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Department of Geological Sciences and Institute for Geophysics, Jackson School of Geosciences, University of Texas at Austin, Austin, Texas, USA
Corresponding author: W. Kim, Department of Geological Sciences and Institute for Geophysics, Jackson School of Geosciences, University of Texas at Austin, Austin, TX 78712, USA. (firstname.lastname@example.org)
 It is now generally accepted that deltas that prograde to the shelf edge are able to transport coarse sediment to deep water either with or without sea level changes. However, it is still unclear how feeder rivers behave differently in the shelf-edge delta case to rivers found in a delta that progrades over the shelf. A series of nine shelf-edge delta experiments are presented to investigate the lateral mobility of the feeder channel at the shelf edge and the associated deep water depositional system under a range of sediment supply rates and shelf-front depths. In the experiments, constant sediment supply from an upstream point source under static sea level led the fluviodeltaic system to prograde over the shallow shelf surface and advance beyond the shelf edge into deep water. The feeder river of the fluviodeltaic system became a bypass system once the toe of the delta front reached the shelf edge. After the delta front was perched at the shelf edge, a submarine fan developed in deep water although remaining disconnected from the delta. In this bypass stage, no regional avulsion or lateral migration of the feeder river occurred and all sediment from the upstream source bypassed the river, delta front, and shelf-front slope. The duration of the bypass stage is proportional to shelf-front depth and inversely proportional to sediment discharge. The combined duration of the shelf-transit phase of the fluviodeltaic system and the bypass phase is the characteristic time scale for the continental margin to “anneal” transgression-inducing perturbation due to high-frequency and/or high-amplitude relative sea level rise. The sequential evolution in the experiment compares favorably to the Eocene Sobrarbe Formation, a shelf-edge delta in Spain, although natural variations are noted. This comparison justifies the application of concepts proposed herein to natural systems and provides insight into interpreting processes from ancient shelf-edge delta systems.
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 Relative lowstands in sea level are regarded as the primary process by which large volumes of coarse-grained sediment are transferred across the shelf and delivered to deep water basins [Posamentier et al., 1988; Galloway, 1989; Posamentier and Allen, 1999; Van Wagoner, 1995; Embry, 2001]. These types of deposits are notable because they constitute over 15% of siliciclastic hydrocarbon reservoirs [Richards et al., 1998]. During relative sea level fall and subsequent lowstand, the fluvial channels incise the antecedent shelf, resulting in incised valleys that localize coarse sediment and restrict lateral shifting of the feeder river system. The depth of the valley scales with the amplitude of sea level fall, and thus, larger sea level falls can produce larger deep water deposits. For that reason, sea level is regarded as the main mechanism to facilitate sediment transport across the shelf through a confined incised valley to a focused deep water basin.
 In an effort to determine the main mechanisms of sediment transport across shelf edges, Burgess and Hovius  estimated maximum times for modern deltas to prograde to the shelf edge and found that the shelf-transit time scales are relatively short and within a range of ~10 to ~100 ka. This led to the conclusion that sea level fall is not essential to transport a significant amount of sediment across shelf edges and into the deep water basin. The calculation considered a balance between space on the shelf and fluvial sediment inputs from modern deltas. Also, the calculation suggested that deltas could arrive at the shelf edge under sea level highstands within a relatively short time period compared with glacio-eustatic cycles over a few million years. However, it remains unknown (1) how the feeder rivers on shelf-edge deltas behave differently from those in deltas that prograde over the shallow shelf surface before reaching the edge and (2) how long the continental margin takes to fully recover from the previous rapid transgression due to high-frequency and/or high-amplitude sea level rise and return to its preperturbation state of overall margin-wide progradation.
 We conducted a series of physical experiments using tank facilities at Nagasaki University to simulate delta progradation across a shelf edge (Figure 1a) in order to address the following two goals. First, we used physical experiments to document a feeder river that pipes large amounts of coarse-grained sediment into a deep water system without lateral changes in the location of the depositional center (compensation) during a highstand in sea level. In the experiment, the delta prograded toward the shelf edge, during which the distributary channels incessantly migrated and avulsed laterally to redistribute the supplied sediment across the entire delta. However, when the system reached the shelf edge, the delta distributaries converged to a single bypass channel. Over this bypass stage, channel avulsion and lateral shifting no longer occurred and all of the sediment supplied through the feeder river was directly piped into the deep water basin. Muto and Swenson [2005b] documented that alluvial grade in a 1-D delta can be attained by damming supplied sediment upstream of a weir. Once sediment fills up to the top of the weir, sediment reaching the weir cannot accumulate on the vertical weir face and exits the domain. In this case, the delta surface has neither net deposition nor erosion. Furthermore, Clarke et al.  documented the formation of single-graded channel on an alluvial fan (i.e., no ponded water downstream) due to a fixed downstream boundary that drains enough sediment to prevent further fan toe progradation. The study in this paper builds upon these findings by expanding the study to include further investigation in regard to delta evolution associated with graded rivers at the shelf edge and interactions with the submarine fan.
