Regional grain size trends in fluvial successions can reveal important information regarding the dynamics of sediment routing systems. Self-similar solutions for down-system grain size fining have recently been proposed to explore how key variables, such as the spatial distribution of deposition, sediment discharge, and sediment supply characteristics, control spatial distribution of grain size in fluvial successions over time scales of 104–106 years. We explore the sensitivity of these solutions to changes in key variables and assess their applicability to ancient fluvial successions. Several sensitivity analyses are presented to investigate the relative control of the key model variables on the spatial pattern of down-system grain size fining in fluvial successions. Sensitivity analyses demonstrate that (1) an increase in the initial value of sediment discharge to a basin causes a decrease in the rate of grain size fining in fluvial successions, an effect that becomes nonlinear for large values of initial sediment discharge; (2) a short-wavelength/high-amplitude subsidence regime generates a greater rate of down-system grain size fining and a long-wavelength/lower-amplitude subsidence regime generates a lesser rate of down-system grain size fining in fluvial successions; and (3) an increase in the spread of grain sizes in the sediment supply generates a greater rate of down-system grain size fining. We apply this modeling technique to grain size data sets collected from two time surfaces within conglomerates of the Upper Eocene Montsor Fan Succession of the Pobla Basin, Spanish Pyrenees. These data sets exhibit approximately self-similar grain size distributions; further, the observed increase in down-system grain size fining associated with smaller depositional system lengths provides support for the application of self-similar solutions to fluvial successions. By applying these solutions to carefully collected grain size data from fluvial successions, we are able to relate explicitly the initial grain size supplied to the system, the spatial distribution of subsidence and the sediment discharge into the basin to the rate of grain size fining in fluvial successions. This method thus offers a powerful means of elucidating sediment routing system dynamics over time.
 In nonglaciated terrain, fluvial systems are the primary mechanism by which sediment is eroded and transported from upland areas to depositional basins. Because fluvial systems are sensitive to long-term changes in boundary conditions, it can be inferred that the sedimentary record represents a time-integrated archive of the erosional-depositional response of fluvial systems to both tectonic and climatic boundary conditions [Heller and Paola, 1992; Whipple, 2004; Allen, 2008]. It is therefore implied that if sedimentary deposits are formed as a result of these boundary conditions, it could, in principle, be possible to decode tectonic and/or climatic signals from time-integrated geomorphological and stratigraphical investigations [Hovius and Leeder, 1998; Allen and Hovius, 1998; Robinson and Slingerland, 1998; Leeder et al., 1998; Weltje et al., 1998; Allen and Densmore, 2000; Sheets et al., 2002; Densmore et al., 2007]. The temporal and spatial patterns of sediment accumulation and the characteristics of material extracted to form fluvial successions are a function of the interplay between tectonic subsidence and sediment discharge [Paola and Seal, 1995; Strong et al., 2005]. One characteristic indicator of this interplay is the spatial distribution of grain size within sedimentary basins [Paola et al., 1992a; Heller and Paola, 1992; Robinson and Slingerland, 1998; Marr et al., 2000]. Temporal and spatial trends in sediment caliber of fluvial successions, together with the migration of specific grain size discontinuities, contain important information about the time-integrated behavior of sediment routing systems and their sensitivity to external forcing mechanisms. A crucial hurdle to overcome in decoding grain size trends in fluvial successions is the requirement for knowledge that cannot be measured or inferred from fluvial successions, such as knowledge of the detailed mechanism(s) that contribute to the transfer of sediment grains at the Earth's surface to the geological record [cf. Paola et al., 2001; Strong et al., 2005; Allen, 2008].
 In contrast to the diffusional approach, Robinson and Slingerland  couple a physically based grain size sorting model [Van Niekerk et al., 1992] to fluvial aggradation in a foreland basin. The model of Robinson and Slingerland  differs from that of diffusional models as, under a prescribed spatial distribution of tectonic subsidence and input grain size supply, it explores the influence of tectonic subsidence rate, sediment discharge, water discharge, hydraulic geometry, and the mechanics of sediment transport on the spatial distribution of grain size in ancient fluvial systems. Each parameter can be varied independently, and hence, outputs (e.g., hydraulic geometry, subsidence rate, grain size) can be tested rigorously against field data [cf. Paola, 2000]. However, the general model results of Robinson and Slingerland  agree with diffusional investigations of gravel progradation in fluvial successions [Paola, 1988; Heller and Paola, 1989, 1992; Paola et al., 1992a].
 These studies show that the rate of grain size fining in fluvial successions is governed by (1) the probability density function of the input grain size supply to the system, (2) the magnitude of sediment discharge to the basin, (3) the spatial distribution of subsidence, which controls the space available for sediment accumulation, and (4) the detailed mechanics of sediment transport and deposition [Paola et al., 1992a; Paola and Seal, 1995; Robinson and Slingerland, 1998]. Both the diffusional approach and the approach of Robinson and Slingerland , although powerful in their application, require knowledge of (4) to predict or interpret grain size trends in fluvial successions. Unfortunately, a full reconstruction of sediment transport capacity through time requires knowledge of the time-dependent distribution of channel discharges and hydraulic geometries, something that is difficult to quantify accurately from fluvial successions. Fedele and Paola  address this by developing self-similar solutions for down-system grain size fining, enabling workers to investigate the controls of grain size trends in fluvial successions in terms of long-term sediment discharge and subsidence rate, parameters that are more readily attainable from stratigraphic field studies. These solutions have been tested and validated over short time scales by field and laboratory experiments but have not yet been applied to fluvial successions over time scales of 104–106 years. In this paper, we first conduct a sensitivity analysis to evaluate the competing effects of these two variables (sediment discharge, subsidence) on grain size fining within stratigraphy. We then compare the output of these similarity solutions to field measurements of sediment caliber collected from the Eocene Montsor Fan Succession, Pobla Basin, Spanish Pyrenees [Mellere, 1993], where key controlling variables are well constrained. Specifically, this contribution aims to assess the application of self-similar grain size solutions to ancient fluvial successions and explore their potential usage in quantifying and predicting key controlling variables using a small sediment routing system as a case study. If validated, the down-system fining model would provide a significant step toward an understanding of ancient sediment routing system dynamics and, inversely, to the prediction of regional grain size trends in fluvial successions if key controlling variables are known.
2. Modeling Procedure
 In this section, we describe our model formulation for the investigation of grain size trends in fluvial successions by incorporating the similarity solutions of Fedele and Paola . We then outline the major assumptions and requirements of the modeling approach.
 A summary of grain size self-similarity and the model derivation is presented in Appendix A of this paper; for a detailed analysis of self-similar solutions for down-system grain size fining, the reader is referred to Fedele and Paola . The starting point of the model is the Exner sediment mass balance for the long term, given as,
where λp is sediment porosity, σδt (x) is spatial distribution of tectonic subsidence (L t−1) over a time interval δt, ∂η/∂t(x) is rate of change of bed elevation (L t−1) at a given down-system location, and ∂qs/∂x is down-system change in sediment discharge. The interval δt is chosen to be long enough to average out intrinsic, flow-controlled (autogenic) fluctuations and high-frequency variability in catchment-related qs behavior, but short compared with the time scales of change in tectonic boundary conditions that impact the large-scale stratigraphic architecture and grain size trends [Paola et al., 1992a; Sheets et al., 2002; Strong et al., 2005]. As we are modeling grain size trends in fluvial successions over geological time scales, the rate of change of bed elevation over time, which is largely a function of short-term channel forming processes, is taken to be zero, i.e., ∂η/∂t = 0 in equation (1). In other words, the overriding control on the long-term rate of deposition in sedimentary basins is tectonic subsidence, so we assume that the gross rate of deposition equals the rate of tectonic subsidence [cf. Paola et al., 1992a].
