To study the importance of autogenic processes relative to external forcings on grain size fining trends, we use a coupled model (GravelScape) that is fully described in . Here, we will only give essential elements and introduce equations that are necessary for the comprehension of the work presented here. The model comprises the Landscape Evolution Model (LEM) solving the Stream Power Law (SPL) enhanced for the effect of sediment transport and deposition : 1dhdt=U-Kp̃mAmdhdsn+Gp̃A∫AU-dhdtdA, where h is surface topography, U is surface uplift or subsidence, K is the erodibility parameter, G is a dimensionless depositional parameter, p̃ represents variations in precipitation rate around a mean value that is included in the definition of K and G, A is drainage area, ∂h∂s is topographic slope in the direction of water flow, and m and n are the area and slope exponents, respectively. G controls whether the system is transport-limited (G>0.4) or detachment-limited (G<0.4). Its value is not well known, but it has been constrained to be moderately transport limited of the order of 0.7 from a wide survey of sedimentary fans. Values of Kf are poorly constrained as its units and value strongly depend on the slope exponent m. The direction of water flow is computed from the surface topography and allows for flow divergence by assuming that, at every node of the model, discharge is distributed to all lower-elevation neighbouring nodes in proportion to slope.

Grain size is computed using the self-similar grain size fining model of for gravel, which assumes that mean grain size, D‾, and its standard deviation, ϕ, vary in a constant ratio during fining that is, in turn, controlled by the ratio between local deposition rate, r, and sediment flux, qs, according to 2D(x*)=D0‾+ϕ0C2C1e-C1y*-1, where x* is a dimensionless distance along flow path and 3y*(x*)=∫0x*R*(x*)dx*, where the primary components controlling the fining are R=r/qs, and D0‾ is the mean grain size where flow initiated. See for a more detailed version of the coupling of the two equations, and see for a description of the coefficients C1 and C2. Note that, in GravelScape, the deposition rate r and sediment flux qs are obtained from the solution of the LEM, contrary to most previous uses of the model of that have made the simplifying assumption that deposition rate can be directly equated with subsidence rate e.g..

We will use a controlled setup similar to that of and also used in , in which sediment is produced in an orogenic area of width LM uplifting at a rate U0 resulting in a sedimentary flux, qs,i=U0LM. Subsidence rate, σ, in the adjacent basin of width LB is assumed to vary as an exponential function of distance, x, from the mountain front, 4σ(x)=σ0e-αx/LB, simulating flexural isostasy under the weight of the adjacent mountain. The elevation at the opposite side of the mountain front (i.e. right hand-side, edge of the basin in our setups) is assumed to be held at a constant elevation, which we will refer to as the base level. As proposed by , we introduce the parameter F, 5F=qs,i/∫0LBσ(x)dx, that measures the degree of bypass of the system. Small F values (i.e. 1<F<10) correspond to low-bypass systems where most of the sediment coming from the mountain is trapped in the basin, whereas large F values (i.e. F>10) correspond to high-bypass systems where most of the sediment coming from the mountain leaves the basin at its outer end. Under these conditions, one can derive an analytical solution to the fining model of , which we will use to estimate, by comparing it to GravelScape's predictions, the contribution from autogenic processes to the grain size fining trend, relative to that resulting from the imposed basement subsidence.

As shown in , fan extent and the subsequent foreland basin long profile are mostly controlled by the distribution of rainfall between the mountain (source) area and the basin (sink) area, with basement subsidence only playing a secondary role in low-bypass (low F) systems. In high-bypass systems, the sedimentary flux remains relatively constant across the basin. At steady state using the Stream Power Law (Eq. ), sedimentary flux is equal to the product between drainage area (to power m+1) and slope (to power n), which implies that the slope must vary as the inverse of discharge. Near the mountain front, discharge is relatively constant and equal to the product of the mountain surface area by the assumed precipitation rate. The slope must therefore be relatively constant, which leads to the formation of a sedimentary fan. Away from the mountain front, rainfall in the basin substantially contributes to the discharge, which therefore increases and causes the slope to decrease to form an alluvial plain. Therefore, the transition between the steep, constant slope fan and the alluvial plain takes place where the contribution to discharge from rainfall in the basin equates the discharge from the mountain area. This explains the broad one-to-one relationship between upstream catchment area and fan area across many scales . It also implies that one of the main controls on the shape of the topography in the basin area is the difference in precipitation rate between the mountain and basin areas . To illustrate this point, we can derive the parameter β: 6β=νMLMανBLB, where νM and νB are the relative precipitation rates in the mountain and basin areas, respectively. β is the ratio of the contribution to discharge from the mountain area, νMLM, and from precipitation in the basin, νBLB, multiplied by the relative wavelength of the subsidence function, α. β is in fact the ratio of the length/size of the fan, νMLM/νB to the size of the subsidence function, α/LB.