 The second goal of the current study is to determine the continental margin “annealing” time scale following rapid transgression and/or backstepping. Simple theoretical sediment-mass balance models are presented here to capture the time scales of (1) the fluviodeltaic system to transit over the shallow shelf surface, similar to the model in Burgess and Hovius  and (2) the development of the submarine fan. The modeling results account for the total duration for the delta to prograde over the shelf and the submarine fan to aggrade in the experiments under various shelf-front water depths and sediment supplies. We also provide a brief discussion about the forcibly sustained bypass of alluvial rivers based on the observation in the current experiments and previous ones [Muto and Swenson, 2005b]. Ultimately, we apply our experimentally achieved understanding to provide insight into the processes that developed the stratigraphy of a shelf-margin delta in the Eocene Sobrarbe Formation, Spain.
2 Tank Experiments
2.1 Experimental Design
 The experiments were conducted in the tank facilities located at Nagasaki University. The experimental tank is 2.8 m long, 1.4 m wide, and 0.65 m deep, but for this experiment, only a 1 m wide by 1 m long horizontal section of the tank was used (Figure 1). The experimental section was composed of two nonerodible, flat basements at different elevations: an upstream half with a higher elevation (termed “shelf” basement) and a downstream half with a lower elevation (termed “deep water” basement) (Figure 1). The difference between the elevation of the shelf and deep water basements defines the shelf-front depth, H. Nine individual experiments, termed Runs 1–9, were carried out under combined conditions of three sediment discharges (Qs) and three shelf-front depths (H) (Table 1). For convenience, the nine experiments were grouped into three series, such that each series had the same H value (i.e., 1 cm in Runs 1–3, 7.1 cm in Runs 4–6, and 14.3 cm in Runs 7–9) and were ordered from low to high ratios of sediment supply to water discharge.
Table 1. Experimental Parameters Used in This Study
Total Run Time (s)
 The sediment used in the experiments was well-sorted natural quartz sand, with uniform grain size of 0.2 mm and specific gravity of 2.65. Sediment was input through an hourglass-like funnel that had an internal weir for sediment overflow, which maintained a constant sediment-feed rate. We used three glass tubes with different diameters (3 mm, 4 mm, and 5 mm) to change sediment supply rate. Water was supplied at a constant rate from a tube connected to multiple weirs that maintained constant water head. Water and sediment were fed into a funnel to mix together and then supplied into the basin. This sediment and water feed system were located at the corner of the two vertical walls opening at 90° from the upstream end (Figure 1). Water depth over the shelf basement during each run was kept constant at 1 cm in Runs 1–3 and at 0.5 cm for the rest of the runs. Each experiment continued until the submarine fan grew large enough to attach itself to the shelf-edge delta.
2.2 Experimental Data
 To document the evolution of the delta, time lapse images were taken every 30 or 60 s during each run. The time lapse images were corrected for distortion and perspective and resized so that 1 pixel represents a 1 mm by 1 mm area of the experimental surface. The brown sediment contrasted against the black floor of the tank (basement surface), which allowed the position of the delta toe to be detected using an automated MATLAB program. The program distinguishes the colors and converts the images to binary, black (basement with no sediment), and white (sediment surface) images. The binary images were then rotated such that the sediment-feed point moves to the bottom-right corner of the image (Figure 2a).
 The MATLAB program outputs the total area of the sediment surface in white (Figure 2a) and thereby the perimeter of the area that represents distal boundary of the delta. So when the delta progrades over the shelf basement, the downstream boundary is the same as the delta toe position, but if the submarine fan develops in the deep water basement, then the perimeter of the area is represented the distal boundary of the submarine fan. The total surface area of the delta and submarine fan was calculated and averaged to detect the location of the downstream boundary for each of the three 30° intervals of the fluviodeltaic system, which are named the “River R,” “Axial,” and “River L” in counterclockwise order (Figure 2b). Whenever the deep water fan aggraded sufficiently such that its apex attached to the delta toe at the shelf-edge position, we measured the horizontal distance (L) between the shelf edge and the submarine fan toe (Figure 1b). The subaqueous slope of the submarine fan, Sf, was then calculated by Sf = L / H. The measured values of L and Sf are shown in Table 1.
