Flow depth estimates and avulsion behaviour in alluvial stratigraphy (Willwood Formation, Bighorn Basin, Wyoming, USA)

The size and geometry of river channels play a central role in sediment transport and the character of deposition within alluvial basins across spatiotemporal scales spanning the initiation of grain movement to the filling of accommodation generated by subsidence. This study compares several different approaches to estimating palaeoflow depths from fluvial deposits in the early Palaeogene Willwood Formation of north‐west Wyoming, USA. Fluvial story heights (n = 60) and mud plug thicknesses (n = 13) are statistically indistinguishable from one another and yield palaeoflow depth estimates of 4 to 6 m. The vertical relief on bar clinoforms (n = 112) yields smaller flow depths, by a factor of ca 0.3, with the exception that the largest bar clinoforms match story heights and mud plug estimates. This observation is consistent with modern river data sets that indicate unit bar clinoforms do not capture the reach‐mean bank‐full flow depths except in rare circumstances. Future studies should use story heights (i.e. compound bar deposits) and mud plugs to estimate bank‐full flow depths in alluvial strata. Additionally, the thickness of multi‐storied fluvial sandbodies (n = 102) and overbank cycles composed of paired crevasse splay and palaeosol deposits (n = 45) were compared. The two depositional units display statistically indistinguishable mean and median values. Building upon previous depositional models, these observations suggest basin rivers aggraded approximately one flow depth prior to major avulsion. This avulsion process generated widespread crevasse splay deposition across the floodplain. Once the main river channel stem was reestablished, overbank flooding and palaeosol development dominated floodplain settings. The depositional model implies river aggradation autogenically generated topography in the basin that was effectively filled during the subsequent avulsion. This constitutes a meso‐timescale (103–104 years) compensational pattern driven by morphodynamics that may account for the high completeness of fossil and palaeoclimate records recovered from the basin.


| INTRODUCTION
Stratigraphy is produced by geomorphological processes operating on timescales from seconds to millions of years preserved, long-term, by accommodation formed in basins (Ager, 1981;Holbrook & Miall, 2020;Paola et al., 2018;Sadler, 1981).Hence, stratigraphy allows reconstructions of palaeoenvironmental conditions across geological time as well as the geomorphological responses to changes in tectonic, eustatic and climate boundary conditions in a diversity of marine and non-marine settings.For example, the cross-set thicknesses preserved from aeolian dune migration in ancient sand seas record palaeo-wind direction, sediment flux, sea level and internally driven (i.e.autogenic) morphodynamic behaviours among other factors (Cardenas et al., 2019;Kocurek, 1981;Swanson et al., 2016Swanson et al., , 2019)).In the marine realm, Allen and Hoffman (2005) estimated palaeo-wind speeds and wave periods during Neoproterozoic deglaciation from the morphology and grain size of giant wave ripples.Estimates on ocean depth, storm hydrodynamics and the relative strength of fluvial, tidal and wave processes are also commonly recovered from marine and marginal marine strata (Dalrymple & Choi, 2007;Immenhauser, 2009;Myrow & Southard, 1996).
Similarly, fluvial strata contain a host of sedimentological features useful for quantitative and semi-quantitative hydrodynamic reconstructions.The prevalence of upper flow regime lithofacies and the morphology of dune crossstratification within channel deposits have been used to characterise the peakedness of river hydrographs and bedform kinematics (Bridge, 1997;Fielding et al., 2018;Jerolmack & Mohrig, 2005;Lyster et al., 2022).Channel depth, width, kinematics and sinuosity can be obtained in some cases from the morphology and geometry of accreting dune cross-beds and barforms (Greenberg et al., 2021;Leeder, 1973;Mohrig et al., 2000;Moody-Stuart, 1966;Paola & Borgman, 1991;Willis, 1993).On larger scales, channelstacking patterns, when coupled with the preservation of bedforms and barforms, yield information on channel belt characteristics, the propensity for aggradation/incision in a system, and channel mobility (Chamberlin & Hajek, 2019;Foreman, 2014;Gibling, 2006;Hajek et al., 2010;Moody-Stuart, 1966).Palaeo-depositional slopes of a basin's sediment routing system can be constrained from grain size and flow depth measurements, which in turn may identify tectonic tilting and subsidence events as well as enable estimates of sediment flux and water discharge (Bhattacharya et al., 2016;Duller et al., 2010Duller et al., , 2012;;Paola & Mohrig, 1996;Trampush et al., 2014).The topography of palaeo-landscapes and related thresholds for channel movement via avulsion have also been recovered from the internal characteristics of fluvial sandbodies and their relationship to adjacent floodplain strata (Barefoot et al., 2021;Hajek & Heller, 2012;Mohrig et al., 2000;Paola & Borgman, 1991).Unifying nearly all these reconstruction methods is the need for an accurate estimate of bank-full discharge in the palaeo-river.
River discharges and flow depths within modern systems vary spatiotemporally with catchment size, tributaries, planform morphology (e.g.braided versus meandering), and the magnitude-frequency of rainfall along with several additional factors (Knighton, 1998;Robert, 2003).Yet the most pertinent flow depth from a longer time-scale fluvial dynamics perspective is that which corresponds to bank-full discharge (Knighton, 1998;Robert, 2003).Somewhat informally, bank-full discharge has been called the interval during which the river performs the most 'geomorphic work' (Wolman & Miller, 1960).Thus, process-oriented interpretations, reconstructions and predictive models for non-marine basin-filling necessitate accurate constraints on bank-full palaeoflow depths as they will be dominantly responsible for the construction of alluvial stratigraphy.
Fluvial sandbodies are a time-integrated account of the channel belt characteristics and are composed of surfaces and structures documenting channel movement by a variety of mechanisms (Gibling, 2006;Miall, 2006).In light of this, three approaches to estimating bank-full flow depth are now in widespread use.The first exploits the scaling relationship between dune heights and river flow depths observed in modern systems (generally 3-20; Allen, 1984;Bridge & Tye, 2000;Paola & Borgman, 1991;Yalin, 1964).This approach requires detailed quantification of preserved bedform heights in ancient strata from cross-bed sets and some assumptions related to the statistical distribution of bedform crest heights, which will invariably be incompletely preserved in the ancient (Bridge & Tye, 2000;Leclair & Bridge, 2001;Paola & Borgman, 1991).While useful for many hydrodynamic-related questions in palaeoenvironmental reconstructions, this approach may be logistically difficult when seeking to understand the evolution of basin filling on timescales of millions of years.Moreover, the preservation quality of dune crossbed sets will not always be sufficient for the robust application of the methodology.In lieu of this approach, other researchers have identified the vertical relief on bar clinoforms as a second potential avenue to extract palaeoflow depths.This proxy follows from the observation that rivers are filled with bar macroforms (e.g.free, fixed, point, midchannel, transverse bars) that generally aggrade vertically to just below the bank-full level (Allen, 1984;Bridge & Tye, 2000;Knighton, 1998;Mohrig et al., 2000;Robert, 2003).Yet these too are subject to reworking during lower-flow stages as well as partial preservation due to erosion during channel movement.Third, some researchers have used channel lithofacies associations and diagnostic surfaces within the sandbodies (i.e.story breaks) to estimate palaeoflow depths (Bridge & Tye, 2000;Miall, 2006).Upsection patterns in grain size and sedimentary structures, although time-integrated, are linked to the hydrodynamics of the channel and can be used to identify a characteristic channel-fill deposit, equivalent to a sandbody story fourth-order surface of Miall (2006).Some channel fills represent compound deposition on bars from bed load as the bar grows and migrates, others represent settling from finer-grained, suspended load within abandoned channels due to a variety of processes such as neck cutoff or avulsion.Again, this proxy may underestimate bank-full flow depth due to incomplete preservation and differential compaction during burial, although it does not necessitate the preservation of a complete bar clinoform toeset, foreset and top-set.Given the importance of bank-full flow depth for sediment transport, geomorphological processes and the temporal completeness of non-marine stratigraphy (Straub et al., 2020), evaluating the coherency of estimates among these proxies is important.