In short, β is a measure of the difference in area (extent) and precipitation rate between the orogen catchment and the sedimentary basin. Combinations of high precipitation and drainage area in the orogen with low basin length and basin aridity result in high, orogen-dominant β values. Inversely, large basin areas, especially with higher precipitation relative to the orogen, result in low, basin-dominant β values. We keep α constant in all our simulations and change only vm to increase or decrease β. To emphasize the impact of changing vm on β, within figures and referring to specific values within the figures, we normalize β by α (Fig. ).

Different topographic profiles predicted by GravelScape for different values of the parameter β are shown in Fig.  . We see that as β increases, the transition point between the steep fan and the curved alluvial plain moves towards the edge of the basin, and the surface topography evolves from concave and steep near the mountain front (low β values) to convex and flat (high β values). High β values correspond to the “constrained systems” in which the distance from the mountain front to the edge of the basin is smaller than the natural width of the fan, i.e. the width it would occupy if it were allowed to develop beyond the base level.

Note that β, through its control on discharge partitioning between the mountain and basin contributions, only affects the position of the transition from steep to curved segments (and subsequent channel profile), whereas the slope at the mountain front (and therefore absolute topographic height) is controlled by additional factors (e.g. G and K) and by the magnitude of the sediment flux from the mountain, qs,i, as explained by the analytical solution for the slope at x=0 given in , 7S(x=0)∝(Gqs,iKLMm+1)1/n, for high-bypass systems (i.e. when subsidence can be neglected). Essentially, the higher the value of G (and the lower the value of K), the steeper the fan with the same fan extent as shown in and . Note that the dependence of basin slope on G and K is not apparent in Fig. , where we chose to normalize the topographic profiles. In the case n=1, K also controls the response time of the system (i.e. the time it takes to reach its final steady-state height), but G does not .

In situations where the surface topography predicted by GravelScape is characterized by very low slopes (e.g. under-filled basins with low F, low G, or high β), local minima can develop that affect the computation of the flow routing needed to solve the modified SPL equation (Eq. ). In these situations, we use the method developed in  to adjust the flow routing and compute the geometry of the resulting lakes forming around each local minimum. In these filled lakes, the algorithm by to solve equation (Eq. ) cannot be used, and sediment is uniformly dumped as a first-order attempt to represent lacustrine deposition, and no grain size can be accurately computed using the grain size fining model of . We checked that all model runs presented in this work were not strongly influenced by the presence of local minima.

In this work, we will vary model parameters G, K, F, and β to assess the impact of subsidence and topography on grain size fining under a near-constant orogen flux at steady state. For simplicity, we will refer to the respective parameters G, K, F, and β as the depositional, erodibility, bypass, and orogen discharge efficiency.

Within our model, the river planform changes over time and space despite constant forcing conditions (see Video supplement), and we refer to this as model autogenic dynamics. These changes arise from the interactions between the depositional and erosional terms in Eq. (). Erosion leads to the formation of channels and deposition to their progressive infilling, which, in turn, affects local slope and the relative distribution of water flow between a node and its neighbours. This may lead, through downstream cascading, to discrete events that reorganize large parts of the drainage network, similar to avulsions that have been observed in laboratory experiments () and natural systems .

describe many autogenic processes, and their associated landforms, on the basis of spatial and temporal scale. Our Landscape Evolution Model can only reproduce autogenic processes that occur over long timescales (102–107 years) as well as large lateral (101–104 km) and vertical spatial scales (100–102 m). This scale matches with the descriptions of for autogenic dynamics such as (1) the regrading of the depositional surface (longitudinal river planform changes); (2) avulsions; and (3) channel convergence, divergence, and, to a limited extent, bifurcations. All of these are observed in the model (as described above). Smaller-scale autogenic processes described in , with vertical scales under 100 m, such as those that involve bedforms (e.g. dunes or bars) or channel reach dynamics (e.g. riffle and pools; cut banks and point bars; meanders dynamics), cannot be reproduced in a Landscape Evolution Model based on Eq. ().