2.3 Experimental Results
 In all of the runs, the delta prograded steadily over the shelf basement with an isotropic arcuate, radially symmetric shoreline until it reached the shelf edge. Active distributary channels sometimes caused rapid local progradation of the deltaic shoreline but also frequently avulsed (and/or migrated laterally) across the delta plain and dispersed the supplied sediment more or less evenly to the entire shoreline. After the toe of the delta reached the shelf edge, the distributary channels ceased avulsing and a dominant channel became entrenched in a narrow zone in the axial part of the delta. A submarine fan developed on the deep water basement while sediment bypassed the delta foreset and the shelf-front slope.
 Figure 3 shows sequential images of Run 7, which had the largest shelf-front depth, H and smallest sediment supply rate, Qs (Table 1). The delta progressively advanced basinward over the shelf basement (Figure 3a) until its toe reached the shelf edge at t ≈ 4890 s (Figure 3b). Slight channel incision occurred in the axial part of the delta, whereby all the supplied sediment bypassed both the deltaic topset and foreset. After this point in time, neither avulsion nor significant lateral migration of the feeder channel occurred (Figure 3c). The channel was fixed in this axial position where the distance between the shelf edge and the sediment-feed point is shortest, and migration of the shoreline of the delta halted. Sediment supplied through the channel accumulated on the deep water basement and developed a submarine fan. The feeder channel was stationary until the submarine fan aggraded to such an extent that its apex became connected to the toe of the delta at t ≈ 9390 s (Figure 3d). Following this critical moment, the channel laterally migrated (or avulsed) and the shoreline resumed prograding basinward, although at a much slower rate than before the delta reached the shelf edge because the sediment was still depositing over the shelf edge. As a result, sediment was again deposited on the River R and River L sides of the shelf and deep water basements (Figure 3e, t = 13,680 s). The submarine fan then laterally widened and was therefore no longer radially symmetric (Figure 3e).
 The progradation of the deltaic shoreline over the shelf basement overall followed the cube root of time relationship suggested in Powell et al. . The progradation rate of the fluviodeltaic system nonlinearly decreased with time as the surface area of the sloped delta top expanded. Initially, the shoreline migration rate was high, but it quickly decreased with time. We note that the delta toe prograded at the same rate with the shoreline in each experiment due to the uniform water depth in the shelf basement and overall constant foreset slope with time. The locations of the delta toe and submarine-fan toe are used here for further analysis. The time series positions of the downstream boundary (i.e., delta toe and submarine-fan toe) measured for each of the three subsections (Axial, River R, and River L in Figure 2b) demonstrate the three stages of sequential evolution: (1) progradation of the delta across the shelf basement, i.e., shelf-transit phase, (2) development of the bypass feeder channel and submarine fan, i.e., bypass phase, and (3) reactivation of the channel to laterally migrate when the submarine fan aggrades to the shelf edge and reconnects with the shelf delta. There were no significant differences in the longitudinal position of the downstream boundary of the Axial, River R, and River L regions until the delta toe reached the shelf edge at 500 mm downstream from the sediment source (Figure 4). This represents that channels in the delta laterally migrate/avulse actively and maintain the shoreline symmetry. After this initial shelf-transit stage, only the downstream boundary of the Axial region prograded and the downstream boundary of River R and River L regions nearly halted, reflecting the establishment of a bypass system to the deep water basement through the Axial section. Figure 4 shows that the Axial plot diverges from the ones for River L and River R due to spatially nonuniform sedimentation associated with the stationary feeder channel in the axial position. Runs 4–9 with H = 7.1 cm and 14.3 cm (Figures 4a–4f) particularly show the strong progradation limited to the Axial region coincidental with bypass to the submarine fan. In the final channel reactivation phase, slow lateral migration of the feeder river caused lateral expansion of the submarine fan beyond the Axial section to River R and River L. Therefore, progradation rates in these regions again become similar, which results in the plots of the Axial, River R, and River L being parallel to each other.