Constraints on bank-full flow depths are particularly critical for understanding the role of avulsion in filling non-marine basins.Many field and experimental basin studies indicate avulsion is the dominant mechanism for distributing sediment across non-marine basins (Bryant et al., 1995;Hajek et al., 2010;Slingerland & Smith, 2004).Avulsion is central to recent distributary fluvial system models (Weissmann et al., 2013), and it is invoked to explain sandstone channel-stacking patterns in traditional 'LAB' models of alluvial stratigraphy (Allen, 1978;Bridge & Leeder, 1979;Leeder, 1978).The geomorphological process of avulsion is commonly described as necessitating two conditions, a setup and a trigger (Mohrig et al., 2000;Slingerland & Smith, 2004).The setup occurs as differential aggradation between the river channel and floodplain causes a critical topographic threshold to be met.This threshold may be related to levee slope, floodplain topographic gradients and/or superelevation of channel-free surface above the floodplain creating relative potential energy (Bryant et al., 1995;Heller & Paola, 1996;Mackey & Bridge, 1995;Slingerland & Smith, 1998, 2004).Once the setup condition is met, a trigger is needed in the form of a large flood, earthquake, log jam or any other event that induces channel margin instability and erosion allowing the diversion of river discharge and sediment to the floodplain environment (Slingerland & Smith, 2004).Avulsions then, in a sense, fill topographic accommodation the river created autogenically.Bank-full flow depths, and associated intervals of geomorphological work, are directly responsible for both generating the pertinent topographic conditions of avulsion setup and represent the most logical scaling parameter allowing comparisons among field and experimental systems (Mohrig et al., 2000;Straub et al., 2009Straub et al., , 2020)).Moreover, constraints on bank-full flow depths are also important for understanding the long timescale behaviour of avulsions, as previous studies have suggested tectonic subsidence and base level may 'steer' avulsions to preferential depozones formed by the differential generation of accommodation (Kim et al., 2010;Peakall, 1998).

| GEOLOGICAL SETTING
The Willwood Formation is exposed in badlands topography across much of the Bighorn Basin in north-west Wyoming, USA (Figure 1A).The formational contact with the underlying Fort Union Formation is defined by the first laterally persistent red bed (Bown, 1980;Gingerich, 2001).In the southern basin, this lithological transition corresponds with the Palaeocene-Eocene boundary (Wing et al., 2005), whereas in the northern portion of the basin, studied herein, the contact precedes the Palaeocene-Eocene boundary by ca 1 Myr (Bowen et al., 2001;Gingerich, 2001;Secord et al., 2006).The formational contact is conformable in the central portions of the basin, however, progressive unconformities are relatively common along basin margins both within the Fort Union and Willwood formations and between them (Bown, 1979(Bown, , 1980;;Bown et al., 2016).The Willwood Formation is conformably overlain by lacustrine strata of the Tatman Formation (Rohrer & Smith, 1969;Sinclair & Granger, 1912), although across much of the basin the latter is missing, likely due to Miocene erosion (McMillan et al., 2006;Roberts et al., 2008).The thickness of the Willwood Formation varies spatially across the basin due to an asymmetric pattern of tectonic subsidence, and there are no single outcrop areas where the full thickness is continuously exposed.However, in the northern portion of the basin Willwood Formation outcrops in the Sand Coulee and McCullough Peaks study areas (Figure 1A) overlap stratigraphically with the lower ca 1,100 m and upper ca 1,500 m of the formation exposed, respectively (Clyde, 2001).The composite total thickness of the Willwood Formation between these two areas is estimated to be ca 1,900 m.
Chronostratigraphic constraints for the Willwood Formation include radiometric dates from altered volcanic ash beds (i.e.bentonite beds), magnetostratigraphic records, biostratigraphic zones and stable isotope chemostratigraphy.Ash beds are exceedingly rare within the main body of the Willwood Formation, but Secord et al. (2006) dated sanidines to ca 59.0 Ma from a bentonite bed in the Fort Union Formation ca 820 m below the formational contact with the overlying Willwood Formation in the northern basin.Whereas Wing et al. (1991), dated sanidines to ca 52.8 Ma from a bentonite bed in the uppermost Willwood Formation ca 50 m below its contact with the overlying Tatman Formation in the southern basin.Within the formation itself an extensive magnetostratigraphic framework is integrated with a densely sampled vertebrate fossil record (Clyde, 2001;Clyde et al., 1994Clyde et al., , 2007;;Gingerich & Clyde, 2001;Secord et al., 2006) and palynologic records (Harrington, 2001;Wing & Currano, 2013;Wing & Harrington, 2001).Of importance here is the subdivision of the Willwood Formation into two landmammal ages, the Clarkforkian (Cf) and Wasatchian (Wa), the contact of which correlates with the Palaeocene-Eocene boundary (Bowen et al., 2001;Gingerich, 2001).These two land-mammal ages are further subdivided into mammal biozones defined by the first and last appearances of taxa.The Willwood Formation in the northern Bighorn Basin was deposited from Clarkforkian-2 (Cf-2) through Wasatchian-7 (Wa-7) (Figure 1B; Clyde, 2001;Gingerich, 2001; Figure 1).Interpolating among these chronostratigraphic constraints, deposition of the Willwood Formation in the northern Bighorn Basin initiated ca 57 Ma in the northern basin and ended just after ca 52.8 Ma.This yields a long-term rock accumulation rate (i.e.compacted sediment) for the entire formation of 452 m/Myr or, equivalently, 45.2 cm/kyr.Within the Willwood Formation rock accumulation rates of 28.8 cm/ kyr, 31.6 cm/kyr, 39.1 cm/kyr and 40.0 cm/kyr have been reported (Abels et al., 2013;Clyde et al., 1994Clyde et al., , 2007)).
The chronostratigraphic constraints within the Willwood Formation have facilitated the development of the highest-resolved, non-marine stable isotopic palaeoclimate records for early Palaeogene strata globally.Most notably, the carbon isotope excursion defining the Palaeocene-Eocene thermal maximum (PETM) at ca 56 Ma is constrained at several locations in the basin (Baczynski et al., 2013;Bowen et al., 2001;Foreman, 2014;Koch et al., 1992;Wing et al., 2005).This hyperthermal event represents the massive release of isotopically light carbon into Earth's oceans and atmosphere and correlates with persistently elevated temperatures (5-8°C) both globally and in the Bighorn Basin for ca 200 kyr (Koch et al., 1992;Wing et al., 2005;Zachos et al., 2001).Moreover, the Bighorn Basin contains the premier nonmarine record of palaeofloral and palaeofaunal changes associated with this abrupt global warming event (Clyde & Gingerich, 1998;Gingerich, 2006;Wing et al., 2005).Recently, several subsequent hyperthermal events later in the early Eocene have been identified including the paired ETM2(H1)/H2 and I1/I2 events (Abels et al., 2012;Westerhold et al., 2018;Widlansky et al., 2022).These smaller, later perturbations of the global carbon cycle also appear to have impacted palaeobiological systems, although lesser in magnitude, and their tempo was influenced by orbital forcings (Cramer et al., 2003;D'Ambrosia et al., 2017;Westerhold et al., 2018;Zeebe et al., 2017).Some hypothesise that even the background Eocene climate state itself was modulated by orbital forcings F I G U R E 1 (A) Map of study areas in the Bighorn Basin, Wyoming, USA (base map from GeoMapApp, www.geoma papp.org).(B) Biostratigraphic framework for the Willwood Formation and uppermost Fort Union Formation.Diagnostic taxa defining the biozones can be found in Clyde (2001), Gingerich (2001), Gingerich and Clyde (2001) and Secord et al. (2006).(C) Representative outcrop photograph of the Willwood Formation (looking northnorthwest in Sand Coulee study area) with three major lithofacies associations identified by Kraus (2001).
sufficient to induce cycles in overbank soil development and river avulsion frequency within the Bighorn Basin (Abels et al., 2013;Aziz et al., 2008;van der Meulen et al., 2020).