 For simplicity, we use an exponential decay function to describe the spatial distribution of tectonic subsidence in our model (Figure 1) as, in the absence of an explicit solution, this approximates the spatial distribution of subsidence in a wedge-top basin of the kind investigated in this study. Other spatial distributions of tectonic subsidence are possible, depending on the underlying basin mechanics [Allen and Allen, 2005], but with this assumption, we have
where x0 represents the position of the beginning of the depositional segment of the fluvial system, and αs parameterizes the down-system decay of the subsidence rate.
 An estimate for value of the time-averaged initial sediment discharge qso could be made from field observations of the spatial thickness of sedimentary units filling the basin or calculated from transport equations [e.g., Tucker and Slingerland, 1996; Paola and Swenson, 1998]. However, if we make the assumption that the basin is closed and is filled at the rate that accommodation is generated, i.e., all sediment supplied to the basin is retained by the basin, then qso (qs at x = 0) is always known, being the sum of the space made available for sediment accumulation during the time interval δt,
As we assume instantaneous basin fill within a closed system (Figure 1), our model is purely geometric and therefore only captures the effects of subsidence, not the time-dependent dynamic response of a fluvial system to climatic and tectonic perturbations [Paola, 2000; Whittaker et al., 2007]. With this assumption, the rate of decay of sediment discharge down-system is described by
where L is the depositional length scale of the system.
 From the above, we are able to calculate the spatial distribution of sediment mass down-system R* (Appendix A, equation (A3)) and hence the dimensionless distance transformation y*(x*) (equation (A4)), which, as Fedele and Paola  show, contributes to the exponent of the solution for self-similar deposit grain size fining. As a result, we can calculate the down-system rate of grain size fining for any spatial distribution of deposition R*,
where and ϕ0 represent the mean and standard deviation of the size distribution of the sediment supply and C1 and C2 are constants that, physically, express the relative partitioning of the variability in gravel supply into that arising from systematic down-system change in mean grain size (C2) and the variation at a site (C1) [Fedele and Paola, 2007]. A full derivation is given in Appendix A.
 The model setup, described above, means that only a small set of input parameters is required: (1) knowledge of the dimensional model space, which in this case equates to depositional system length; (2) knowledge of the rate of subsidence at x = 0 (σo) and the spatial decay of the rate of subsidence (σ(x)); (3) an estimate of the initial sediment discharge qso; and (4) knowledge or reconstruction of the input grain size distribution (mean and standard deviation ϕo).
2.2. Model Requirements and Assumptions
 It is important to recognize that a number of assumptions and simplifications are either implicit or explicit in the grain size fining model. We assume that (1) the system must be depositional along its entire length if down-system fining is to take place and (2) deposits must be the product of streamflow processes and not debris flow processes so that selective transportation of individual particles can be unambiguously inferred. The following limitations are noted: (3) the self-similar solutions used are strictly only valid for unimodal grain size distributions, (4) the mechanical breakdown of particles (abrasion) is unaccounted for, (5) there can be no additional lateral inputs of sediment other than that at the upstream boundary, and (6) predictions are limited to two-dimensional distributions of grain size; the model does not replicate the full three-dimensional lateral variation of grain size nor describe the details of facies partitioning of grain sizes down-system.
 Some of these assumptions are more limiting than others. It is generally easy to identify fluvial deposits in stratigraphy that have a unimodal grain size distribution, and most workers agree that selective deposition/transport dominates abrasion in setting regional trends in sediment caliber [e.g., Paola and Seal, 1995; Hoey and Bluck, 1999]. In contrast, long length scale fluvial systems may have substantial tributary junctions that reset fluvial grain size trends and display significant facies partitioning between depositional environments [Rice, 1998; Frings, 2008]. In such cases, a simple application of the model may be problematic unless explicitly incorporating additional sediment inputs or recognizing that the model only outputs “average” grain sizes for down-system stratigraphy. See section 8 for further discussion. To avoid these potential complications, we have chosen to apply the fining model to fluvial successions of a relatively simple sediment routing system, a streamflow-dominated alluvial fan. Such deposits are ideal for an initial field application because (i) they receive sediment from a point source and are not usually in receipt of additional sediment down-system, (ii) the two-dimensional application of the model is a reasonable simplification for the radial distribution of sediment found in such settings, and (iii) there is little or no facies partitioning in the gravel units.
 In addition to the points above, we note further simplifications in our model formulation: (7) an exponential decrease in subsidence is used because this is a simple way of varying the amplitude and wavelength of tectonic subsidence, as well as being typical for describing accommodation generation in foreland and wedge-top basins [Allen and Allen, 2005]; (8) a mass balance approach is adopted to model our field data and so we assume a closed system, which is supported by the localization and geometry of the Montsor Fan Succession within the structurally restricted Pobla wedge-top basin; and (9) our geometric modeling approach invokes a fluvial system that is always at steady state, so the transient dynamics of sediment routing systems and transient stratigraphy are unaccounted for.
3. Sensitivity Analysis
 Before applying the model to field data, we first conducted a sensitivity analysis on the model by varying several of the key parameters to explore their relative effects on the resultant spatial trend of mean grain size within stratigraphy. The parameters varied were (1) the initial value of sediment discharge qso, (2) the spatial distribution of subsidence σ(x), (3) the constant of variance partitioning C1 (Appendix A, equation (A6)), and (4) the standard deviation of the sediment supply ϕo.
3.1. Varying the Initial Value of Sediment Discharge and Spatial Distribution of Subsidence
 In order to explore the effects of varying sediment discharge and subsidence distribution on time-averaged grain size within stratigraphy, we first evaluate the case of a basin with a fixed distribution of subsidence σ(x) and a fixed system length L and vary initial values of sediment discharge qso. For the two-dimensional system considered here, we define a dimensionless parameter Fqs, which represents the ratio of qso to accommodation creation due to tectonic subsidence, i.e.,
 In a two-dimensional closed system, time-averaged sediment discharge is such that it exactly balances the rate of accommodation generation by the spatial distribution of tectonic subsidence and qs = 0 at the point where the subsidence rate falls to zero. In this scenario, the fraction of sediment supplied to the basin Fqs is equal to 1. The case Fqs < 1 represents a situation where basin accommodation is in excess of sediment discharge (underfilled basin) and Fqs > 1 represents basin sediment discharge being in excess of basin accommodation (overfilled basin). Figure 2a presents modeled grain size for a fixed negative exponential distribution of subsidence, for varying values of Fqs. Down-system distances are plotted relative to a normalized system length x*. For the case of a perfectly filled basin, as modeled in section 7.1, Fqs= 1. For runs with Fqs> 1, there is necessarily sediment left over at x* = 1 (Figure 2b).