 Initiation and termination of sediment bypass to the submarine fan were identified using sequential photo images, while carefully considering the following: (1) sediment accumulation on the deep water basement, (2) whether or not the subaerial delta remained in the same size, and (3) whether or not radial symmetry of the submarine fan was retained. Measured time data related to the initiation and termination of the bypass phase are listed in Table 2. The data are also shown in Figure 4 and agree with the time series trend related to the three stages explained above. The time interval for the bypass phase (Tb) depends on magnitudes of Qs and H in the manner summarized as follows: In the comparison between Runs 4 and 6, as Qs increases from 1.118 cm3/s (Run 4) to 4.238 cm3/s (Run 6), the time periods for the bypass phase (Tb) decreased from 480 s to 240 s. The same trend occurred in the results of Runs 7 and 9, where the bypass phase (Tb) became shorter, from 2700 s to 900 s, as Qs increased from 1.604 cm3/s (Run 8) to 4.238 cm3/s (Run 9). This trend was mainly due to higher Qs values, which facilitated rapid aggradation of the submarine fan and thus rapid connection of the fan apex with the delta toe. The shelf-front depth (H) functioned in the way opposite to sediment discharge. Comparing Runs 2 and 5, which were conducted with the same Qs (1.604 cm3/s) and approximately the same Qw (10.147 − 10.272 cm3/s), bypass periods in Run 2 (H = 1.0 cm) and Run 5 (H = 7.1 cm) were 180 s and 360 s, respectively. Clearly, a high H requires a bigger submarine fan to connect to the feeding delta and thus a longer time of sediment accumulation.
Table 2. Measured and Calculated Times of Beginning and Ending Bypass System and Continental Margin Recovery
Delta Toe at Shelf Edge (s)
Bypass, Beginning (s)
Bypass, Ending (s)
Observed Tb (s)
Calculated Tb (s)
Calculated TmR (s)
3.1 Alluvial Grade Attained in a Fixed-Boundary System
 The concept of grade, originally advocated by Gilbert , is referred to as an equilibrium state of a river that conveys sediment downstream with no net deposition and net erosion of sediment [Schumm, 1997; Muto and Swenson, 2005a]. Since a graded river is a perfect sediment bypass system, it experiences neither aggradation nor degradation. When considering a particular graded section of a river, the influx of sediment across the upstream end of the section is balanced with the outflux of sediment across the downstream end on a relatively long time scale.
 Most sequence stratigraphic models suggest that (1) rivers aggrade during base-level rise and degrade during base-level fall, (2) when base level is stationary, rivers eventually attain grade, and (3) grade therefore represents the final stable state of a river system [e.g., Posamentier and Vail, 1988; Galloway, 1989; Shanley and McCabe, 1994]. The concept of subaerial accommodation is derived from this notion. Recent renewal of the grade concept has brought a totally different understanding, which is that the downstream alluvial reach of a river system can attain and sustain grade only during base-level fall. The necessary pattern to attain and sustain grade depends on geomorphic conditions of the basin and the depositional system [Muto and Swenson, 2005a, 2006; Petter and Muto, 2008]. However, these proposed solutions are based on the assumption that the downstream end of the feeder alluvial river (i.e., shoreline) is a moving boundary [Lorenzo-Trueba et al., 2009] that can migrate basinward in association with the delta's progradation. If the downstream end is a fixed boundary (e.g., a weir), base-level fall may not be a necessary condition for the attainment of grade [Muto and Swenson, 2005b]. Although this fixed boundary hypothesis was tested in the previous experimental study [i.e., Muto and Swenson, 2005b], questions still remain because the previous finding was based on the results of 1-D experiments conducted in a narrow flume.
 In the present experiments, the feeder river on the 3-D delta that prograded in all radial directions attained grade during stationary base level because the shelf-front slope was steeper than that of the delta foreset thereby functioning as a weir of a sort. We did not take the fluvial topographic data to verify no net erosion and deposition in the bypass channels, but the significantly suppressed lateral migration in these channels strongly hinted that they were likely to attain grade. During the time the delta prograded over the shelf basement, the shoreline progradation caused deposition along the river as it tried to maintain an equilibrium slope. Aggradation of the river resulted in lateral migration and/or avulsion in effort to find a new route. However, when the delta toe was situated at the shelf edge and was physically disconnected from the apex of the submarine fan, the feeder river could not either prograde or aggrade and neither lateral migration nor avulsion could occur even within the open 3-D tank. The feeder fluvial system did not experience nodal (or regional) avulsions (in the sense of Leeder ), which originate at the delta apex and generate channels with length scales comparable to that of the entire fluviodeltaic system. However, in natural systems, local avulsions (in the sense of Leeder , Heller and Paola , and Mohrig et al. ) could occur. The experimental sediment mixture lacked cohesion and minimized substantial levee development. Even without major levee development, the mobility of the river was significantly hampered during this bypass stage. This pattern reflects a process that removes sediment from the river mouth, which minimizes shoreline progradation and therefore prevents significant deposition on the feeder/riverbed, facilitating the maintenance of a grade river with minor lateral mobility. This is somewhat analogous to alongshore transport in wave-dominated deltas, which effectively removes sediment from the delta front, suppresses bifurcation, and increases the duration of the interavulsion period [Swenson, 2005].