Yet these three lithofacies associations only capture the broadest depositional elements within the Willwood Formation.Several studies have noted spatiotemporal variability in the characteristics of fluvial sandbody deposition within the basin.For example, Owen et al. (2017Owen et al. ( , 2019) ) classified fluvial sandbodies into five subdivisions and quantified their spatial distribution across the basin.The south-western portion of the basin contains distinct, thick (>20 m) conglomeratic fluvial units (Kraus, 1985;Owen et al., 2017) whereas the southeastern and eastern portions contain isolated, thinner (ca 4 m) fluvial units and cut-and-fill channel deposits (Owen et al., 2017;Wing et al., 2005).In the central and northern basin, a preponderance of multi-storied, sand-dominated fluvial sandbodies crop out (internally amalgamated and semi-amalgamated sandbodies sensu Owen et al., 2017).This spatial variation in fluvial deposition is attributed to a system of transverse river systems draining the adjacent Laramide uplifts that fed an axial river system draining northward into southern Montana (Foreman, 2014;Kraus & Middleton, 1987;Neasham & Vondra, 1972;Owen et al., 2017Owen et al., , 2019;;Seeland, 1998;Welch et al., 2022).Recently, Welch et al. (2022) provided an extensive provenance analysis for the Willwood Formation and suggested that sediment source played a strong role in the observed fluvial characteristics.Specifically, larger conglomerate-dominated fluvial systems of the south-western basin drained quartzite and crystalline basement sources in westernmost Wyoming, Idaho, and south-western Montana whereas smaller, isolated rivers on the eastern side of the basin drained catchments in the Bighorn Mountains with exposures of Mesozoic siliciclastic strata (Welch et al., 2022).The axial and northern rivers represented a mixture of all sources as well as input from the Beartooth Mountains (Welch et al., 2022).The most significant temporal change in fluvial deposition occurs during the PETM where an anomalously thick and laterally persistent multi-story sandbody crops out in the northern Bighorn Basin (Foreman, 2014;Kraus, 1980).The internal lithological characteristics of this sandbody are similar to others in the northern basin leading Foreman (2014) and Kraus et al. (2015) to hypothesise a variety of forcing mechanisms that altered channel mobility and avulsion behaviour.Greenberg (2017) recently provided data that suggest the major shift in fluvial deposition during the PETM is an increase in avulsion reoccupation events.
Heterolithic crevasse splay units and mudrock overbank units are linked genetically to one another and, in many cases, are stratigraphically organised into 'overbank cycles' (Abels et al., 2013;Aziz et al., 2008;Figure 3A).These cycles appear linked to the avulsion behaviours of the major river systems in the basin.Widespread deposition of heterolithic crevasse splay units document channel breaching and advection of fine sands and silts to the floodplain in unconfined flows (Kraus, 1996(Kraus, , 2001)).Subsequently, scour into underlying splay deposits and channelisation may occur and form small, lenticular, 'ribbon' sandstones, which contain palaeocurrent indicators of transport oblique to perpendicular from larger, multi-story sandbodies (Kraus, 1996(Kraus, , 2001;;Kraus & Davies-Vollum, 2004).If an avulsion is successful, ribbon channels divert an increasing amount of sediment and water from the main channel stem causing the river to switch course (Kraus, 1996(Kraus, , 2001)).Once this occurs, floodplain deposition reverts to being dominated by overbank floods and soil development (Abels et al., 2013;Kraus, 1996Kraus, , 2001)).Across the basin, variations in overbank lithofacies patterns have been attributed to a combination of palaeo-topographic features on the floodplain, subsidence rates, local palaeoenvironmental conditions and overarching palaeoclimate (Kraus, 1992;Kraus et al., 2013Kraus et al., , 2015;;Kraus & Aslan, 1999;Kraus & Gwinn, 1997).Generally, the uppermost portion of the overbank cycle is characterised by well-drained, red bed palaeosol development with thicker horizonation (Abels et al., 2013;Aziz et al., 2008;Kraus, 1996).However, a similar cycle pattern is observed in portions of the Willwood Formation characterised by more drab, hydromorphic palaeosols (Davies-Vollum & Kraus, 2001).

| METHODS
Fluvial stratigraphic data were obtained from two study areas, Sand Coulee and McCullough Peaks (Figure 1A).Data come from all the major biozones identified within the Willwood Formation (Figure 1B).Bar clinoform relief, story height, mud plug thickness and sandbody thickness were derived from multi-storied fluvial sandbodies that are interpreted as the main conduits of sediment and water in the basin by previous studies (Kraus, 2001;Kraus & Gwinn, 1997;Kraus & Middleton, 1987;Owen et al., 2017Owen et al., , 2019)).In the classification developed by Kraus and colleagues these multi-storied sandbodies have been termed 'trunk' or 'axial' sandbodies (Kraus, 2001;Kraus & Gwinn, 1997;Kraus & Middleton, 1987).In the more recent classification system developed by Owen et al. (2017Owen et al. ( , 2019) ) these multi-storied sandbodies in the northern basin are termed 'internally amalgamated' and 'semi-amalgamated'.Most important to this study is that these fluvial sandbodies are the primary means of sediment transport and distribution, and they are distinct from single-story ribbon sandbodies (sensu Kraus, 2001 or floodplain ribbon channels sensu Owen et al., 2017) that are commonly associated with heterolithic intervals linked to crevasse splay deposition.Herein the term 'splay channels' is used to emphasise the lithofacies association with floodplain crevasse splay deposits.
In the majority of cases, sandbodies could be assigned to a mammal biozone using the framework of Gingerich and Clyde (2001) (Figure 1B).Measurements of sandbody thicknesses, story heights and bar clinoform relief were collected using Jacob's staff and laser range finder and represent both new data and compiled data from Foreman (2014).The accuracy of the laser range finder is ±0.1 m; confirmed by paired Jacob's staff and range finder measurements.Reported thickness values from laser range finder measurements represent the average of three independent measurements.Jacob's staff measurements were obtained by measuring bed-by-bed thicknesses through the thickest portions of the sandbodies and overbank cycles present along an outcrop.Descriptions included grain size, sorting, sedimentary structures, pedogenic features and the presence of trace/body fossils.In locations where outcrops were physically inaccessible, the laser range finder was used and bed-by-bed descriptions not taken.
Data are presented in both compacted form (i.e.raw field measurements) and decompacted form based on estimated burial depth.For each fluvial measurement, the estimated burial depth was obtained from the total thickness of sediment accumulated above the top of the Willwood Formation plus the thickness between the stratigraphic midpoint of the mammal biozone(s) (obtained from table 3 in Clyde, 2001) in which the sandbody is found and the top of the Willwood Formation.Sandbody measurements near the contact between biozones are assigned a stratigraphic position of that biozone contact.Late Miocene erosion removed the vast majority of post-Willwood Formation strata from within the Bighorn Basin, however, previous studies have estimated the Willwood Formation was buried to a depth of 1,200 to 1,500 m (McMillan et al., 2006;Roberts et al., 2008).The midpoint of these post-Willwood Formation estimates, 1,350 m, was used for the decompaction procedure.The decompaction equation of Sheldon and Retallack (2001) was used for non-marine sediments where burial compaction (C, the fraction of original thickness) is estimated by, where, S i is the initial solidity and is found by the quotient of the average dry bulk density and solid grain bulk density, F 0 the initial porosity as a fraction, D the burial depth (km) and k an empirically derived constant.For sandbodies, 0.692, 0.308 and 0.12 were used for S i , F 0 and k, respectively, based on modern measurements of alluvial sediment (Nadon & Issler, 1997;Sheldon & Retallack, 2001).For floodplain mudrock strata, 0.635, 0.365 and 0.16 were used for S i , F 0 and k, respectively, based on modern environment measurements (Nadon & Issler, 1997;Sheldon & Retallack, 2001).Individual thickness measurements, stratigraphic position, compaction estimates and calculations are provided in the Supporting Information.It should be recognised that the decompaction procedure only yields approximate estimates of the true, uncompacted deposits since precise burial depths are unknown.The major goal in the decompaction procedure was to reduce the systematic error introduced by differential compaction of sandstone units and mudrock units.Thus, while the decompacted values are still imprecise, an underlying, systematic bias is substantially reduced.In general, decompacted sandstone units are typically ca 10% thicker and mudrock units ca 20% thicker than raw field measurements.