 Sensitivity results show that increasing the initial value of sediment discharge to a basin relative to a fixed spatial distribution of tectonic subsidence has the effect of causing a decrease in the rate of grain size fining. This effect is approximately linear when Fqs ∼ 1, as can be seen when we plot the grain size at an e-folding length scale L*/e against Fqs (Figure 2c) for the model runs presented in Figure 2a. For the subsidence distribution used, these results also show that by increasing qso to an order of magnitude greater than the reference case (Fqs = 1), the rate of down-system fining becomes negligible and is insensitive to subsequent increases in the value of Fqs. This type of behavior may explain sheet-like sedimentary units that characterize far-field gravel transport [cf. Heller and Paola, 1989; Paola et al., 1992a] and suggests that preserved grain size trends in these units should not be significantly affected by fluctuations in the absolute magnitude of qso.
 In our second set of model runs, initial sediment discharge is kept constant and wavelength and amplitude of tectonic subsidence are varied (Figure 3a). In each case, Fqs = 1. Here the normalized system length x* relates to the system length where qs = 0 for the longest-wavelength/lowest-amplitude subsidence in Figure 3a. It can be clearly seen that the rate of down-system fining is greater under a short-wavelength/high-amplitude subsidence regime and that this rate decreases under a progressively longer-wavelength/lower-amplitude subsidence regime (Figure 3b). Relatively slow, quasi-linear grain size fining is associated with conditions of very low amplitude/long-wavelength tectonic subsidence, such as that characterized by cratonic basins [Armitage and Allen, 2010]. Under short-wavelength/high-amplitude subsidence regimes, sediment is exhausted (i.e., P* = 1) at progressively shorter distances, manifested as shorter system lengths (Figure 3c).
 Clearly, both the spatial distribution of tectonic subsidence and sediment discharge strongly influence the rate of down-system fining in sedimentary basins, and the behaviors described above replicate the results of previous works that have modeled grain size variation in sedimentary basins using more complex hydraulic and sediment dynamics approaches [e.g., Paola et al., 1992a; Robinson and Slingerland, 1998; Marr et al., 2000]. Importantly, our results highlight the role of sediment extraction in setting the rate of grain size fining within basins. In our model, the relative roles of qso and σ(x) in down-system fining in a variety of geodynamic settings is described by the spatial distribution of deposition R*(x*) (see Appendix A, equation (A3)) or in other words by the rate of down-system extraction of sediment, dP*/dx. The controls on R*(x*) can be appreciated from Figures 2a and 3b, where the rate of grain size decrease for given values of qso and σ(x) mirrors the rate at which sediment is extracted from the system (Figures 2b and 3c). These model results confirm that the spatial distribution of deposition and hence spatial distribution of tectonic subsidence exert a first-order control on the rate of down-system grain size fining in time-integrated stratigraphy preserved in sedimentary rocks.
3.2. Changing C1 and the Standard Deviation of the Input Sediment Supply
 Down-system fining amounts to the partitioning of the total variance in the sediment supply into local variance (at a site) and the variance down the whole system [Paola and Seal, 1995; Fedele and Paola, 2007]. For example, if the hydraulic processes permitted all sizes to be deposited at each site in the same proportion as in the supply, then, if input size distribution were uniform, so would be the distribution at every location, and hence, there would be local size variability (high variance at each location) but no down-system fining. On the other hand, if local hydraulic selectivity was perfect, there would be strong down-system fining but a single grain size at any site (zero variance at each location). Within the model, two constants, C1 and C2, describe the relative partitioning of this variance in supply, between down-system changes in mean grain size (C2) and in standard deviation at a location (C1) (equation (5)). The ratio C1/C2 is called the coefficient of variation Cv (equation (A6)). It has been observed that Cv takes only a narrow range of values: 0.7 < Cv < 1 (see Appendix A). In the case of gravels, the mean and standard deviation are observed to decay at approximately the same rate down-system; hence, the value of Cv must remain approximately constant for a fluvial system [Fedele and Paola, 2007].
 Using a constant value of Cv, we vary the value of C1 throughout the numerically derived allowable range (0.55 < C1 < 0.9) to assess the control of C1 on the rate of grain size fining within the model. Again, model runs are for a negative exponential distribution of subsidence and are computed for the case where Fqs = 1. The constant C1 can be envisaged as describing the transfer of material into the deposit, whereby higher values of C1 cause more rapid rates of down-system fining (Figure 4). In comparison to the effect of varying qso and σ(x), the value of C1 exerts a weaker control on the rate of down-system grain size fining with little difference in predicted grain size in the proximal and distal parts of the system. For intermediate-system lengths, e.g., x* = 0.4, the normalized mean grain size varies from 0.15 to 0.3 across the range for C1 modeled here.
 Another parameter that affects the spatial distribution of grain size is the standard deviation of the input sediment supply ϕo [Robinson and Slingerland, 1998; Paola et al., 1992b; Paola and Seal, 1995; Seal et al., 1997; Fedele and Paola, 2007]. The value of ϕo is an important consideration in the modeling approach as it is independent of the closed system mass balance assumption. The rates of down-system grain size fining are shown for a range of ϕo/Do scenarios in Figure 5. All runs were subjected to the same distribution of tectonic subsidence and initial sediment discharge as in the analysis of C1. For larger values of ϕo relative to Do, the rate of grain size fining increases; physically, this represents the preferential and thus more rapid extraction of the larger population of clast sizes, which are more abundant in a high ϕo/Do scenario. This demonstrates that by widening the spread in grain sizes in the sediment supply, an increased rate of down-system grain size fining results. This is important because the distribution in grain sizes supplied to sedimentary basins is not usually assumed to be a dominant control on trends in sediment caliber preserved in stratigraphy [cf. Whittaker et al., 2010] and has not been explored in models of down-system grain size fining in fluvial stratigraphy [Paola et al., 1992a; Robinson and Slingerland, 1998]. It therefore highlights the need for a more accurate quantification of initial grain size distribution of the sediment supply from modern catchments under a range of tectono-climatic conditions in order to apply this kind of modeling approach more effectively. However, for our purposes, since it is unlikely that down-channel sediment transport in gravel bed rivers could introduce more variance in grain size of the deposits than is present in the source material, it can be assumed that the standard deviation of the input sediment supply grain size must be greater than the standard deviation of the grain size of the deposits. In the absence of direct measurements of variance in sediment grain size at source, the most proximal upstream deposits can therefore be used to constrain a minimum value of ϕo/Do.