 The reactivation of channel migration following the connection of the delta toe with the apex of the submarine fan documented in this study provides insight into the change in morphology between the main elongated river and the downstream bird's-foot portion of the Mississippi Delta. This spatial change in morphology formed after the elongated delta prograded to the modern shelf edge [Coleman et al., 1998]. After the Mississippi Delta ceased to advance into the shelf slope to deep water, distributary channels were developed laterally in the along-shelf direction inboard of the shelf edge [Kim et al., 2009]. At the final stage of the current experiments, the feeder river resumed lateral migration, although at a considerably slower rate. This slow transverse building and associated deeper water deposition continued without major avulsions of the feeder system away from the Axial section. Natural compaction and/or relative sea level rise acting on the current Mississippi Delta is significantly different from the experimental shelf-edge deltas presented here. However, the experimental results point toward possible causes for the “flaring” of the downstream portion of the Mississippi Delta inboard of the shelf-slope break when comparing to the previously advancing elongated delta and thus the overall bird's-foot morphology.
3.2 Mathematical Modeling of Channel Reactivation Time
 The termination of the bypass stages in each experimental run coincided with the termination of the radial symmetry of the submarine fan as the fluvial channel started to laterally migrate and laterally move the depocenter. A simple geometrical model that explains the observed bypass durations is developed herein. Consider a radially symmetric submarine fan occurring during the bypass stage. The fan has a cone shape (Figure 1b). As the fan grows, its apex onlaps the vertical shelf-front slope. Assuming no sediment accumulation basinward of the fan toe, the amount of sediment fed by the river during the graded stage balances with the volume of the submarine fan, the shape of which can be approximated with a half cone. The time for the bypass stage, Tb, can be represented as
 The calculated values of Tb using the experimental input parameters and the submarine-fan surface slope Sf based on the measured L are shown in Table 2. The calculated Tb values are in agreement with those measured (Figure 5), although discrepancies in the case of Runs 1, 2, and 3 are noticeable (H for these Runs is shortest in comparison to the others).
 The measured Tb values in Runs 1, 2, and 3 are about 2 orders of magnitude larger than the calculated ones. A possible reason to this discrepancy is that the transition from a nonbypass stage to a bypass stage is not instantaneous but had a time interval of several minutes. This transient state is not considered in the geometrical model. Another possible reason for this discrepancy is that the relatively longer duration in internal channel processes associated with the switch of the channel location (i.e., avulsion time scale) than Tb that is required for the delta to overcome the shelf-edge topography. Reitz et al.  observed that the backfilling of sediment in a channel takes place during a major portion of the interavulsion period in their experiments and suggested that the characteristic avulsion time scale (TA) may be estimated as the filling time of a channel belt with typical depth (h) and width (B) at the supplied sediment discharge (Qs) (also reported in recent studies, e.g., Kim and Jerolmack , Van Dijk et al. , and Powell et al. ). The characteristic avulsion time (TA = shB(Qs)−1) in Run 3, for example, is approximately 112 s using estimated values for a channel that was extended to the shelf edge as h = 0.5 cm, B = 5 cm, s = 50 cm, and Qs = 1.118 cm3/s. TA in Runs 1–3 is much longer than Tb calculated using (1). The measured Tb values in Runs 1, 2, and 3 are most likely reflecting the internal channel dynamics.