We now present results obtained with the coupled model to quantify the relative contributions from external forcings and autogenic processes to the control of grain size fining trends. For this, we define ΔD the difference at the basin outlet between the multi-channel (2D) grain size GravelScape solution computed in the largest channel and that predicted for a single channel using the method of , i.e. assuming that deposition rate is equal to the imposed subsidence rate. We will call this quantity the grain size fining deviation and define it as 8ΔD=<D‾(xmax,y=yMC)-D‾D(xmax)>SS, where D‾(xmax,y=yMC) is the mean grain size computed by GravelScape at the exit of the basin within the largest main channel (i.e. at the y location of the maximum discharge), D‾D is the mean grain size predicted by , and the symbols <>SS indicate a temporal average, once the system has reached steady state. We computed ΔD for a large number of simulations varying G, K, and β, which control the surface topography, and varying F (through σ0 while keeping a constant U), which controls the degree of bypass of the system. The results are shown in Fig. , where each panel shows how ΔD varies as a function of β and F. The dependence on K and G can be appreciated by inspecting the different panels from left to right and top to bottom, respectively. All values are computed when the system has reached steady state (constant apex topography) or is approaching it (i.e. within 90 % of their steady-state topography, for model runs characterized by a low value of K). The regions of model space where local minima dominate are left blank to exclude them from our interpretation of these results.

We see that the grain size deviation, ΔD, is controlled by all four factors, with ΔD increasing with increasing values of F, β, and G but decreasing with increasing values of K. If we discard the model runs with low values of K that have not yet reached their steady state at the end of the experiments, i.e. those shown on the left column in Fig. , we see that the dependence on K is minimal.

In absolute terms, the maximum values of ΔD are around 15 % to 20 % and are reached for high F and high G values, corresponding to systems in high bypass and in transport-limited conditions. Conversely, systems that are in low bypass (low F) or in detachment-limited conditions (low G) show less than a few percent deviation in grain size fining compared to the predictions of a one-dimensional model, assuming that fining is controlled by subsidence only.

In Fig. , we show the grain size predicted by GravelScape at the exit of the basin, which we call D‾e. A value of 1 indicates no fining, and a value of 0.5 corresponds to 50 % fining from the original source distribution. Each panel corresponds to the same specific values of K, G, β, and F as in Fig. . As demonstrated by , we see a strong dependence of grain size fining on F, with high-bypass (high F) systems showing the least fining (smaller value of D‾e). Contrary to , we also see a strong dependence on β and G and thus on topography. Finally, we see little to no dependence of D‾e on K, demonstrating that the apparent dependence of ΔD on K is indeed an artifact due to the fact that the model experiments with low K values have not reached steady state. Figure  also demonstrates that similar downstream final values of fining can be observed by changing multiple parameters (e.g. β and G) for the same F values.

For completeness, in the Supplement, Figs. S3 and S4, we show plots of predicted grain size and other autogenic quantities derived from the model as a function of downstream distance.

Combining the results from the Figs.  and , we see that the greatest grain size deviation is produced under high bypass (high F) because GravelScape predicts much more fining than expected from . This is because of a strong dependence of grain size fining on topography, which is not predicted by , as high values of G corresponding to more transport-limited systems producing higher fans, and low β values producing shorter and steeper fans, cause more fining and thus grain size deviation from . The dependence on K is less important, except that systems that are characterized by low values of K will take longer to reach steady state and are therefore likely to produce more fining than expected from their basement subsidence.

To demonstrate this last point, we ran the low-K experiment (i.e. where K=2×10-6 m^1−2 m yr⁻¹) for longer than the reference 25 Myr used in all model runs presented in Figs.  and . We show the results in Fig. . In Fig. a we show the evolution of the basin apex topography as a function of time with the different time periods over which we computed the deposition rate and grain size fining shown in different colours: dark blue when the system has reached 90 % of its final topography, light blue when it has reached 95 %, and orange and yellow when it has reached steady state. In Fig. b and c, we compare the predicted deposition rate (averaged in the y direction and over the time span indicated in panel a) to the imposed subsidence rate and the predicted grain size fining (similarly averaged) to the predictions of , respectively. We also show in Fig. c the grain size fining obtained at steady state with a larger value of K (i.e. K=2×10-5 m^1−2 m yr⁻¹) for reference. We see that the mean deposition rate converges towards the subsidence rate, and the fining rate converges towards the solution predicted with a higher K value, as the system moves towards steady state (i.e. from dark blue to yellow). This clearly demonstrate that K has no influence on the grain size fining at steady state but shows an impact during the transient steady state (i.e. even with values within 90 % of steady state showed deviation) where more fining was produced than expected (similar to a lower F) and there is more deviation from the constant, imposed subsidence rate. This is because, before the system reaches steady state, the erosion rate in the mountain is smaller than the imposed uplift rate, and, therefore, the flux coming out of the orogen, qs,i, is smaller than the value, U×LM, which we have used to compute F and impose a constant subsidence rate. In , we will further develop this point for natural systems and, in particular, for foreland basins, where subsidence rate and erosion rate in the source area are intimately linked by flexural isostasy.

