one of the rafts of material determined its organic material to be approximately 6,980 years old, DX 3 (Larson Report) 17, roughly the same age as the organic material in the thin, organic-rich layer, see id. at 16-17 (stating that radiocarbon dating of this layer revealed the *167 presence of organic materials approximately 6,630 to 6,675 years old).

The geologic history described by Dr. Mickelson strongly suggests that the southern reach of plaintiffs’ zone is composed primarily of sand, with a thin layer of organic-rich material deposited approximately 6,600 years ago. Tr. 2029:19-2033:8 (colloquy between Dr. Mickelson and defendant’s counsel); see also DX 293 (Mickelson Report) 3-6 (describing the geological history of the southern reach). The geologic history is less probative of the shoreline composition in the northern reach. The stratigraphy of the northern reach is more complex, containing two layers of till layered with other sediments. See DX 3 (Larson Report) 34, Fig. 9 (geological map and cross section of plaintiffs’ zone).

b. Grain Size Distribution in the Northern and Southern Reaches

In addition to describing the geological history of the area, Dr. Mickelson calculated the sand content of the nearshore lakebed as a percentage of its overall composition. 72 DX 293 (Mickelson Report) 10, 20. Dr. Mickelson restricted his analysis to sediments located between the level of the lake surface and a level twenty feet below the lake surface 73 Id. at 20, 34. Dr. Mickelson states that whether a sediment is cohesive depends in part on grain size:

Gravel and sand are not cohesive. Coarse silt, if well sorted[,] is usually not cohesive. Medium and fine silt and clay typically have cohesion. A mix of sizes, such as occurs in till, can be cohesive depending on the amount of clay and silt it contains. Both till units (Saugatuck and Ganges) have cohesion. Most of the lake sediment between the till is not cohesive.

Id. at 10. 74 Plaintiffs do not argue otherwise. 75 See Pis.’ Br. passim; Pis.’ Resp. passim. As sources of data, Dr. Mickelson relied upon well logs, his own observations, sediment samples that Dr. Mickelson and a graduate student under Dr. Mickelson’s supervision collected and analyzed, Dr. Larson’s observations and “various reports and papers including USGS reports and journal articles.” DX 293 (Mickelson Report) 6-7. 76

To calculate the sand content of the lake-bed in the southern reach and the northern reach, Dr. Mickelson employed a two-step method. 77 Dr. Mickelson’s first step entailed *168 calculating the number of feet of each sediment type that each driller encountered, as recorded in the well logs, and determining the percentage of the total described as sand. See, e.g., id. at 22, Table 11 (tabulation of sediment types and thicknesses for the northern reach) (stating that, at the depths Dr. Mickelson observed, drillers in one section encountered forty-four feet of sand, zero feet of clay, thirteen feet of interbedded sand and silt, and 25 feet of Ganges till). Dr. Mickelson calculated that the well logs described 42% of the sediment in the northern reach between the level of the lake surface and a level twenty feet below the lake surface as sand and 18% as interbedded sand and silt. See id.; Tr. 2062:9-15 (colloquy between Dr. Mickelson and defendant’s counsel). Dr. Mickelson calculated that the well logs described 89% of the sediment located at the same range of depths in the southern reach as sand. DX 293 (Mickelson Report) 35, Table 13 (tabulation of sediment types and thicknesses for the southern reach); see Tr. 2087:10-11 (Mickelson).

Dr. Mickelson’s second step accounts for the fact that the individual sediment layers contain not a single grain size of material, such as sand or silt, but a distribution of grain sizes. See Tr. 2088:6-8 (Mickelson) (stating that “what I’ve tried to do then is adjust these numbers ... for the realistic sand content in those sediment types”). Dr. Mickelson explained that “when a driller— when somebody describes something as sand[,] it isn’t necessarily 100 percent sand[,] and so what I tried to do here is not count sand as being 100 percent sand____There’s going to be some silt [and] clay in it probably.” Tr. 2087:25-2088:6 (Mickelson). Using his own observations and reviewing the measurements made by other researchers, Dr. Mickelson examined the sand content of the various types of sediment present in plaintiffs’ zone. See DX 293 (Mickelson Report) 11 (discussing sources of data). 78 Dr: Mickelson determined that the Saugatuek till contains 29.7% sand, the Ganges till contains 31% sand and the unit of interbedded sand and silt between the till layers contains 81.5% sand. Id. at 13-14. Dr. Mickelson treated sediments described by the well drillers as “sand” as containing only 90% sand and treated any layers described as “clay” as containing no sand. See id. at 20. Applying the sand content of each sediment type to the amount of each sediment type present, Dr. Mickelson calculated that that the material between the lake surface and a level twenty feet below the lake surface—and therefore, the material in the nearshore lakebed—consists of 65% sand in the northern reach, id. at 20, and consists of 80% sand in the southern reach, id. at 38.