 The model has simplified geometries for the shelf-front and submarine-fan surface. Natural shelf-front slopes are much shallower [see O'Grady et al., 2000] than the vertical cliff set up in the experiment. Submarine fans are generally constructed by deep water processes, such as turbidity currents, which operate on lower surface slopes than those developed in this experiment (i.e., sediment avalanche processes). However, the conclusion from the experiment and geometrical model can still be deemed valid if (1) there is any sediment transport process that removes sediment from the feeder river mouth and bypasses it to the deep water and (2) the shelf-front slope is steeper than the angle of the upper submarine-fan surface, which leads to onlapping of the submarine stratigraphy onto the shelf-front slope. Therefore, the absolute high value for the shelf-front slope used in the experiments is not a necessary condition, but the relative differences between the upper submarine-fan surface and shelf-front slopes are the key condition for attaining sediment bypass and maintaining a magnitude of sediment transport to deep water. In fact, the shallower surface slope of natural submarine fans would operate to increase the duration of Tb as the denominator in equation (1) decreases and aid in sustaining a bypass system to produce a larger submarine fan deposited in a focused location.
3.3 Continental Margin Annealing Time Scale
 The current experimental results provide a framework for quantifying the time required for a continental margin to return to an overall (relatively strike independent) mode of progradation following relative sea level rise. Evolution of a progradational continental margin is subject to perturbation in the trinity of geological controls (sea level, tectonics, and sediment supply) [Kim et al., 2006]. This perturbation includes rapid transgression and the development of a ramp-like, sediment starved backstepping surface, parallel to the shelf basement in the current experiment. Prior to the perturbation, the margin filling geometry and evolution are largely two-dimensional, i.e., largely independent of strike. Following this perturbation, the margin begins the process of returning to its unperturbed state; this return is characterized by the two shelf-transit and bypass phases.
 The first phase involves fluviodeltaic filling of space on the shelf. The morphodynamics in this stage is highly three-dimensional depending on the mobility of the coastal-plain rivers and the influence of long-shore drift by waves. This shelf-transit phase in the experiment is in the same line with the study in Burgess and Hovius . We expand the Burgess and Hovius  model further and formalize the time scale for shelf-transit Ts as follows:
where θd denotes the deltaic open angle, s represents the total width of the shelf, and St denotes the delta topset slope. In the experimental system, where St is likely to be large (~0.1) and w is small, fluvial storage is significant (i.e., the second term in the parentheses). In natural systems where the fluvial slopes are much smaller, this term is likely to be negligible.
 The second phase of continental margin recovery includes the sediment bypass duration described in the previous sections. As soon as the feeder channel in the shelf-edge delta encounters the shelf edge at a point, the locus of deposition shifts to the continental slope and deep water basin. During this bypass phase, the margin morphodynamics and geometry associated with the submarine fan continuously show a strong three-dimensional signature. Equation (1) indicates the time scale required for the submarine fan to aggrade sufficiently such that its apex is reconnected to the feeder river. From this point in time onward, the margin has in large part returned to its preperturbation state of overall progradation, and the morphodynamics and overall evolution geometry becomes primarily two-dimensional and thus independent of strike. The total margin recovery time TmR after a high-amplitude and/or high-frequency relative sea level rise can scale with
where θf denotes the opening angle for the submarine fan. Figure 6 shows the comparison between the measured and calculated margin recovery time scales in Runs 1–9. Since there were no topographic measurements, we uniformly applied St = 0.18 to all experiments, which produced the minimum total errors. Although there was overall good correlation, the significant fluvial storage and its variation in different experiments caused the discrepancies. The errors caused by the exaggerated fluvial storage in the experimental delta would be diminished since natural deltaic slopes are significantly shallower and thus the final slope term in equation (3) is negligible. The continental annealing time provides a useful information about how much time is required for a margin to return to the preperturbed progradation without significant sea level rise events.
3.4 Outcrop Example—Sobrarbe Delta, Spain
 The Eocene Sobrarbe Formation, Ainsa Basin, Spain contains an ancient delta that prograded to the shelf edge and supplied sand-grade sediment to the slope and basin floor with minor-to-no relative changes in sea level. The Sobrarbe system followed a similar sequential evolution to that of the tank experiment, but several important distinctions between this natural example and the experimental results are noted. Below, we discuss similarities and differences in an effort to improve the application of our understanding gained in this experiment to natural cases.
 The Sobrarbe Formation was deposited in the Ainsa Basin, a subbasin of the larger Tremp-Ainsa-Jaca Basin (Figure 7a). The basin developed from a foreland piggyback basin located south of the axial zone of the South Pyrenean Central Thrust [Puigdefabregas et al., 1992; Fernandez et al., 2004]. The Sobrarbe Formation is the youngest marine formation in the basin-fill succession and records the progradation of a linked shelf-slope-basin system across the basin [Dreyer et al., 1999; Moss-Russell, 2009; Pickering and Bayliss, 2009; Silalahi, 2010; Pyles et al., 2010b].