Bar clinoforms are recognised by elongate, sigmoidal surfaces that commonly display ripple cross-laminations and dune cross-bedding between the surfaces and in some cases display mud drapes (Figure 2A,B).Top-sets and toesets of the bar clinoforms are typically very low-angle (<3°) to flat, commonly displaying roll-over and toe-out, respectively.Foresets are more variable in dip (5°-15°).Kraus (1980) grouped these units as low-angle stratified sandstone lithofacies.Grain size tends to be medium to fine sand, with coarser grains more common within the toe sets.Isolated pedogenic carbonate nodules and siltstone rip-up clasts are not uncommon along foresets and toe sets.Palaeocurrents derived from dune cross-beds are typically perpendicular (±20°) to the dip direction of the bar clinoform foreset, although other orientations are common suggesting both upstream, lateral and downstream bar accretion mechanisms (Foreman, 2014).Bar clinoforms can be traced laterally for upwards of 25 m in some cases, preserving a minimum bar migration distance.The bar clinoforms are equivalent to the third-order surfaces of Miall (2006) and are interpreted as unit bars.Mean compacted and decompacted bar clinoform thicknesses (n = 112) are 1.4 ± 0.8 m (±1σ) and 1.5 ± 0.9 m, respectively.Median compacted and decompacted bar clinoforms are 1.2 m and 1.3 m, respectively.
Fluvial stories are recognised by fining upward sandstone lithofacies that display sedimentary structures of weakening/shallowing flows, and are found above lenticular-shaped, broadly convex surfaces (Figure 2C).These underlying convex surfaces are fourth-order surfaces of Miall (2006).The sandstone lithofacies that compose a story are variable, but broadly can be placed into two types (Foreman, 2014).One type is composed of medium-grained to fine-grained sandstone beds, that fine up-section.The basal beds also tend to display sets of large-scale trough cross-bedding, occasionally with scattered siltstone rip-up clasts and pedogenic carbonate nodules.Middle portions display multiple sets of smallscale trough cross-beds and uppermost beds can display ripple cross-laminations, although convolute bedding is very common.In some cases, the cross-bedded units are associated with low-angle stratified facies and minor massive sandstone units.The other type of fluvial story is composed near exclusively of low-angle stratified facies with variable dip and dip directions.These lithofacies are separated from one another by sharp bed contacts oriented at oblique angles, with more steeply dipping beds near the middle portions of the story.These stories are filled with lithofacies similar to the toesets, foresets and top-sets of the bar clinoforms described above, but a continuous, sigmoidal surface is absent due to reactivation surfaces.Both of these story types are interpreted as compound bar deposits that filled the palaeo-channel.In some instances, stories were observed to be filled with abundant massive or convolute beddings.These were not included in the thickness analysis due to an inability to recover primary depositional features.Mean compacted and decompacted story heights (n = 60) are 4.1 ± 1.6 m (±1σ) and 4.7 ± 1.8 m, respectively.Median compacted and decompacted story heights are 3.7 m and 4.2 m, respectively.
Mud plugs are recognised as lenticular units of finergrained lithofacies typically nested within the uppermost parts of fluvial sandbodies although they can be found hosted within floodplain strata as isolated cut-and-fill channels (Figure 2D,E).These are the Type 3 channel fills of Foreman (2014), the mudstone lithofacies group of Kraus (1980), and the Type 1 carbonaceous deposits of Wing (1984).The basal contact is typically sharp and broadly convex.The lithofacies that compose mud plugs are typically a series of interbedded fine-grained sandstones, siltstones and claystones, whose thickness and grain-size fine upwards.The dip of the layers tends to match the geometry of the underlying convex contact, and sandstone layers are usually thickest near the lower, central portions of the mud plug.In some instances, basal sandstone layers are coarse-grained and contain siltstone rip-ups and pedogenic carbonate nodules.While sandstone layers can display weak planar laminations in some cases, most are massive.Siltstones and claystones are typically massive, especially towards the top of the mud plug unit, although lithofacies margins tend to display fine laminations.Organic material can be abundant within the claystones, although quality varies from unidentifiable organic debris to very well-preserved leaf macrofossils (Figure 2F).Mean compacted and decompacted mud plugs thicknesses (n = 13) are 4.5 ± 1.8 m (±1σ) and 5.5 ± 2.2 m, respectively.Median compacted and decompacted mud plugs are 4.8 m and 5.8 m, respectively.Widths of mud plug lithofacies associations can be up to 50 m.More commonly they are ca 20 m wide, although in many cases it is difficult to determine true width perpendicular to flow direction due to the lack of palaeocurrent indicators within the unit.
Splay channel deposits are lithofacies associations hosted within laterally persistent heterolithic overbank facies (see description further below) that display a sharp, convex lower contact and contain a variety of sandstone lithofacies (Figure 3A,B).These units have been called ribbon sandstones by Kraus (2001) and floodplain channels by Owen et al. (2017).The ribbon descriptor is related to the width-to-thickness ratio of less than 10 that the units display.Splay channel deposits are composed of medium-grained, but more commonly fine-grained, sandstone lithofacies that slightly fine upwards into silty sandstones and siltstones.The lithofacies display small-scale planar cross-bedding, small-scale trough cross-bedding, and occasionally ripple cross-lamination near the top.Several instances of massive, bioturbated or extensively soft sediment deformed examples can also be found.Mean compacted and decompacted splay channel thicknesses (n = 50) are 3.1 ± 1.3 m (±1σ) and 3.5 ± 1.4 m, respectively.Median compacted and decompacted splay channel thicknesses are 2.9 m and 3.3 m, respectively.
Multi-story sandbodies are amalgamated units of channel fill sandstone lithofacies.These sandbodies display sheet geometries with width-to-thickness ratios above 100, and, in many cases, can be traced laterally for several hundreds of metres with many exceeding 1 km in width (Foreman, 2014;Kraus, 1980Kraus, , 2001;;Owen et al., 2017Owen et al., , 2019)).The degree of amalgamation varies across the basin, likely related to palaeodrainage and catchment conditions (Owen et al., 2017(Owen et al., , 2019;;Welch et al., 2022), with some sandbodies containing 'windows' of floodplain strata between stories (Figure 2C) and others not.Three to four vertically stacked stories are common in a given outcrop, although up to six have been observed (Foreman, 2014;Kraus, 2001;Owen et al., 2017Owen et al., , 2019)).The bases of the multi-storied sandbodies are commonly sharp and erosive on top of crevasse splay deposits.The stories themselves are defined by sharp, broadly convex, throughgoing surfaces (fourth order surface of Miall, 2006) and composed of channel fill facies previously described above.Mean compacted and decompacted multi-storied sandbody thicknesses (n = 102) are 9.7 ± 5.1 m (±1σ) and 10.8 ± 5.7 m, respectively.Median compacted and decompacted multi-storied sandbody thicknesses are 8.3 m and 9.3 m, respectively.