Dr. Mickelson’s calculations indicate that plaintiffs’ zone has significant sand content. The well logs described 60% of the sediment at the depth of the lakebed in the northern reach as either sand or interbedded sand and silt. See id. at 22, Table 11 (tabulation of sediment types and thicknesses for the northern reach). The well logs described 89% of the sediment at the depth of the lakebed in the southern reach as sand. Id. at 35, Table 13 (tabulation of sediment types and thicknesses for the southern reach). Applying the sand content of each sediment type to the observations documented in well logs, Dr. Mickelson determined that the sand content of the lakebed in the northern reach is 65%, id. at 20, and that the sand content of the lakebed in the southern reach is 80%, id. at 38.

c. Grain Size Distribution on a Property-by-Property Basis

Dr. Nairn analyzed the sand content of the shoreline to determine, on a property-by-property basis, which of plaintiffs’ properties *169 are located along a sandy shoreline. Recognizing that there is a “spectrum of shore types between purely cohesive shores and fully sandy shores,” DX 155 (Nairn Composition Report) 5, Dr. Nairn argues that, for three reasons, the sand content of the shoreline is “the over-riding factor” in determining whether the shoreline should be considered sandy or cohesive for purposes of erosion:

1. the definition in the literature of a limit of 10% to 30% [sand] for a cohesive shore;

2. the fact that the rate of removal of sand is limited by the available wave energy whereas the rate of removal of cohesive [material] (silt and clay) ... offshore is not; and

3. the fact that without removal of the sand the erosion of any underlying exposures of cohesive and sandy sediment could not continue,

id. at 17.

Dr. Nairn undertook an extensive literature review to determine the maximum sand content of cohesive shorelines described in other research. See id. at 5-15. Dr. Nairn states that “[a] large body of scientific understanding of cohesive shores on the Great Lakes emerged from the Port Burwell Litigation in the early 1980’s,” which involved erosion caused by harbor structures. Id. at 5. Beginning with the scholarly papers that emerged from the Port Burwell litigation in 1983, Dr. Nairn examined a number of papers and articles—including several of his own that were published before this litigation, id. at 5-15—and concluded that “[t]he literature unanimously puts the upper limit for sand content of a cohesive shore no higher than 30% (although for one part of one cohesive shore bluff in the Toronto area[,] the sand content was found to be 36%),” id. at 16-17.

Using the stratigraphic map developed by Dr. Miekelson, Dr. Nairn calculated the sand content at a number of “representative locations” along the shoreline to a depth of eight meters below chart datum. 79 Id. at 30. Dr. Nairn selected the eight-meter cutoff because he determined—in the analysis of historic shoreline erosion contained in the report he prepared for the trial of liability— that the nearshore profile is eroding to a depth of approximately eight meters. 80 Id. For each representative location and for each of plaintiffs’ properties, Dr. Naim calculated the sand content using the Wentworth scale applied by Dr. Miekelson and most geologists, on which sand is defined as sediment larger than .0625 millimeters. See id. at 28; see supra note 74 (discussing the Wentworth scale). Dr. Nairn also calculated the sand content using a cutoff of .1 millimeters, which, although larger than the grain size cutoff normally utilized by geologists and engineers, 81 is “more consistent with the grain size of sand that might be expected to exist in the littoral zone (i.e.[J whereas finer sediment would move offshore outside the littoral zone).” DX 155 (Nairn Composition Report) 28. Of course, the .1 millimeter cutoff results in findings of lower sand content, which is favorable to plaintiffs’ argument that the shoreline is cohesive.

Regardless of the grain-size cutoff Dr. Nairn used, he determined that all but one of plaintiffs’ properties 82 were adjacent to areas of lakebed with sand content that is consistent with a sandy shoreline. See id. at 35, Fig. 4.3b (chart of lakebed sand content) (showing one property with lakebed sand content equal to or less than 30%). The *170 northernmost property, 83 belonging to plaintiffs Gregory and Candice Bovee, has “a sand content between 61.5% and 67.5% (using the minimum sand size of 0.0625 mm and 0.10 mm, respectively).” Id. at 31. The next property, belonging to plaintiff Marcia Wine-berg, has a sand content between 66% and 72%. Id. The third property, belonging to plaintiffs Kent and Margaret Werger, has a sand content between 41% and 48%, the lowest sand content of the plaintiffs’ properties. Id. Dr. Nairn concludes that, although the sand content of the Werger property “is above the sand content limit for cohesive shores derived from the available literature (i.e., 30% with one instance of 36%), it is close enough to the limit[] that in this expert’s opinion, it is classified as a cohesive shore.” Id.

Continuing to the south, “The properties between Okonski and Nueser all fall in a range where sand content of the lakebed sediment is always greater than 59% and usually greater than 70%[,] with averages across this zone of 67.5% to 73%.” Id. The properties in this area are owned by plaintiffs Okonski, Bodnar, Wilsehke, Jyung, Varga, Jackson, Greenbriar, Marzke and Neuser. Id. The sand content of the lakebed adjacent to the next property, which belongs to the Del Mariani plaintiffs, is “just above 75%.” Id. The lakebed adjacent to the remaining properties

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