4. Study Area: La Pobla Basin, Spanish Pyrenees
4.1. Basin Stratigraphy
 The Pobla Basin is a wedge-top basin within the south verging fold and thrust belt of the Spanish Pyrenees. It formed in the hanging wall of the Boixols thrust and is bounded in the southeast by the San Cornelli Anticline (Figure 6a). In the north, the basin is partially closed by the Morreres back thrust, which displaces the Lower Cretaceous carbonates over the Triassic sequences of the Nogueres zone [Mellere, 1993]. The formation of the wedge-top basin and associated movement of tectonic structures was contemporaneous with late Eocene-Oligocene compression of the Pyrenean chain [Muñoz, 1992; Mellere, 1993; Meigs et al., 1996; Vincent, 2001]. The stratigraphy of the Pobla Basin fill is divided into five groups (Pessonada, Ermita, Pallaresa, Senterada, and Antist; Figure 6b), all of which represent terrestrial sedimentation in the wedge-top basin [Mellere, 1993]. The age of the complete basin succession is well constrained by mammalian assemblages and magneto-stratigraphic analysis and spans 15 Ma, from 42 to 27 Ma [Beamud et al., 2003, and references therein]. We present grain size data collected from conglomerates of the Upper Eocene Montsor Fan Succession of the Pallaresa group (Figure 6b), which ranges in age from 40 to 34 Ma. It represents the larger (20–30 km in length) [Mellere, 1993] of two time-equivalent streamflow-dominated sediment routing systems in the basin, the other being the Collegates–Roca de Peso fan system (Figure 6b). Together, these constitute the Pallaresa group of the Pobla Basin and overlie the Ermita group [Mellere, 1993] (Figures 6b and 6c). The transition from the Ermita group to the Pallaresa group is represented by an erosional surface, which marks a period of change in the palaeogeography in the basin and in overall sediment routing system dynamics [Mellere, 1993]. Stratigraphically, this transition is expressed by the upward passage of a thinly bedded (∼5–10 m) fan delta-lacustrine-floodplain succession, composed entirely of locally derived Mesozoic limestone material [Mellere, 1993], to a thickly bedded and amalgamated alluvial fan succession (Figure 6c) that contains clasts indicative of a source area within the Nogueres and Axial zones (Figure 6a). The scales of the depositional systems also differ significantly, while the Ermita depositional system was represented by small alluvial fans prograding into small lakes over a period of ca. 1.5 Ma, the Montsor depositional system was represented by the progradation of spatially extensive alluvial fans over a period of ca. 6 Ma, largely filling the wedge-top basin. Consequently, the Montsor Fan Succession represents a major period of change in sediment routing system dynamics, related to an intense episode of thrust activity within the antiformal stack of the Nogueres and Axial zones [Mellere, 1993; Vincent, 2001]. Activity of this sort played a crucial role in governing temporal patterns of sediment yield and supply to adjacent basins [Morris et al., 1998; Vincent, 1999] and also influenced patterns of tectonic subsidence in adjacent basins [Beamud et al., 2003].
 The Montsor Fan Succession is divided by Mellere  into three major units ( M1, M2, and M3), each one ∼200–400 m in thickness and bounded by a major lacustrine interval (Figures 6b and 6c). Here we focus on the lower unit M1, and we further divide it into two separate units, M1T1 and M1T2, which are separated by a lacustrine horizon, traceable across the length of the basin. The conglomerates of the Montsor Fan Succession are cobble to boulder grade, clast supported, and moderately sorted to well sorted and show well-developed imbrication. These local features, together with the presence of upward-fining units and common gutter casts, indicate the dominance of turbulent streamflow processes, a prerequisite for applying the selective deposition model of Fedele and Paola .
4.2. Constraints on Sediment Deposition Rates
Beamud et al. , using magneto-stratigraphic data from samples collected at a location 5 km north of Pobla de Segur (magneto-stratigraphic transect shown on Figure 6c), calculate sediment accumulation rates for the Montsor Fan Succession on the order of 0.05 mm yr−1 from ∼40 to 35.5 Ma, coinciding with the deposition of units M1T1, M1T2, and M2 (Figure 6d). In this paper, we present data from units M1T1 and M1T2 only, with thicknesses of ∼75 and ∼150 m directly north of Pobla de Segur (Figure 6c) and separation by a thickness of stratigraphy of ∼25 m. The deposition of unit M1T1 took place synchronously with a marked decrease in the rate of tectonic subsidence (Figure 6d) in this part of the Pobla Basin during an episode of major tectonic reorganization [Beamud et al., 2003]. A detailed comparison of our time lines with this magneto-stratigraphic data suggests that time surface M1T1 has an absolute stratigraphic age of 40.5 Ma (Beamud, personal communication). Using the published sediment accumulation rate of 0.05 mm yr−1 for this time period, the thickness of ∼25 m that separates M1T1 and M1T2 suggests an intervening time period of 0.5 Ma, giving an age of ca. 40 Ma for M1T2 (Figure 6d).
 The Montsor Fan Succession provides an excellent natural laboratory to investigate the dynamics of time-integrated grain size variation in fluvial stratigraphy because of (1) excellent exposure over the entire length of the fan system, allowing collection of grain size data and the inference of system length; (2) the presence of laterally extensive lacustrine and paleosol horizons that enable correlative horizons or “time surfaces” to be traced across the basin; (3) a constraint on local sediment accumulation rates (and hence tectonic subsidence rates) from magneto-stratigraphy [Beamud et al., 2003]; (4) the geometry and structural restriction of La Pobla wedge-top basin, as inferred from the distribution of the Montsor Fan Succession [Mellere, 1993], which enables us to be confident in our closed system modeling assumption for the gravel fraction; (5) the recognition of sedimentary features that are associated with turbulent, fluvial sediment transport and deposition; (6) the wedge shape geometry of major sedimentary units (fan lobes) and palaeoflow directions, which supports a point source sediment delivery to the system with no additional sediment input down-system; and (7) knowledge of the location of the fan apex (x = 0) over time in order to establish an absolute distance for each grain size measurement (Figure 6e).
5. Methods: Data Collection and Analysis
 As this study attempts to explain the trends in grain size in preserved fluvial stratigraphy, it is necessary to collect grain size data that (1) is representative of a significant period of geological time and (2) follows units along discrete time horizons or time surfaces. The former requirement means that the details of individual flows and high-frequency climatic variations are integrated and therefore “smoothed out” over geological time, essential if the quantitative model is to be applied. For documented sediment accumulation rates in the Montsor Fan Succession on the order 0.05 mm yr−1, ∼5 m of stratigraphy would be accumulated in 105 years. We therefore chose to consider grain size variations within ∼10 m of vertical stratigraphy at each sample site, therefore averaging grain size in stratigraphy over time periods of ∼0.2 Ma. To gain an accurate representation of the spatial arrangement of grain size trends in stratigraphy without incorporating short-term temporal variations, it is essential that sample locations follow key time surfaces. Although the term “time surface” is not be applicable over the short term, e.g., <104 years, due to diachroneity, it is applicable here as we are sampling grain size over a substantial period of geological time and so limit potential diachroneity. Time surfaces were defined several meters above the base and below the top of major stratigraphic units in order to limit the uncertainty in grain size values that may result purely from fan lobe establishment. We present spatial grain size data for two time surfaces at the base of unit M1T1 and the base of unit M1T2 (Figure 6e).
 Sampling the outcrop for grain size distribution involved the random selection and measurement of the long and short axis of 100 clasts, > 2 mm, within a 1 m2 sample area [Wolman, 1954; Stock et al., 2008]. This was undertaken both manually and digitally, using scaled photographs [Buscombe, 2008; Whittaker et al., 2010], with both methods producing statistically similar distributions. There are various ways to estimate a time-averaged grain size distribution curve for the 10 m2 area without sampling the entire area. Given a physically based assumption for a theoretical probability distribution, in this case lognormal, the best robust estimate is a least squares fit to estimates of the most widely spaced points on the distribution. To obtain a three-point estimate, we therefore measured the visually finest, intermediate, and coarsest 1 m2 areas within the 10 m2 area and fitted a lognormal pdf (probability density function) to generate a representative curve for the 10 m2 area.