Overbank cycles are lithofacies associations of paired heterolithic intervals and mudrocks dominated by palaeosol development.Heterolithic intervals consist of interbedded, tabular fine-grained sandstones, siltstones and claystones.Individual beds can reach over a metre in thickness but are more commonly tens of centimetres thick or less.Contacts between beds can be sharp to gradational.In many cases the three lithologies form fining-upwards patterns.Fine-grained sandstone beds may contain relict bedding in the form of ripple cross-laminations and smallscale trough cross-bedding.However, massive and bioturbated structures are more common.All three lithologies commonly display evidence for incipient or weak pedogenesis.This includes mottling, root traces, mud-filled burrows, clay cutans and pedogenic carbonate nodules.The heterolithic intervals constitute the lower portion of overbank cycles and are interpreted as sequences of crevasse splays (Kraus, 1987(Kraus, , 2001)).The upper portion of overbank cycles is composed of palaeosols.Palaeosol horizons are identified by the prevalence of combinations of mottling, bioturbation, clay cutans, slickensides, rhizoliths, pedogenic nodules (siderite or carbonate) and ped structures.Palaeosols can occur as compound, composite or cumulative varieties, which describe the degree of overlap of B horizons.Well-drained, intermediate-drained and poorly drained palaeosols have been identified and characterised as well as a diverse suite of soil profile types such as alfisols, spodosols and vertisols, among others (Bown & Kraus, 1987;Davis-Vollum, 2001;Kraus, 2001;Kraus et al., 2013;Smith et al., 2008).However, arguably the most useful analysis of palaeosol development to emerge is a semi-quantitative soil development index that ranks the intensity of pedogenesis based on mottling colour and chroma, nodule type and prevalence, B horizon thickness and rubification (Abels et al., 2013;Adams et al., 2011).From this analysis, researchers (Abels et al., 2013;van der Meulen et al., 2020) were able to more rigorously quantify overbank cycles identified by Kraus (1987).From these methods, overbank cycles were demonstrably shown to consist of heterolithic intervals overlain by palaeosoldominated mudrocks that increase in pedogenic alteration up-section.Mean compacted and decompacted overbank cycle thicknesses (n = 45) are 7.7 ± 1.6 m (±1σ) and 9.2 ± 2.0 m, respectively.Median compacted and decompacted overbank cycle thicknesses are 7.5 m and 9.1 m, respectively.
The overall distributions of thickness measurements for these depositional units are displayed in violin plots in Figure 4A (compacted) and Figure 4B (decompacted).
Visually, most bar clinoform thicknesses are smaller than all other depositional units.Story heights and mud plugs display similar distributions with splay channel thicknesses slightly thinner overall.Multi-story sandbodies and overbank cycles are thicker than other depositional units and mostly overlap with one another except for the thickest of multi-storied sandbodies.These measurements were submitted to t-tests for equal means using a Monte Carlo permutation (due to unequal sample size and variance), the Mood test for equal medians, and the Kolmogorov-Smirnov test for one-dimensional probability distributions with Monte Carlo permutations (Table 1).The statistical tests reject the null hypothesis that the means, medians and probability distributions are drawn from the same underlying populations in nearly all cases.The exceptions are pairwise tests of mud plug and story height thickness wherein the null hypothesis cannot be rejected in all cases as well as the tests for the thickness of multi-story sandbodies and overbank cycles.In these cases, it suggests the mud plug and story thicknesses are drawn from the same underlying population and the multi-story sandbodies and overbank cycles are drawn from the same underlying population.The Mood test for equal medians cannot reject the null hypothesis for splay channels and mud plugs thicknesses, but the t-test for means and Kolmogorov-Smirnov test for probability distributions reject the null hypothesis.

| Flow depth estimates
Bank-full flow depth is an important parameter in developing predictive depositional models of alluvial stratigraphy.The models themselves are integral for understanding meso-term and long-term behaviours of earth surface processes as well as constructing accurate hydrocarbon reservoir models.Bank-full flow depth controls bed shear stress within river systems and thereby influences local capacity and competence, downstream fining and sorting rates, and the resultant porosity and permeability characteristics of the sedimentary deposits.When combined with methods to estimate channel width, it may be possible to constrain the planform morphology of palaeorivers (Greenberg et al., 2021;Willis, 1993;Zaleha, 2013).Thus, locally constraining palaeoflow depths within a basin transect may allow the prediction of upstream and downstream channel geometries and sandbody size.Accurate flow depth constraints are also necessary for estimating palaeo-depositional slopes and sediment flux, which themselves aid in source-to-sink models of sediment transport that can be integrated into stratigraphic models that incorporate sediment mass balance (Dade & Friend, 1998;Hampson et al., 2014;Paola & Martin, 2012;Sincavage et al., 2019;Strong et al., 2005).The frequency and magnitude of bank-full flow depth also appear to influence channel kinematics in both experimental and field studies, which have consequences for channel-stacking and connectivity patterns of potential hydrocarbon reservoirs as well as the completeness of the palaeobiological and palaeoclimate record (Esposito et al., 2018;Straub & Foreman, 2018).Additionally, several recent studies have argued a distributive fluvial systems model for sediment transport may be appropriate for most alluvial basins (Hartley et al., 2010;Weissmann et al., 2013).These models make testable predictions regarding systematic changes in channel geometry downstream as the river bifurcates and avulses.Constraining flow depths in a proximal-todistal transect as well as other aspects of fluvial sandbody deposition allows the recognition of such a system (Hartley et al., 2010;Weissmann et al., 2013).Finally, and perhaps most importantly, bank-full flow depths may be the most meaningful and applicable scaling parameter for comparisons of experimental and field-scale basins (Paola et al., 2009;Straub et al., 2020).
The data set presented herein suggests that reliance on bar clinoforms may bias flow depth estimates and, by extension, depositional models that incorporate them.Among the three approaches to recovering bank-full flow depths, bar clinoforms, story heights and mud plugs, two yield statistically indistinguishable means, medians and distributions.Story heights herein represent amalgamated bedforms and bar forms that construct fining upward channel fills (Bridge & Lunt, 2006;Bridge & Tye, 2000;Ethridge & Schumm, 1977;Leeder, 1973;Miall, 2006).Reactivation surfaces are related to bar form movement, amalgamation and fluctuating flow conditions (Bridge & Tye, 2000;Herbert et al., 2020;Miall, 2006) over the course of several years.Thus, they represent repeated events of bank-full flow.Assuming modern rivers as a guide, this would be multiple instantiations of the 1 to 5 year recurring floods (Allen, 1965;Knighton, 1998;Robert, 2003).In contrast, mud plug and channel abandonment facies provide the near instantaneous preservation of channel  (Miall, 2006).While the triggering events for such a process are varied, they likely coincide with periods of bank-full flow (Mohrig et al., 2000;Slingerland & Smith, 2004).Bar clinoforms, however, can form at a variety of discharge levels and locations within a channel (Bridge & Tye, 2000).Unit bars can occur in isolation, adjacent to or on the margins of compound bars in modern river systems (Herbert et al., 2020;Miall, 2006).Their formation tends to correspond with a specific depositional event, although studies also note significant reworking is possible after initial deposition (Herbert et al., 2020).Based on these genetic origins, story heights and mud plugs are interpreted as more accurate estimates of bank-full flow depth in the Willwood Formation.Conservatively, the major rivers in the northern Bighorn Basin at that time displayed bankfull flow depths of 4 to 6 m.This interpretation is bolstered by an independent method from dune cross-bed heights by Wang et al. (2022) who estimated flow depths of 4.3 m in the northern Willwood Formation.Bar clinoforms appear to under predict flow depth by ca 0.3 of other approaches (ca 0.32 of mean story height and ca 0.27 of mean mud plug thickness), and this mirrors observations of modern rivers.For example, Alexander et al. (2020) found unit bars under predicted the reachmean bank-full flow depth by a factor of three in the Missouri River (USA).Only in the rarest of cases (i.e.maximum bar clinoform relief) did bar clinoforms capture the full reach-mean bank-full flow depth.This occurred when the bar clinoform slipface base (i.e.toeset) terminated in or near the channel base.Similar observations were made in other modern rivers (Bridge & Lunt, 2006;Herbert et al., 2020;Reesink et al., 2014;Sambrook Smith et al., 2009).The Willwood Formation matches the underprediction relationship.Moreover, the maximum bar clinoform relief observed in the Willwood Formation was 4.9 m, the 99th percentile was 4.5 m, and the 95th percentile was 3.6 m.The coherency of three of four methods, and a clear geomorphological reason and modern analogue for underprediction suggests basin river systems were 4 to 6 m in depth.