To further synthesize our results and facilitate their use for the interpretation of grain size fining data, we have developed a generalized framework to determine under which basin-wide configurations grain size fining is dominantly controlled by subsidence (i.e. mean deposition rate) or by autogenic dynamics (i.e. depositional divergence). The framework is based on two maps shown in Fig. a and b, one of the grain size fining (Fig. a) and one of the grain size fining deviation (Fig. b) as a function of two variables, F and β. These maps are obtained by averaging the results shown in Figs.  and over G and K, respectively. This averaging is justified by the low dependence of these results on K and the fact that the value of G is difficult to assess but likely to be relatively close to 1 as many alluvial, continental sedimentary systems are predominately, moderately transport-limited . In Appendix  and Fig. , we show maps similar to those shown in Fig.  but using the slope, S, along the vertical axis.

In these maps (Fig. a and b), we define three main regimes for grain size fining based on the grain size divergence computed by GravelScape: (1) a subsidence-dominated regime, where grain size deviation is insignificant (under 5 %); (2) an autogenic-dominated regime (including transient and steady-state systems), where grain size deviation is larger than the fining induced from subsidence (i.e. that predicted from Duller); and (3) a mixed regime, where grain size divergence is non-negligible (over 5 %) but not greater than the subsidence-induced fining. Note that the top-left region in both panels (a) and (b) of Fig.  corresponds to model parameter values that lead to unrealistically flat topographies such that the solution is dominated by the presence of numerous local minima. We do not include modelling results in this region in our framework as it represents situations that are inadequate for measuring grain size.

We see that where total fining is larger than 30 % (bottom-left corner), the system is consistently in the subsidence-dominated regime. In other words, autogenic grain size fining alone cannot explain fining above 25 %–30 %. These subsidence-dominated systems are in low-bypass and steep slopes (low β values). At the diagonally opposite side of the map, i.e. in systems characterized by high bypass and low slopes (high β), there is little to no fining (less than 2.5 %, top-right corner) as neither subsidence nor autogenic processes can produce fining. In our new framework, the subsidence-based interpretation for grain size fining of is equivalent to a trajectory in the subsidence-dominated regime from these two locations, i.e. from the bottom left to the top right. This means that in order for grain size to be interpreted as a function of subsidence only, increasing values of F must be accompanied by decreasing values of slopes or increasing β.

The presence of an autogenic-dominated regime in the bottom-right corner of the Fig.  map, corresponding to high bypass and low β (high slopes), has two implications. Firstly, the one-to-one relationship between grain size fining and F breaks down in systems that are characterized by high slopes. In other words, following a horizontal trajectory in our framework at constant, high slope values, we do not observe the decrease in grain size fining with increasing F due to the importance of autogenic processes. Secondly, high-bypass systems may experience substantial fining in proportion to their slope. This situation corresponds to a trajectory along a vertical path in our framework in the high-bypass regime, where progressive fining corresponds to increasing slopes or an increasing importance of autogenic processes, in systems that are all characterized by high bypass.

These findings have important implications for the interpretation of grain size fining. Firstly, there is a danger to interpret grain size fining trends only as a direct measure of present-day or past subsidence only, whereas some of the fining may be due to autogenic processes. To avoid this over-interpretation of data, one should compare the subsidence derived from the grain size fining trend to preserved sedimentary thickness, where possible. Another option suggested by our work is to consider the topography and, in particular, the surface slope. Secondly, we postulate that in a high-slope, high-bypass system, grain size fining is likely to be dominated by autogenic processes and should therefore not be used to constrain subsidence patterns.

To illustrate the use of the framework, we have compared our modelled sedimentary systems to the natural systems (Fig.  c) according to their bypass characteristics and their surface slope and confinement, which we assume to be indicative of β.

Extension in the Basin and Range area has led to a high subsidence and accommodation rate in the Death Valley graben , resulting in an under-filled sedimentary basin with an active depositional surface that presently lies 86 m below sea-level and where large lakes frequently form . This indicates low F conditions. The relatively large size of the catchment, especially along the western side of Death Valley, compared to the extent of the fans, as well as the relatively linear slopes of the fans, is indicative of a β/α value near or just above 1, which positions the Death Valley fans in the subsidence-dominated regime (the square in Fig. b). This is consistent with the fining observed by and interpreted as reflecting the spatial distribution of subsidence suggested by the underlying stratigraphy .