6. Grain Size Data
 The spatial grain size characteristics are represented by plots of D50 and D84 against the down-system distance of the system for each of the M1T1 and M1T2 time surfaces. The ratio of D84 to D50 remains approximately constant (∼1.8) for each of the time surfaces. Data sets for each of the time surfaces are best represented by an exponential regression curve. Figure 7 shows these curves with associated errors of the coefficient (initial mean grain size, ) and exponent (fining rate, αg). The initial mean grain size associated with time surfaces M1T1 (Figure 7a) and M1T2 (Figure 7b) are 87.6 (±23.2) and 166.1 (±41.7) mm and the observed rates of fining for each are -0.09 (±0.03) and -0.25 (±0.05) km−1, respectively. The initial grain size and the fining rate for M1T2 and M1T1 are significantly different (no overlap of 95% ranges). An additional important observation is that greater values of and of αg are associated with a smaller system length L. We also note that both the initial grain size and the fining rate for the two units are significantly different.
6.2. Similarity Variable and Values of Constants
 A physical phenomenon or property is said to be “self-similar” if it appears to be temporally or spatially invariant, i.e., it looks the same at each point, or equivalently, if there exists a (spatial or temporal) transformation which makes this happen (Appendix A, equation (A1)). It is important to assess whether the grain size distributions in stratigraphy are actually self-similar because this assumption forms the basis of the modeling of fluvial sediment fining introduced by Fedele and Paola . By plotting the similarity variable ξ against the fraction f of a particular grain size D of a grain size population for a sample site along the time surfaces, a general agreement between observed and theoretical deposit grain size distributions of time surface M1T1 (Figure 8a) and time surface M1T2 (Figure 8b) is demonstrated. However, variability between individual sample populations of each time surface is still apparent. This variability is associated with the value of Cv (equation (A6)), which is assumed to be constant within the self-similar solutions and is a crucial step to the definition of ξ. Grain size data collected from the gravel fraction for both time surfaces show that Cv, calculated directly as the coefficient of variation, has a mean value of 0.7 and associated standard error of ±0.16, placing the data set in the lower end of the empirically and theoretically derived range of Cv values: 0.7 < Cv < 1 [see Fedele and Paola, 2007]. However, deriving C1 and C2 inversely from the best fit to the mobility function (Figure 8) leads to values of C2 < C1 (i.e., Cv > 1) as opposed to Cv < 1, which we would expect based on results of flume and numerical experiments on idealized pure gravel input. We interpret this as being due to the presence of a significant sand fraction (> 0.2) [Wilcock, 1998] within the gravel-dominated fluvial system [cf. Fedele and Paola, 2007]. The presence of a mixed gravel sand system generates variability in the value of dimensionless critical Shields stress due to “enhanced” surface sorting and hiding effects [Wilcock, 1998; Robinson and Slingerland, 1998]. As the similarity solution is based on the assumption of constant dimensionless Shields stress for gravels (Appendix A), any variability will cause variation in Cv, indicating departure from perfect similarity. Our methods of data collection are sufficiently robust to refute the notion that data collection contributed to the observed variability in Cv and ξ, although the data collection focused exclusively on the gravel fraction.
 Notwithstanding the significant differences from idealized end points we have noted, the behavior of deposit grain size distributions from time surfaces M1T1 and M1T2 of the Montsor Fan Succession is inferred to be a good approximation to self-similarity (Figures 8a and 8b), lending support to the usage of ξ as an appropriate scaling parameter for longitudinal grain size profiles in ancient fluvial successions. The best fit analytical curve [see Fedele and Paola, 2007, equation (23)] to the similarity plots for time surfaces M1T1 (Figure 8a) and M1T2 (Figure 8b) yields a value for Cv of 1 < Cv < 1.1, from values of 0.5 for C1 and 0.45 < C2 < 0.5 for C2.
 Irrespective of differences in between each of the time surfaces, the consistency of the ratio of D84/D50 (∼1.8) for each of the time surfaces suggests that the input sediment characteristics remained similar in terms of standard deviation, i.e., the coefficient of variation Cv remained approximately constant. This is ideal in terms of model application and comparison of model results between each of the time surfaces, as the sorting of the initial sediment input ϕo (of the gravel fraction) is unlikely to have varied significantly. Additionally, this shows that one of the key assumptions of Fedele and Paola  (i.e., approximate constancy of the coefficient of variation throughout the gravel system) holds in this particular field case.
7. The Role of Tectonics
7.1. Model Results
 We applied the down-system fining model, described in section 2 and Appendix A, to quantitatively assess the spatial distribution of tectonic subsidence and sediment discharge within the Pobla wedge-top basin during the deposition of time surfaces M1T1 and M1T2. Under the assumption of a filled but closed system, the magnitude of initial subsidence rate σo has no effect on the spatial distribution of mean grain size , because by definition the increase in accommodation generation must be balanced by a rise in sediment discharge to the basin. Instead, the variable that controls is the exponent αs in equation (2), which governs how subsidence is distributed in the down-system direction. Thus, it follows that if data are collected to show the spatial distribution of grain size in a sedimentary basin then the spatial distribution of subsidence can be found either by curve fitting or by numerically solving the set of equations in Appendix A and section 2. In this case, we determine a best fit negative exponential subsidence curve and associated sediment discharge, assuming Fqs = 1, for two end-member grain size curves deduced from the data presented in Figure 7. These end-member curves represent (1) maximum initial , maximum fining rate (dotted curve, Figures 9a and 9b) and (2) minimum initial , minimum fining rate (dashed curve, Figures 9a and 9b).
 From magneto-stratigraphic data [Beamud et al., 2003], we estimate temporal sediment accumulation rates for both time surfaces M1T1 and M1T2 of ∼0.05 mm yr−1 at a down-system distance of 7 km from the fan apex (Figures 6c and 6d). We make two first-order assumptions: (1) the Pobla wedge-top basin was a closed system during “Montsor times” and (2) derived sedimentation rates are equivalent to subsidence rates at that point. These assumptions and the published data constraints on sediment accumulation rates within the basin allow us to deduce absolute values of σ(x) and qso. Model results for grain size profile M1T1 generate values for initial subsidence rate of 0.08 < σo < 0.16 mm yr−1 and initial sediment discharge of 655 < qso < 688 m2 kyr−1 (Figure 9a). Modeled rates of subsidence for the M1T2 grain size profile generate initial subsidence rates of 0.29 < σo < 0.8 mm yr−1 and initial sediment discharge of 797 < qso < 1433 m2 kyr−1 (Figure 9b). The modeling results therefore suggest that the amplitude of maximum subsidence within the Pobla Basin increased by a factor of at least 2 between the deposition of M1T1 and M1T2 and sediment discharge to the basin increased by a minimum of 17%.
 Our approximation of an exponential decay of subsidence rate over the depositional length means that sediment is extracted at an exponentially decreasing rate down-system. Following the discussion in section 3.1, the rate of down-system grain size decrease must exactly match that of the exponential down-system decrease in the rate of sediment mass extraction from qso. The down-system decrease in mean grain size observed along time surfaces M1T1 and M1T2 is best described by a negative exponential function, consistent with the use of an exponential function to describe the distribution of tectonic subsidence.