 Figure 7b contains a photopanel of the outcrop of the Sobrarbe system. The shelf edge is located where the dip of the transgressive surface that overlies the deltaic strata abruptly increases from flat to ~2° northward. The strata located landward (south) of the shelf edge contains basinward-dipping foresets composed of fine-grained, structureless sandstone with abundant nummilites (Figures 7b and 8a). These units are interpreted as delta front foresets that were deposited at the location where the distributary channels met the standing body of water. The surfaces record the progradation of the delta across the shelf through time.
 At the shelf edge, three local erosional surfaces are evident (E1, E2, and E3; Figure 8a). The surfaces locally erode into underlying strata and are mantled with medium-grained sandstone with clasts of reworked gastropods (inset photo in Figure 8a). These strata are coarser grained than any other strata on the delta front (inset in Figure 8a). The erosional surfaces can be correlated landward to the delta plain where they are overlain by trough-cross stratified sandstone (e.g., surface E3 in Figure 8a) interpreted to be the coevally deposited river channel. The topsets of the three erosional surface-bounded units are at nearly the same elevation, indicating that sea level did not change measurably during deposition. The erosional surfaces can also be correlated basinward to channels that are located on the proximal part of the slope (Figure 8b). These channels contain lateral accretion packages (LAPs) (in the sense of Abreu et al. ) composed of shale-clast conglomerate and structureless sandstone couplets (inset photos in Figure 8b) and are similar in architecture to an outcrop of a sinuous submarine channel documented by Pyles et al. [2010a, 2012] in the Brushy Canyon Formation. The surfaces can be further correlated 4 km to the middle part of the slope (Figure 8c) where submarine channels are interbedded with gray slope mudstone. Due to thick vegetation and local slumping of younger units, the erosional surfaces cannot be directly correlated to the basin floor, although the larger stratigraphic unit and the flooding surface can. At least two lobe elements are located on the basin floor (e.g., Figure 8d). They are composed of laterally continuous, fine-grained sandstone beds.
 Similarities between the Sobrarbe system and the tank experiment are summarized here: As the delta prograded to the shelf edge, basinward-dipping foresets were deposited across the shelf (Figures 8a and 9a), similar to when the experimental delta prograded to the shelf edge (Figures 1b and 3a). When the delta reached the shelf edge, a bypass surface developed (surface E1, Figures 8 and 9b). The bypass surface correlates basinward (north) to a slope channel and slope mudstone, indicating a punctuated spatial shift in deposition from the delta front to the slope and basin. At this time, the shelf edge was operating as a fixed boundary and the slope channel was operating as a weir. The submarine channel contains coarse-grained lateral accretion deposits, indicating sustained bypass [Pyles et al., 2012] because the grain size of the LAPs and of the strata overlying the erosional surface on the delta front are interpreted to reflect coarse-grained sediment that was directly piped from the coeval river channel. This river is comparable to the graded feeder river in the experiment shown in Figures 3c and 3d. A submarine lobe composed of fine-grained sandstone was deposited on the basin floor (Figures 8d and 9b) over this stage, analogous to those deposited in the experimental deep water basement (Figures 1b and 3c). After the slope channel filled, deposition resumed on the shelf edge, although mudstone was deposited basinward of the delta front allowing the delta to prograde farther (Figures 8 and 9), similar to the final stage shown in Figure 3e.
 It is evident that the Sobrarbe system evolved similarly to the tank experiment, yet there are differences that we associate with natural variability. First, in the Sobrarbe system, the angle of the delta front foresets (e.g., Figure 8a) reaches a value of up to 15° in the upper portions, whereas the angle of the shelf-front slope in the Sobrarbe is ~2° (compare Figures 1 and 9). The Sobrarbe does not meet the slope conditions of the experiment conducted herein. However, the slope channels developed in the Sobrarbe system work as a weir/steep shelf-edge slope, similar to the set up in the experiment, by removing sediment from the delta and directly connecting the sediment source to the deep water depositional system across the shelf edge.
 Second, uniformly coarse grains in the experimental sediment mixture entirely bypass the shelf-edge slope to the submarine fan, which causes no topographic changes or aggradation of the slope between the delta toe and the submarine-fan apex. Contrastingly, the Sobrarbe system has a wide grain-size distribution (clay to coarse-grained sand). As such, mudstone is deposited at all positions basinward of the shelf edge, allowing the slope to aggrade as submarine fans are constructed (Figures 8 and 9). In this natural system, Tb would be smaller compared to that estimated using equation (1) due to the mudstone deposit that fills the space between the submarine fan and shelf-edge delta.