Importantly, flow depths did not substantially change spanning the PETM if they are based on bar clinoform (Foreman, 2014;this study).Thus, Foreman (2014) suggested palaeoflow depths were relatively insensitive to changes in the hydrologic regime during the global warming event.Within the Bighorn Basin, palaeofloral and palaeosol geochemistry suggest decreases in mean annual precipitation by ca 40% during the early phases of the event (Kraus et al., 2013(Kraus et al., , 2015;;Kraus & Riggins, 2007;Wing et al., 2005).This proposed decrease in mean annual rainfall is difficult to separate from changes in the seasonal and interannual distribution of rainfall and drainage that plant taxonomic and palaeosol morphological changes indicate occurred (Adams et al., 2011;Smith et al., 2008;Wing et al., 2005).Recent climate models suggest an increase in the frequency, yet not necessarily magnitude, of extreme rainfall events (Carmichael et al., 2018).Assuming these rainfall events corresponded with bank-full flow depths then the frequency of bank-full depth increased but not necessarily the magnitude during the PETM.In contrast, bank-full flow depths estimated from story heights suggest an increase from, on average, 4.3 m in the Palaeocene to 6.5 m during the PETM and a return to 4.2 m depth in the Eocene.This would be more consistent with an overall increase in extreme rainfall events during the PETM that would generate larger discharge events and bankfull flows.However, the existing data set for story heights during the PETM is small (n = 11) compared to the data set for bar clinoforms during the PETM (n = 39).Additional work is needed to assess if the increase in story height thicknesses during the PETM is robust.

| Avulsion behaviour
Key lithofacies associations have been documented by a number of authors within the Willwood Formation and used to develop a depositional model for basin river systems and floodplains (Bown & Kraus, 1987;Kraus, 1987Kraus, , 1996Kraus, , 2001;;Kraus et al., 2013Kraus et al., , 2015;;Kraus & Middleton, 1987).This depositional model can be summarised as follows.Trunk river systems represent the main conduits for sediment and water discharge within the basin.Adjacent to these trunk rivers are levee systems composed of heterolithic intervals subject to palaeosol development.Distal from these levee systems, floodplains are dominated by laterally extensive and thick palaeosol development often indicative of well-drained conditions.Floodplains aggrade slowly as fine-grained silts and clays accumulate via overbank floods.However, pedogenesis outpaces the sedimentation rate on the floodplain.As the system evolves and aggrades, eventually a channel breach occurs advecting fine sand and silt to the floodplain and depositing widespread crevasse splays.If the breach stays open and draws more water and sediment, the previously unchannelised splays will spontaneously construct a network of smaller channels on the floodplain.If the avulsion is successful, then one of these channels will become the new main stem of the trunk river system.This pattern is representative of the aggradational avulsion pattern identified by other studies (Mohrig et al., 2000;Slingerland & Smith, 2004).Abels et al. (2013) built upon this model noting that floodplain deposition occurred as cycles of paired crevasse splay-rich and well-developed palaeosol profiles.
Importantly, they noted these cycles can be laterally persistent for several kilometres.They posited two phases of deposition, an overbank phase and an avulsion phase, that potentially affected the majority of the basin width.The overbank phase occurs when most fluvial water and sediment discharge is confined to the river channel itself and there is minimal advection to floodplains.During the avulsion phase the channel is breached at one or more locations and a network of smaller floodplain channels and splays distribute sediment across the basin.Once a new location is found, the system rechannelises and a new overbank phase begins.Based on time series analysis, Abels et al. (2013) argued this occurred on precessional timescales and that major avulsion events were driven by orbitally controlled climate oscillations, a result reproduced by other researchers (Aziz et al., 2008;van der Meulen et al., 2020).However, recent numerical models produce a similar shift from widespread splays to fewer, concentrated splay outlets, and eventual avulsion within fluvial systems with high suspended sediment loads (Nicholas et al., 2018).The higher suspended load facilitate 'healing' of main stem channel breaches, diverting flow to fewer and fewer (but larger) splay outlets.Given that Willwood Formation river deposits contain provenance signatures indicative of Mesozoic shale erosion, as well as reworked Cretaceous marine fossils (Welch et al., 2022), it is likely they had a comparatively high suspended load.This suggests the avulsion pattern may be autogenic in origin and the precessional timescales a coincidence.The hypothesis could be tested in the future by performing a time series analysis on overbank strata in the south-westernmost portion of the Bighorn Basin wherein provenance signatures indicate minimal contribution from Mesozoic shale units (Welch et al., 2022).Regardless of forcing mechanism, Kraus' model bears a striking resemblance to the avulsion processes observed in real-time in the Saskatchewan River and documented in a number of studies (Morozova & Smith, 2000;Pérez-Arlucea & Smith, 1999;Smith et al., 1989).
Herein additional nuances to the Willwood Formation depositional model are offered.Specifically, the overbank and avulsion phases of Abels et al. (2013) are both split into early and late phases, channel kinematics are integrated, and a vertical threshold for the setup of the avulsion cycle is proposed (Figure 5).The early avulsion phase is characterised by widespread crevasse splays across the floodplain that generate ribbon sandbodies hosted within heterolithic crevasse splay deposits (top of Figure 5).Splay channels measured herein are ca 3.5 m thick, smaller than trunk river depths, and indicate only a portion of water and sediment discharge passed through them.The splay channels represent 'failed' avulsions; in that they did not evolve into main river stems.However, they do preserve evidence of an initial scour of a few metres into underlying heterolithic crevasse splay units.Moving into the late avulsion stage, a single splay channel becomes dominant and inherits the initial scour depth (base of Figure 5).As this splay channel obtains a greater portion of sediment and water from the parent river channel, levee construction begins leading to more common pedogenesis across the floodplain, although rapid sedimentation events in the form of splays are still common.Once levees are established the system shifts into the early overbank and late overbank phases (Figure 5).These represent the time interval during which the river constructs its channel belt, the main zone of local avulsion and meandering.This process leads to the amalgamated and semi-amalgamated sandbodies of Owen et al. (2017Owen et al. ( , 2019)).In some outcrops, sandbody stories are offset from one another forming a compensational stacking arrangement (Figure 6), and in some instances it is possible to trace a clear increase in palaeosol development away from these individual channel stories (Figure 6).Preferential deposition near the channel during the early and late overbank phases leads to river channel perching and increased palaeosol development in the floodplains, as the latter receives only rare overbank floods of silts and clays.Once the free surface of the channel reaches a superelevation of one bank-full flow above the floodplain, creating an alluvial ridge, the system is set up to begin the avulsion process.The early avulsion phase breaks the adjacent banks, potentially leaving behind an abandoned mud plug in places, and sediment and water discharges across the floodplain.This effectively fills the topography lows generated originally by differential aggradation of the river channel, and the process begins again.The commensurate thicknesses of the multi-storied sandbodies and overbank cycles suggest a shifting back and forth of sediment provision between floodplain and channel systems.Overall, the aggradation of the channel one flow depth above the floodplain before avulsion is similar to estimates of 0.6 to 1.1 times flow depth of superelevation in other systems (Mohrig et al., 2000).This study's observation that fluvial sandbodies are commensurate in vertical scale with overbank cycles complements recent work in modern through Pleistocene alluvial systems that examine alluvial-ridge development and avulsion length-scales.Recent studies along the northern Gulf of Mexico coastal plain note the presence of alluvial-ridge drainage basins between major river systems such as the Brazos and Trinity rivers among several others (Swartz et al., 2022).Alluvial ridges were generated primarily by differential sedimentation rates from the channel to distal floodplain and the alluvial ridges represent the main topographic features across these lowrelief landscapes (Swartz et al., 2022).Importantly, the vertical relief of this topography appears to correspond with river flow depths (Swartz et al., 2022).In the subsurface, seismic data sets of Pleistocene sandbodies in the northern Gulf of Mexico coastal plain display orientations and scales consistent with avulsions driven by morphodynamics; preferentially filling associated alluvial-ridge basins (Cardenas et al., 2023).Pleistocene sandbodies display a 'shingled' stratigraphic pattern (i.e.partial stratigraphic overlap of laterally offset units) that Cardenas et al. (2023) interpreted as compensational in origin.These fluvial systems constructed a sandbody within their channel belt until reaching superelevation, at which point they avulsed into the alluvial-ridge basin and began constructing a new fluvial sandbody, offset and distinct from the previous.The Willwood Formation data suggest this behaviour sets a characteristic vertical scale of aggradation for both fluvial and overbank strata.Once a river avulses to a given location it must first fill one flow depth's worth of accommodation in the alluvialridge basin, in effect smoothing previous topography.It then begins to aggrade until another flow depth's worth of topography has been generated through construction of the alluvial ridge.