7.2. Are These Model Results Reasonable?
 The Pobla Basin sediments were deposited during the period of movement on both the Boixols thrust to the south and the Morreres thrust bounding the north of the basin (Figure 6a) [Mellere, 1993; Sinclair et al., 2005]. Our grain size data and modeling results suggest that proximal subsidence rates in the Pobla Basin increased by a factor greater than 2 from the deposition of unit M1T1 to the deposition of M1T2, and at the same time the length scale of the sediment routing system shortened by a similar factor from ∼40 to <20 km, consistent with higher-amplitude, shorter-wavelength subsidence in the basin. A key question is therefore to evaluate whether these results are geologically reasonable and supported by sedimentological evidence?
 As a perched piggyback basin within the southern Pyrenean fold and thrust belt, the local distribution of subsidence depended upon deformation and movement on the basin bounding faults, which warped the “basement” Mesozoic carbonates (Figure 6c) rather than being controlled by the regional flexural response of the crust to the topographic load of the Pyrenean orogen over length scales of ∼100 km [Gaspar-Escribano et al., 2001; Allen and Allen, 2005]. The modeled increase in amplitude of the proximal subsidence rate in the north of the basin from M1T1 to M1T2 by a factor of at least 2 argues for an increase in the rate of movement of the Morreres back thrust at ca. 40 Ma [Beamud et al., 2003]. A number of strands of independent evidence support this interpretation. First, the observed thicker, more prominent appearance and shorter system length of unit M1T2 relative to M1T1 explicitly demonstrates increased rates of sediment accumulation in the proximal part of the Pobla Basin, consistent with an increased rate of tectonic subsidence to provide the accommodation space required. Second, the subsidence wavelengths and subsidence values estimated from the modeling are consistent with the length (10–50 km) and sediment accumulation rates of other wedge-top basins in the Spanish Pyrenees [Hogan and Burbank, 1996; Sinclair et al., 2005]. Moreover, we know that sediment accumulation rates varied in the Pobla Basin through the Upper Eocene (Figure 6d), increasing by a factor of 4 in the middle of the basin from 0.05 to 0.18 mm yr−1 during the deposition of the thick basin filling M3 unit (Beamud, personal communication). These observations are best explained by a change in tectonic subsidence rates [Mellere, 1993]. Additionally, the statistically significant increase in the initial grain sizes preserved in proximal deposits for unit M1T2 attests to a clear temporal change in sediment caliber released into the basin. Larger values of proximal grain sizes necessarily reflect the export of coarser sediment from upstream, typically because the hinterland fluvial network is adjusting to a change in the rate of uplift upstream [e.g., Whittaker et al., 2010]. The data are also consistent with the documented cessation of shortening on the Boixols thrust at or around 40 Ma [Sinclair et al., 2005], while activity increased on the Morreres back thrust [Mellere, 1993; Vincent, 2001]. We therefore consider that our inversion of the grain size fining curves for tectonic subsidence rates is reasonable.
 On the basis of the observation that greater values of and of αg are associated with a smaller system length L, we may infer a dynamic coupling between , αg, and L. This relationship is based primarily on tectonics, because the rate of hinterland uplift and rate of basin subsidence generate a dynamic gradient along the sediment routing system [e.g., Whipple and Trayler, 1996; Dade and Verdeyan, 2007]. Values of αg and represent measurable proxies for this dynamic gradient at the interface between the erosional engine and the depositional system of a small-scale sediment routing system. Absolute dating of the time surfaces as described in section 4 tells us that the reorganization of the Montsor sediment routing system, in response to increased thrust activity at ca. 40 Ma, took place over a time period of ca. 0.5 Ma from the time of deposition of time surface M1T1.
 Although fluctuations in climatic boundary conditions over this time period may have influenced sediment discharges to the basin on a timescale of 104–105 years, we believe that tectonic activity was the primary control on down-system grain size fining of time surfaces M1T1 and M1T2. There is no evidence for a dramatic climate shift in the Pyrenees at 40 Ma, while a large increase in sediment flux in the absence of a concomitant increase in accommodation generation would result in the progradation of thinner, laterally extensive units, which we do not see [cf. Heller and Paola, 1989, 1992].
8. Discussion and Future Work
 The ability to directly couple fluvial successions, expressed here in terms of down-system variation in grain size, with formative time-averaged surface processes is a crucial step toward our understanding and quantification of sediment routing system dynamics. Such insights are required if we are to predict the distribution of grain sizes in sedimentary basins or, conversely, if we are to invert grain size trends for key surface process and tectonic variables in a wide range of settings. Self-similar solutions for down-system grain size fining (Appendix A) are an important step toward fulfilling this goal, as they only require input parameters that are readily attainable from stratigraphic field studies. While the development of the similarity solutions used here depended on the results of fluvial process models over short time periods in order to define a relative mobility function, it is import to reiterate that these solutions are developed for fluvial successions over significant periods of geological time.
 By undertaking a sensitivity analysis of key parameters within the model, we are able to show the impact of sediment discharge, the spatial distribution of tectonic subsidence (and hence the spatial distribution of deposition), input sediment supply characteristics, and “sediment transfer” constants C1 and C2 on the spatial trends in down-system grain size. These results highlight the extent to which spatial trends in sediment caliber in sedimentary basins can be interpreted in terms of key controlling variables and provide a basis for future application to a range of basin types with significantly different regimes of tectonic subsidence, sediment discharge, and sediment supply characteristics. Of particular note is the fact that modeled grain size trends are very sensitive to changes in the magnitude of sediment supply when the basin is approximately full (i.e., Fqs ∼ 1) but are relatively insensitive to fluctuations in sediment discharge when Fqs is large (i.e., > 10), and there is already significant bypass from the system. Such results are important when considering sedimentation in areas where there is little accommodation generation. In general, the results of the sensitivity analysis are consistent with the results of more complex numerical models of down-system grain size fining [e.g., Parker, 1991; Robinson and Slingerland, 1998] and highlight the need to explore in more detail how the distribution of sediment size, in addition to the magnitude of supply, affects grain size trends in sedimentary basins [cf. Whittaker et al., 2010].
 In particular, more theoretical and field data are needed to further constrain the values of the “constants” C1 and C2, and hence Cv. In the original model, a range of numerical values for C1 (0.55 < C1 < 0.9) was determined from numerical experiments and limited field data [see Fedele and Paola, 2007]. Values of C1 and C2 within a given fluvial system are likely to be characterized by a specific section of this range, i.e., the values of C1 and C2 are unlikely to be universal [cf. Fedele and Paola, 2007]. Analytical best fits to our gravel (field) data set are provided by values of C2 < C1 (i.e., Cv > 1) as opposed to C1 < C2 (Cv < 1), derived from flume and numerical experiments on idealized pure gravel input. This is in contrast to a value for Cv of 0.7 ± 0.16 obtained from the raw analysis of the field data set, which was undertaken on the gravel fraction only. We attribute this difference as being due to the presence of a significant sand fraction. While our field data are consistent with an approximately self-similar distribution of grain size down-system, the impact of departures from this assumption requires further investigation. In addition, a combined theoretical and field study that investigates the form of the similarity variable and the form of the relative mobility function between the two idealized end-members, pure gravel and pure sand, is needed.