Megafans exiting the Himalayas (e.g. the Kosi megafan) are of similar extent than the flexure wavelength of the underlying crust resulting in a large β values with low slopes. This is confirmed by their convex or linear surface topography . The resulting low slopes and high β are likely to minimize autogenic deviation and positions them in the upper portions of the framework, where there is little fining unless there is subsidence. This is in agreement with the findings by of fining trends limited to parts of the Kosi fan characterized by higher subsidence. However, attribute some of the fining observed to abrasion, a process not yet included in our model setup.

The Iglesia basin is a piggy-back basin fed by crustal shortening and uplift in the Argentine Frontal Cordillera . The basin is in moderate to high bypass . The flexurally controlled subsidence is likely to be much greater than the fan extent leading to small values of β. This is confirmed by the fan concave up topography . This should position the system in the mixed to autogenic-controlled regime (the triangle in Fig. b). This is consistent with the internal reworking described by .

High-bypass systems can also be found near mature orogenic settings such as the Alberta Basin of southern Canada, where subsidence and in-filling rate has greatly decreased since the onset of collision in the Jurassic and again in the Cretaceous . The sedimentary fans that form adjacent to the Canadian Cordillera have much smaller extent than the flexural wavelength of the underlying old cratonic lithosphere , implying a small β value and high slope. These systems are likely to be in the autogenic-dominated fining (the star in Fig.b) regime as suggested by the well-documented importance of autogenic processes within the post-glacial fans of southern Alberta .

For completeness, we also positioned the Kenyan rift as a lake-dominated, under-filled system where deposition is dominated by lacustrine processes in the local minima-dominated regime (the circle in Fig. b), where the self-similar grain size fining model of does not apply.

Past studies have used grain size fining trends extracted from the stratigraphic record as a tool to estimate past histories of subsidence in sedimentary basins. This can be effective in situations where topography is low (high β) or the basin is filling (low F), but we have also shown that this can lead to overestimating the subsidence rate in transient basins or systems where autogenic processes contribute significantly to grain size fining. Thus, there is a need to identify when autogenic induced grain size fining is most likely dominating the record.

The general relationship we have evidenced between depositional divergence, rugosity, and slope (Fig. b and c) is useful for this, as it could be used by field geologists to estimate the magnitude of the autogenic processes controlling the depositional divergence by measuring rugosity (or channel depth). In turn, because we have shown that, in most situations, grain size deviation is proportional to depositional divergence (Fig. a), a measure of rugosity can be used to estimate whether grain size fining is affected by autogenic processes and/or whether a grain size fining trend can be used to constrain basement subsidence.

A combined approach of estimating stratigraphic thickness (F), while also considering paleoslope, assessing reworking, or measuring maximum channel to interfluve depth (rugosity), would be most ideal to indicate the general relevance of autogenic induced fining within the system (because slope and rugosity correlate with grain size deviation as shown in Fig. b). This means that if a system has high paleochannel depths and slopes, combined with evidence for reworking, it is likely to be strongly influenced by autogenic processes, and grain size fining estimates may overestimate the subsidence rate. Conversely, paleo systems with thick stratigraphic packages characterized by low paleochannel depths and slopes with relatively uniform infilling would have grain size fining rates more closely controlled by basement subsidence.

Further consideration should be taken specific to grain size and its response and recovery to perturbations. have described how sediment flux returns back to the value set by the tectonic forcing after a climate perturbation and that tectonics therefore determine the underlying sedimentary record over long enough timescales and constant conditions. The same trend is predicted for grain size signals , with subsidence rate controlling the long-term trend and climate-driven perturbation producing only relatively short-lived deviations from that trend. Our new findings show that, firstly, in high-bypass systems, long-term grain size fining can be set by the autogenic dynamics, especially in systems characterized by steep surface topography (low β) and in a transport-dominated state (high G). Secondly, we have also shown that, under high bypass, grain size fining becomes a function of β, which, in turn, is related to the size of the source catchment (or the sedimentary fan) relative to the subsidence pattern (or flexure wavelength), weighted by the relative precipitation rate in the source and basin areas. This implies that variations in precipitation between the basin and catchment (in space or time) could impact the long-term sediment recorded through grain size fining trends beyond a short-lived perturbation.