These inferences are strengthened by Edmonds et al. (2016)'s observations in both the modern Himalayan and Andean foreland basins that rivers tend to avulse a characteristic distance away from their main river stem.They suggested this non-random avulsion behaviour was driven by alluvial ridge development dictating the avulsion flow path.Edmonds et al. (2016) proposed scaling laws for avulsion that appear robust even under different boundary conditions of tectonism and prevailing climate.These scaling laws for characteristic avulsion distances, combined with commensurately thick sandbody and overbank cycles, offer a mechanism to explain the stratigraphic 'shingling' of sandbodies observed by Cardenas et al. (2023).Overall, it appears morphodynamics play a central role in the construction and organisation of alluvial stratigraphy, imparting non-random behaviours on the system.Future studies are predicted to find similar commensurate packages of fluvial sandbodies and overbank packages in other formations and basins beyond the Willwood Formation.However, accurate constraints on bank-full flow depth as well as detailed assessments of overbank depositional patterns are needed for these to be recognised.Cardenas et al. (2023) suggested major shifts in boundary conditions may disrupt this underlying morphodynamic control on avulsion patterns.Sincavage et al. (2022) presents evidence that the Bahmaputra River has preferentially filled Holocene palaeochannels generated by outburst floods rather than the topographically favourable course into the Sylhet Basin of Bangladesh.In the Bighorn Basin, the extreme outlier in the multi-story sandbody data set (Figure 4) is found within the PETM climate event (Foreman, 2014).This sandbody is both anomalously thick (30+ m) and laterally extensive (ca 12 km).Previous studies attributed its size to a transient increase in channel mobility driven by some combination of river discharge and/or enhanced floodplain erodibility (Foreman, 2014;Kraus et al., 2015).Although this study cannot provide insight into the underlying mechanism, it does imply the driver must override the default morphodynamics of the system.
F I G U R E 6 Uninterpreted and interpreted photograph panel of coeval fluvial channel and proximal-to-distal floodplain strata (image from Google Earth Pro).Fluvial sandbodies highlighted in yellow, mud plug in blue and palaeosols outlined in white.Palaeoflow direction is approximately out of the image towards reader (photograph oriented looking south).Note mud plug occurs at top of the lowermost fluvial sandbody and the next stratigraphically higher sandbody occupies a position above the well-developed, overbank palaeosols coeval with the previous fluvial sandbody.This is an example of a local compensational deposition pattern.

| completeness
Recently there have been renewed efforts within the sedimentary scientific community to quantify the completeness of the stratigraphic record (Holbrook & Miall, 2020;Paola et al., 2009;Romans et al., 2015;Straub et al., 2020).Much of this is driven by the foundational work of Sadler (1981) wherein he noted a reduction in sedimentation rates as the time interval of observation increased (i.e.'Sadler effect') as well as hypotheses suggesting many sediment flux signals driven by allogenic forcings may be effectively 'shredded' by the sediment transport system (Allen, 2008;Jerolmack & Paola, 2010).The prevailing suspicion within the sedimentary community is that stratigraphy is both incomplete and insensitive to the preservation of environmental signals.The observed stratigraphic patterns within the Willwood Formation indicate otherwise.
The Sadler effect is attributable to the presence of unconformities generated by both erosion and stasis (i.e.nondeposition) on palaeo-landscapes.Recent work within experimental basins indicates a relative increase in the role erosion plays compared to stasis within a given stratigraphic record as the time interval of interest increases, but also that the overall completeness increases as the time interval of interest lengthens (Straub & Foreman, 2018).Completeness in this context is defined as the proportion of time intervals (i.e.how a researcher chooses to discretise time) in a succession for which at least some sediment is both deposited and preserved in the resultant stratigraphic column.Studies note that the strength of the Sadler effect is reduced, and completeness increased if deposition in the directions parallel and perpendicular to dominant sediment transport are taken into account (Jerolmack & Sadler, 2007;Mahon et al., 2015;Straub & Foreman, 2018).Finally, with all other factors held constant, higher rates of basin accommodation lead to more complete stratigraphy (Straub et al., 2020).Thus, incompleteness is a function of the spatiotemporal heterogeneity in geomorphological activity as well as underlying subsidence characteristics, whether tectonically or otherwise generated.
Experimental data indicate that 100% completeness is achieved in stratigraphic successions at timescales approaching that of compensational depositional behaviours of the geomorphological system, denoted T comp (Straub et al., 2020).This is the timescale at which subsidence has removed sediment from the zone of geomorphological reworking and the sediment routing has evenly distributed sediment to fill accommodation (Straub et al., 2020).T comp corresponds with the ratio of maximum topographic roughness divided by the long-term sedimentation rate (Sheets et al., 2002;Straub et al., 2020;Wang et al., 2011).In experimental basins, topographic roughness is linked with the maximum channel flow depth.This result emerges from a statistical analysis of topographic variability across the experimental landscape surface, but it is also an intuitive result.Channels are both erosional and depositional.Their scouring behaviour causes loss of sediment (i.e.erosional unconformities) and differential aggradation between the channel and overbank settings causes areas of reduced or non-deposition in the basin (i.e.stasis unconformities).
In the Willwood Formation there is clear evidence of channel-stacking in a compensational manner (Figures 5  and 6), and nearly all depositional elements (e.g.channel incision depths, splay channels, crevasse splays, palaeosol profiles, incised mud-filled channels) exhibit thicknesses equivalent or smaller than bank-full flow depths of the basin rivers.Moreover, the structured nature of overbank cycles and their similarity in thickness to multi-storied sandbodies suggests the two are depositional complements to one another, and there is a clear genetic link through the avulsion process.Thus, it is likely that the topographic roughness across the palaeo-landscape during deposition of the Willwood Formation was equivalent to the bankfull flow depth of basin rivers (4-6 m).Given a long-term accumulation rate of 45.2 cm/kyr for the formation, this yields a T comp of 8.8 to 13.2 kyr.A more conservative estimate may be obtained by positing that the thickness of multi-storied sandbodies represent topographic roughness (9.3-10.8m, median and mean respectively), which yields a T comp of 20.6 to 23.9 kyr.In both cases, the estimates suggest significantly shorter compensation timescales perhaps by an order of magnitude than previous field-scale studies in other basins that rely on the preserved stratigraphic surfaces within fluvial sandbodies exclusively (Chamberlin et al., 2016;Trampush et al., 2017;Wang et al., 2011).Incorporating more detail and quantification of overbank depositional patterns in other field areas and geological formations may reveal shorter compensation timescales than previously estimated.This is particularly true if the aforementioned morphodynamic controls on avulsion dominate, as appears to be the case in modern foreland basins (Edmonds et al., 2016).Moreover, compensation timescales of 8.8 to 23.9 kyr imply a high degree of completeness and resolution from palaeobiological and palaeoclimate records in the Willwood Formation.