 We apply and validate the similarity solutions for down-system grain size fining using the Montsor Fan Succession, Spanish Pyrenees, as a case study. This ancient fluvial succession is ideal as it adequately satisfies all modeling assumptions (section 2.2). However, some of these assumptions are more limiting than others when applied to “nonideal” case studies: (1) the self-similar solutions used are strictly only valid for unimodal grain size distributions (discussed above) and (2) the system must be depositional along its entire length if down-system fining is to take place (this is not always the case; however, it must be assumed to be so if one is to “trace” a grain size time surface across the system length). The similarity solutions for grain size account for short-term erosion during flood events but cannot account for basin scale incision over the long term due to base level fall. However, as long as a grain size time surface is defined within a discrete sedimentary package bounded by basin scale erosion surfaces, the model may still be applied. There are two situations where this assumption is most limiting: (1) when the time frame of periodicity of incision at a site is less than the stratigraphic time frame investigated (δt) and the depth of incision is greater than the chosen vertical dimension of the sampled outcrop area; (2) when incision takes place in proximal regions and time-equivalent deposition takes place farther down-system. In the latter case, the length of the entire fluvial system will increase, and the onset of deposition is forced basinward. It is at the point of depositional onset that the solutions recognize the start of the depositional fluvial system. Additionally, as proximal incision liberates previously deposited grains, the sediment supply characteristics to this new fluvial system must be modified; (3) the mechanical breakdown of particles (abrasion) is unaccounted for; down-system fining in natural fluvial systems is driven by some combination of selective transport deposition and abrasion, with the mechanism of selective transport deposition dominating [Parker, 1991]. Although abrasion alone cannot reproduce observed down-system fining trends in experimental and field data [Seal et al., 1997], its impact on the rate of down-system fining when coupled to a model of selective transport deposition is potentially large [Parker, 1991]. The relative contribution of abrasion to the rate of down-system fining will increase with increasing distance from source, and so larger length scale fluvial systems are more likely to be affected by abrasion. This must be accounted for in the similarity solutions as it will indirectly affect the values of “constants” C1 and C2 down-system; (4) there are no additional lateral inputs of sediment other than that at the upstream boundary; although this can be justified for the simple catchment-fan sediment routing system used in this investigation, the assumption is limiting for larger length scale fluvial systems where tributaries deliver additional sediment to the main trunk rivers. In such cases, a simplistic application of the model may not be valid without explicitly incorporating additional sediment inputs or recognizing that the model outputs are representative of an “average” down-system grain size trend in fluvial successions. If sediment inputs via tributaries are geographically stable over substantial periods of geological time, there can be no question that these inputs will affect the time-averaged rate of down-system grain size fining in fluvial successions, but the degree of this influence will vary considerably [cf. Rice, 1998]. If sediment input (discharge and caliber) released from tributaries is significant over geological time periods, this will inevitably lead to underestimates or overestimates of the spatial distribution of tectonic subsidence, depending on tributary spacing and density down-system [cf. Rice, 1998]. It should be recognized, however, that any study that attempts to predict down-system fining in fluvial successions will always encounter this problem; (5) the relative partitioning of grain size into distinct fluvial facies is unaccounted for; the model output is represented by a single value of mean grain size at any given point in space and time. In the absence of a self-similar solution for facies proportions down-system, this value represents the mean of all grain size distributions in each facies type across a river channel cross section. This is a serious limitation that must be addressed if the similarity solutions are to be applied to the successions of larger sand-rich fluvial systems where the degree of facies partitioning is greater than that of small length scale gravel systems. A key challenge is therefore to understand quantitatively how particles are transferred and partitioned into different sedimentary facies and how these sedimentary facies are apportioned at given distances down-system under different spatial distributions of deposition/tectonic subsidence. In keeping with the simplicity of self-similar solutions, it has been shown experimentally that the spatial distribution of deposition determines not only the spatial trends of grain size in basinal fluvial successions [Paola and Seal, 1995] but also the spatial organization of sedimentary facies in these successions [Strong et al., 2005]. This is the most logical direction of future research in this area.
 It should be noted that the severity of the limitation imposed by one or more of the modeling assumptions is strictly dependant on the particular field case under investigation. Furthermore, it is important to recognize that even the most complex process models that strive to accurately represent every natural fluvial phenomena, although ideal for understanding fluvial system behavior, are still hindered by the fact that flow phenomena, such as tributary inputs, cannot be measured from the fluvial rock record. Self-similar grain size solutions therefore provide an elegant and simple approach for the direct comparison of theoretical grain size distributions with real-world grain size distributions in fluvial successions.
 Understanding the controls on time-averaged grain size preserved in sedimentary deposits in the geological record is a key challenge within the Earth Sciences. In this paper, we assess and explore the applicability of self-similar solutions to down-system grain size fining [Fedele and Paola, 2007] in fluvial successions, as they provide an elegant and powerful means to quantify the key geological controls that can be measured from fluvial successions. A sensitivity analysis of the grain size solutions shows that predicted time-averaged grain size trends in sedimentary basins are controlled to first order by (1) the sediment discharge into the system, where an increase in the initial value of sediment discharge to a basin causes a decrease in the rate of grain size fining in fluvial successions, an effect that is nonlinear at large values of initial sediment discharge; (2) the amplitude and wavelength of tectonic subsidence, where short wavelength/high amplitude subsidence regimes generate a greater rate of down-system grain size fining and vice versa; and (3) the variance in grain size supplied to the system, where an increase of the spread in grain sizes in the sediment supply generates a greater rate of down-system grain size fining. Knowledge of these three primary controls are all that is required to predict the spatial distribution of grain size in fluvial successions, raising the prospect of its application to a wide variety of geodynamic settings. We calibrate and apply this model to the Upper Eocene Montsor Fan Succession of the Pobla Basin, Spanish Pyrenees, where key variables are well constrained. Our observation that larger system lengths correspond to lower rates of down-system grain size fining and that spatial grain size data can be approximately collapsed via the similarity variable for gravels offers support for the usage of self-similar grain size solutions in fluvial successions. From detailed measurements of the spatial distribution of time-averaged grain size along two time surfaces within the succession, we observe a vertical change of (1) an increase in the rate of down-system grain size fining, (2) an increase in the caliber of material supplied to the basin, preserved in proximal deposits, (3) a decrease in depositional system length, and (4) a twofold increase in proximal unit thickness. This vertical change in system properties between the time surfaces took place over a period of ca. 5 × 105 years from 40.5 to 40 Ma, coincident with a period of major thrust activity and uplift. Using the self-similar solutions, we decode each of the grain size trends for the spatial distribution of tectonic subsidence and the time-averaged sediment discharge to the basin. This suggests an increase in the absolute subsidence rate by a factor of >2 and an increase in sediment discharge to the basin of >17% and is consistent with distribution of sediment thicknesses found in the basin.
 The combination of careful grain size data collection, knowledge of system length, and good stratigraphic age constraints provides a good working framework for the identification of distinct periods of sediment routing system reorganization from fluvial successions. Self-similar grain size solutions incorporate this simplicity and present a powerful means of decoding and quantifying key controlling variables of grain size trends in fluvial successions. We have shown here that, through detailed and careful collection of data on grain size trends in fluvial successions, a quantitative picture of the dynamics of sediment routing systems through time can be decoded from fluvial successions. Key challenges for the future are to apply this methodology to basins where accommodation generation is known to be constant and where the dominant control on the deposition of sedimentary packages is sediment discharge from hinterland catchments and to apply the model to larger sand-dominated fluvial systems.