 Third, once the submarine channels backfilled to the delta, the delta built out a few more hundreds of meters until a new bypass surface developed (Figures 8 and 9c). As a result, the sequential pattern described above cycles 3 times. These stratigraphic cycles are interpreted to result from a threshold related to a maximum relief that the delta front can attain—below which deposition on the delta front occurs and above which bypass occurs. The filling of submarine channels and associated deposition of slope mudstone reduced the amount of relief on the delta front thereby allowing it to prograde basinward until the relief threshold was met once again. These cycles could also simply result from a three-dimensional effect associated with natural lobe switching, which would be analogous to the last stage of the experiments conducted in this study (after the submarine-fan apex became connected to the toe of the delta; Figure 3d). In this scenario, the new feeder channel shifted from the previous location and delivered sediment to the deep water basement, gradually covering the initial submarine fan from a different angle. This could result in the cyclic pattern with a reactivation surface in the 2-D section shown in Figure 8.
 Although all natural variability and transport mechanisms cannot be explained using a few physical experiments, physical experiments produce spatial structures and kinematics that are justifiably similar to those in the natural systems [Paola et al., 2009]. The experiments described here are not a perfect reduced-scale model for the Sobrarbe system, but the simplified delta, shelf geometries, and sediment transport mechanism work as an analogous model that provides insight into the evolution of the Sobrarbe system and can be applied to other linked shelf-slope-basin systems.
 Sediment can be transported to deep water basins across the shelf edge without aid from base-level fall. However, if the feeder alluvial river is forced to become graded, i.e., a perfect sediment bypass system, a large volume of sediment can be focused into a narrow area of the deep water basin. The alluvial grade can be attained with a stationary base level when a prograding delta reaches a shelf edge when the basinward slope is steeper than that of the delta front such that the delta toe cannot downlap over the slope. During the grade stage, streams on the delta plain are confined to a single major channel and entirely feed the supplied sediment directly onto a submarine fan. As long as this graded state is sustained, the submarine fan continuously grows without change in the sediment feeder location.
 The bypass stage is terminated when the aggrading fan apex connected with the delta foreset. At that time, the feeder alluvial river (now distributary channels) resumes avulsing and migrating laterally. The period for the sustained bypass is longer with smaller magnitudes of sediment supply and with larger magnitudes of shelf-front depth.
 The measured time durations for the submarine fan development by a forcibly sustained sediment bypass are consistent with predictions made by a simple geometrical model, except in the case of short shelf-frontal depth. Channels laterally migrate and/or avulse into new locations, following a characteristic internal time scale. If this internal time scale is longer than the time associated with the topographic forcing, the model cannot capture Tb correctly. Overall consistency between the experimental observations and predictions confirms the forced sediment bypass stage as a potential source for transporting sediment to deep water in large shelf systems over long periods of time.
 This study provides a framework for quantifying the time required for a continental margin to return to an overall mode of progradation following relative sea level rise. This continental margin annealing time is composed of the shelf-transit phase and the shelf-edge bypass phase, which is mainly correlated with the shelf-edge height, shelf width, and shelf depth, and inversely related to sediment supply rate.
 Although some conditions for the physical experiment and Sobrarbe system are mismatched, the reduced-scale tank experiment adequately captured the sequential evolution of the shelf-edge delta in the Eocene Sobrarbe Formation. The shelf geometry in the Sobrarbe system is not consistent with the experiment, but the slope channels effectively bypassed sediment from the fluvial feeder river to the deep water fan. Natural variations in sediment transport, short-term basinal forcing, and grain size should be considered in order to meaningfully apply the understanding from physical experiments to natural cases.
 This study was supported by a Jackson School of Geosciences at the University of Texas at Austin research grant and a postdoctoral fellowship in Japan Society for the Promotion of Science (JSPS) to W.K. T.M. appreciates financial support by a 2008–2010 Japanese grant-in-aid for Scientific Research (JSPS-B2034140). Funding for fieldwork in the Sobrarbe Formation was provided to D.P. by Chevron. We would like to thank Andrew Ashton, George Postma, Elizabeth Hajek, Meredith Reitz, Chris Paola, Carolina Baumanis, and an anonymous reviewer for their constructive comments on the paper.