The Willwood Formation of the Bighorn Basin has been the focus of intense palaeobiological study for over a century (Gingerich, 2001;Gingerich & Clyde, 2001).Hundreds of vertebrate and floral fossil sites have been found that span the full stratigraphic thickness of the formation.Vertebrate sites are dominated by reptile and mammal fossils and are primarily found associated with palaeosol horizons (Bown, 1979;Bown & Beard, 1990) although in some cases near complete skeletons are recovered, entombed in lacustrine carbonates (Bowen & Bloch, 2002).vertebrate fossil localities are associated with periods of relative stasis on the ancient floodplain and accumulate during the early and late overbank depositional phases when the river channel is primarily restricted to local avulsion and migration.These fossil localities are subsequently entombed and removed from the zone of reworking during the early avulsion phases as widespread crevasse splays blanket the floodplain.Locally there is clear evidence for cut-and-filling of some palaeosol mudrocks (Kraus & Davies-Vollum, 2004), but these display similar scour depths (3-5 m) as river flow depths and represent only a few tens of metres of lateral removal from the floodplain which was at least several tens of kilometres wide.Vertebrate fossil localities are accumulated and preserved on the timescale of an overbank cycle and likely represent time-averaged concentrations of less than ca 24 kyr.Davies-Vollum and Kraus (2001) suggested a similar process of preservation for organic debris and plant fossils in laterally persistent backswamps present in some areas within the Willwood Formation.In contrast, macrofloral sites are commonly contained with the mud plug and channel abandonment units (Hickey, 1980;Wing, 1984;Wing et al., 1995).These represent an initial scour surface by a river system (i.e.erosive unconformities) and comparatively rapid abandonment.The presence of most mud plugs and channel abandonment units near the top of multi-storied sandbodies suggests their formation and preservation occur during the early avulsion depositional phase.As such the conditions for preservation in the stratigraphic record would reliably occur every ca 24 kyr or less.The macrofloral fossils themselves would not be time-averaged to the same extent of vertebrate fossil localities.Although further quantification and analysis is needed, the depositional patterns within the Willwood Formation suggest the potential that a palaeobiological record could be constructed with 100% completeness on the ca 24 kyr, if not the ca 10 kyr, time scale.Similarly, geochemical proxies of climate change such as stable isotope records and major and trace element composition of palaeosols derived from these same deposits would have a similar completeness or finer.This creates the possibility of paired reconstructions of palaeoenvironmental and palaeobiological change in the early Palaeogene of similar temporal resolution to those available for the Plio-Pleistocene and Holocene.

| CONCLUSIONS
Fluvial dynamics and channel-floodplain interactions are significantly driven by bank-full flow conditions.The alluvial stratigraphic record provides the opportunity to understand how fluvial systems evolved under a diversity of background conditions and driving mechanisms.As such, several approaches to estimating bank-full flow depths from fluvial deposits have been proposed in the literature including dune cross-bed thickness, relief on bar clinoforms, story heights and mud plug thicknesses.Three of the four methods yield coherent means and medians in the early Palaeogene Willwood Formation of north-west Wyoming of 4 to 6 m, which are statistically indistinguishable from one another.The exception is the relief on bar clinoforms, where mean and median are approximately one-third of other estimates.However, the thickest bar clinoforms fall within the bank-full flow depth estimates of other methods.This is consistent with observations in modern rivers where only rarely do unit bars capture the reach-mean bank-full flow depth.It is suggested that palaeoflow depths are more reliably preserved by compound bar deposition that fills stories in fluvial sandbodies, channel abandonment facies such as mud plugs, and dune cross-bed thickness than by the relief on bar clinoforms.Furthermore, the data collected herein suggest Willwood Formation river systems in the Bighorn Basin aggraded, on average, approximately one flow depth above the floodplain before major avulsions.Local, compensational avulsions were more common and likely responsible for the multi-storied character of fluvial sandbodies in the basin.Based on these observations, the autogenic behaviour of basin river systems likely generated floodplain topography of 4 to 6 m in relief, but subsequently filled this space during regional avulsions.These processes likely facilitated the unusually well-preserved fossil and palaeoclimate records present in the basin with the potential resolution finer than 10 4 years.

DATA AVAILABILITY STATEMENT
Data used in this research are available in the Supplementary Information associated with the article as well as upon reasonable request from the corresponding author.

ORCID
Brady Foreman https://orcid.org/0000-0002-4168-0618 and (B) Examples of bar clinoform deposits.Hammer is ca 30 cm long and yellow field notebook is ca 20 cm tall.(C) Uninterpreted and interpreted photographs (looking north-northeast in Sand Coulee field area) of fluvial sandbody in outcrop showing sequence of stories numbered oldest (1) to youngest (5).(D) Uninterpreted and interpreted photographs of mud plug channel fills in outcrop (looking east in Sand Coulee field area).(E) Example of mud plug lithofacies in outcrop, Jacob's staff in background is 1.5 m long (looking north in Sand Coulee field area).(F) Examples of stratigraphic sections through sandbody stories (left section from Wasatchian-5 biozone in McCullough Peaks study area and right section from Clarkforkian-3 biozone in Sand Coulee study area) and mud plug (from top of Wasatchian-0 biozone in Sand Coulee study area).

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I G U R E 3 (A) Outcrop photograph showing crevasse splay channel and overbank cycles (looking north-northeast in Sand Coulee study area).Overbank cycles are composed of a heterolithic lower interval of interbedded drab, weakly developed palaeosols and crevasse splay deposits and an upper interval of red or purple, well-developed palaeosols.(B) Representative stratigraphic sections through splay channel deposits (both sections from Clarkforkian-3 biozone in the Sand Coulee study area).

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I G U R E 4 Violin plots with inset box-and-whisker plots of compiled Willwood Formation fluvial and overbank data; (A) field, compacted measurements and (B) decompacted measurements.Outlier data points that fall outside 1.5 times the interquartile range shown as open circles.

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Depositional model for avulsion in the Willwood Formation with overhead, map view illustration of the major intervals of channel and floodplain behaviour (left) and an associated cross-section of the constructed stratigraphy (right).Darker blue channels in map view are active and lighter blue channels are previous locations.Point bars and mid-channel bars are denoted by yellow lines and polygons with black stipples.Numbers correspond between map view fluvial channels and deposits generated by them.Grey zones in overbank areas are intervals dominated by crevasse splays and incipient pedogenic development, orange zones are areas dominated by moderate pedogenic development, and red zones are areas dominated by significant pedogenic development.Thicknesses of orange and red bands on crosssection denote the thicknesses of palaeosol horizons, which tend to increase away from fluvial sandbodies.
This research was made possible by funds from the David B. Jones Foundation (to B.Z.F.), Geological Society of America Graduate Student Grants (to G.M.S. and D.J.T), a Colorado Scientific Society grant, an SEPM Student Grant, a National Science Foundation Student Travel Grant, and a Tobacco Root Geological Society grant (to G.M.S.) as well as internal grants from Western Washington University (to G.M.S., D.J.T., K.D.P. and A.S.).E.F.Lalor, B. Ward, A. Schluneger, A.K. Lesko, D.M. Rasmussen, P.Z.Christenson, K.E.Fristad, M.D. D'Emic, C. Hoegger, C. Rowan-Arnett, S. Reynolds, R. Reynolds and especially the late D. Foreman are thanked for field assistance.H. Beaumont and R. Sincavage are thanked for reviews that improved the manuscript substantially.
Summary of p-values for statistical tests of decompacted thicknesses.
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