A1. Self-Similar Grain Size Distributions and the Similarity Variable
where D is a given grain size and and ϕx* are the local mean and standard deviation of the mixture at a normalized longitudinal location x* (x* = x/L) along a depositional system of total length L. The shape of the grain size distribution at any position along the length of the depositional system is the same when scaled by the similarity variable ξ for gravels [Fedele and Paola, 2007].
A2. Summary of Self-Similar Solution Formulation for Down-System Grain Size Fining in Fluvial Successions
 To describe the rate of down-system grain size fining in fluvial successions, we must define (1) appropriate functions that describe the precise mechanism and nature of particle transfer between the transporting system and the underlying deposit and (2) how rapidly sediment mass is extracted along the length of the depositional system. This is described by the fractional Exner sediment mass balance for the long term [Paola and Seal, 1995], takes the form
where x* = x/L, f = fraction of a given sediment size in the deposit, J = p/f (p = fraction of a given sediment size in the transporting system) defined as the relative mobility function, and R* = spatial distribution deposition down-system, given as
where λp = sediment porosity, L = length of depositional system or basin, σ(x) = spatial distribution of tectonic subsidence (L t−1), and qs(x) = spatial variation in sediment discharge (L t−1).
 The time-averaged behavior of sediment hydraulics, sediment transport, and particle transfer from transporting system to the underlying deposit through J (equation (A2)), which defines the local selectivity and sorting of particles in transport p versus that in the deposit f [Fedele and Paola, 2007]. The definition of the relative mobility function is an important step to obtaining a practical solution for grain size fining in fluvial stratigraphy, as geologists are only faced with the character of the deposit f and are only able to speculate on the character of the transporting system p. Numerical models show that the form of the J is approximately self-similar [Fedele and Paola, 2007, Figure 1], supporting the idea that time-averaged behavior of particle transport, particle segregation, and selective deposition from transport to deposit can be collapsed via the definition of the relative mobility function. If a self-similar solution for f of equation (A2) is to be found, both f and J must be expressed as functions of ξ (equation (A1)): (f(ξ) and J(ξ)).
 It is important to note that the formulation of the relative mobility function J by Fedele and Paola  differs from the formulation of the relative mobility function by Paola and Seal . Paola and Seal  develop their relative mobility function from first principles for reach scale patchiness in gravel bed rivers. One assumption in their formulation is that the transfer of sediment from the active layer to permanent storage in the deposit is free from any bias, in particular, that arising from vertical position in the active layer. Fedele and Paola  define their own value for J over the long term, using p and f (where p is the fraction in transport of a given size and f is the fraction in the substrate of that size). They used ACRONYM4 [Parker, 1991] to determine f from known values of p, thereby calculating J. A large number of experimental runs of ACRONYM4 were carried out under different conditions of sediment feed, initial size distribution, spatial distribution of tectonic subsidence, and base level changes. The aim of this was to find an underlying behavior of grain size mobility for the long term. The limitations of the “long-term” relative mobility function, as defined by Fedele and Paola  model, are therefore due to the limitations (or resolution) of ACRONYM4, in which some bias is introduced through its own transfer functions and transport relations. As ACRONYM4 accounts for detailed fluvial behavior, sediment transport, and deposition, including the bias of particle transfer from the active layer to permanent storage, so the relative mobility function of Fedele and Paola  also accounts for these.
 The distribution of deposition R* represents the relative distribution of sediment mass down-system (equation (A3)) resulting from deposition and is determined independently of the solution for the relative mobility function J due the invariant property of equation (A2) [Paola and Seal, 1995; Fedele and Paola, 2007]. Fedele and Paola  generate a solution of equation (A2) for a reference case, i.e., where they define R* = 1. A value of R* = 1 is necessary as a reference case as it physically represents a spatially uniform distribution of deposition across an undeformed surface or perfectly flat topography. To solve for deposit grain size f for any value of R*(x), the invariant property of equation (A2) means that a solution can be found under a dimensionless distance transformation y*(x*) [Fedele and Paola, 2007],
As noted by Fedele and Paola , equation (A4) describes the details of the pattern of sediment mass distribution down-system and represents the idea that greater rates of down-system fining are associated with greater rates of down-system sediment mass removal from the transporting system.
 A final and self-similar solution of equation (A2) for the deposit grain size distribution, following the definition of the relative mobility function J for the reference case of R* = 1, is given by equation (23) of Fedele and Paola . As a result of this solution, it is possible to calculate the down-system rate of grain size fining for any spatial distribution of deposition R* [Fedele and Paola, 2007],
where and ϕ0 represent the mean and standard deviation of the size distribution of the sediment supply and, as noted in section 3.2 above, C1 and C2 are constants that physically express the relative partitioning of the standard deviation in the gravel supply into that controlled by the systematic down-system change in mean grain size (C2) and that of the intrinsic variation or standard deviation at a site (C1) [Fedele and Paola, 2007]. The constants C1 and C2 are needed for a self-similar solution of equation (2) for the deposit size distribution, f = f(ξ), which depends only on the local characteristics of the sediment in transport (mean and standard deviation ϕ),
From numerical calculations, Fedele and Paola  found that a range of values exists for C1 (0.5 < C1 < 0.9) while values of C2 can be calculated from equation (A6), provided values for the coefficient of variation Cv are known. Field investigations of bed surface material in modern gravel bed rivers demonstrate a range of values for Cv (0.7 < Cv < 0.87), with an upper theoretical limit for both bed surface and deposit Cv of 1. It has been found that the rates of down-system decay of and ϕx* for gravels, both surface and subsurface, are comparable [Paola and Wilcock, 1992; Paola and Seal, 1995], i.e., Cv, remains approximately constant for a fluvial system, providing additional support to the assumption of self-similarity for grain size distributions.
apex of fluvial system
interval chosen to time-average
rate of change of bed elevation
initial sediment discharge
exponent to the decay function used to describe the spatial distribution of tectonic subsidence
depositional system length
spatial distribution of deposition
dimensionless distance transformation
rate of tectonic subsidence at x = 0
spatial distribution of tectonic subsidence
mean grain size of the sediment supply
mean grain size
down-system distribution of grain size
standard deviation of the sediment supply
standard deviation of gain size at
ratio of qso/σ(x)
a measure of the rate of decay of qso (P* = 1: sediment exhaustion)
expresses the relative partitioning of variance in the gravel into down-system change in standard deviation
express the relative partitioning of variance in the gravel supply into the down-system change in mean grain size
coefficient of variation = C1/C2
exponent to the decay function used to describe the rate of grain size fining subsidence exponent to the fining rate
fraction of a given sediment size in the transporting system
fraction of a given sediment size in the deposit
relative mobility function = p/f
characteristic grain size
bed shear stress
critical Shields stress
 We are extremely grateful for the financial support given to the sediment routing systems group at Imperial College by StatoilHydro. Springett, Smithells, and Fordyce carried out fourth year undergraduate projects at the Department of Earth Science and Engineering at Imperial College London. We greatly appreciate the insight and advice of Chris Paola (Minnesota) and Cai Puigdefàbregas, and we especially thank Bet Beamud (Barcelona) for providing us with further details on the locations of dated sections in the Pobla Basin, thus helping us locate our time surfaces within these sections. We thank Barrie Wells for a thorough presubmission review. Finally, our thanks go to Rob Ferguson, Paul Heller, and Rudy Slingerland for their reviews.