## Abstract:

Knowing the position of an apex of a distributive depositional system can provide important spatial constraints on paleogeographic reconstructions, and thus can greatly help facies predictions, at both a system and a basin scale. To date, predicting the position of an apex of a sedimentary system is often limited to generalized statements based on facies mapping and qualitative analyses of paleocurrent readings. This paper presents a user-friendly quantitative methodology based on the von Mises distribution and uses the method of maximum likelihood to obtain an estimated apex and associated confidence regions for a dataset. The methodology presented has been applied to two modern distributive fluvial systems (DFSs), the Taquari DFS, situated in southwestern Brazil, and the Gilbert DFS, situated in northwestern Queensland, Australia. The position of each apex is known for the two systems, thus allowing the accuracy of the methodology to be tested. A range of datasets, within which the amount and spatial distribution of localities were selected independently, was analyzed. The predicted apices came within encouraging proximity of the true apices, ranging in distance from 2.7 km to 40.3 km (1.6 to 23.4% of the total DFS length) away, with accuracy generally increasing with increasing dataset size and proximity to the apex. Data collected from the Late Jurassic Salt Wash DFS were also analyzed using the code. Results have helped to give better geographical constraints on the system and apex location as well as on the southern margin of the Morrison depositional basin. Although tested on modern and outcrop-based datasets from DFS, the methodology can be applied to any dataset, subsurface or surface, in which dispersion occurs from a point source, thus unlocking the potential for better paleogeographic constraint on a broad range of sedimentary environments such as deltas and submarine fans.

## Introduction

Understanding the position of an apex for a point-sourced sedimentary system, such as a distributive fluvial system (DFS), delta, or submarine fan, can significantly aid paleogeographic reconstructions, as it allows an understanding of where sediment entry points into sedimentary basins are located. This gives an understanding of sediment distribution, and therefore facies distribution, in continental, deltaic, and deep-water settings through the utilization of facies models of different depositional systems, i.e., DFS, deltaic, and deep-marine-fan facies models. The distribution of reservoirs and aquifers, prior to tectonic movement, is primarily controlled at the time of deposition, and therefore understanding the location of an apex may aid exploration efforts for key commodities. At a larger scale, it allows the basin limit to be delineated better, as the apices of distributive systems are commonly found either at or close to the basin margins, i.e., the distributive pattern marks the point at which the systems aggrade, as well as aiding the identification of areas of reservoir-rich or reservoir-deficient areas, i.e., megafan versus interfan areas.

To date, constraining the location of an apex has been achieved largely through mapping of facies distributions and analysis of paleocurrent datasets (e.g., Howard 1966; Hirst and Nichols 1986; Nichols 1987). However, facies-mapping efforts are often hampered by the quality and extent of outcrop exposures, or by limited borehole and seismic-reflection data coverage when analyzing subsurface systems. This has resulted in many authors being able to speculate only on the general area in which the source area of a system resides (e.g., Van de Graaff 1972; Turner-Peterson 1986). The analysis of paleocurrent data allows an understanding of flow direction to be gained, and thus provides a reference point for upstream and downstream facies predictions. When observations are taken at several localities, it may be possible to estimate the position of the apex and source area for a sedimentary system.

Jupp et al. (1987) presented a methodology that estimated the position of an apex statistically from paleocurrent readings for a distributive fluvial system (DFS) (Weissmann et al. 2010; Hartley et al. 2010). They applied this methodology to the Miocene Huesca and Luna DFSs situated in the Ebro Basin, northeastern Spain (Hirst and Nichols 1986) and successfully estimated the apex of the Luna DFS. The results complemented estimates based on facies distributions. The approach described by Jupp et al. (1987) is, however, an underutilized quantitative method for paleogeographic reconstruction, one that is now being pursued as it provides objective analyses based on quantitative directional statistics to estimate the position of an apex.

This paper aims to build upon the framework of Jupp et al. (1987) by revising and simplifying the models used. The robustness of the methodology is tested on two modern DFSs, for which the apex locations are known, and a rock-record example from the Salt Wash Member of the Upper Jurassic Morrison Formation, SW Utah, U.S.A. The methodology uses R (R Core Team 2014), a freely available statistical program. The information required by the end user includes paleocurrent directions and a grid reference in a simple tabular format. The code and step-by-step manual for implementation can be downloaded at http://www.mcs.st-andrews.ac.uk/∼pej/apex.html. The original data used in the analyses can be found as a supplementary file (see Supplemental Material).

## Mathematical Principles

In order to estimate statistically the location of an apex for a distributive system, and subsequent precision of any estimates made, it is necessary to assume that the data are generated by a particular statistical model (see Equations 1 through 3). Such a model will include the position of the apex, an unknown vector parameter denoted by , as well as unknown concentration parameters and mean directions in Equations 2 and 3. Standard mathematical techniques can then be used to estimate the unknown parameters and so provide an estimate and confidence regions for the apex, the details of which are given in the following section.

Field observations, i.e., paleocurrent readings, can be reduced to a set of pairs for and . Here *s* is the number of sites, is the number of directions observed at the th site, denotes the position vector (in the Cartesian plane) of the th site, and is the direction (in the interval (0, 360°)) representing the th direction at the th site. Reasonable statistical models for the directions can be based on the von Mises distribution (Section 3.5.4 of Mardia and Jupp 2000; Section 3.3.6 of Fisher 1993; Section IV(ii) of Gaile and Burt 1980; and under the name “circular normal distribution” in Section 2.2.4 of Jammalamadaka and SenGupta 2001; and Section 10.3 of Blaesild and Granfeldt 2003). We assume that (i) the directions (angles) are independent observations, (ii) at each site , the angles are observations on a von Mises distribution with probability density function (p.d.f.) of

where (with 0 ≤ ≤ 360°) represents the mean direction at the th site, (with 0 ) is a measure of concentration about , and (·) is the modified Bessel function of the first kind and order zero. A plausible statistical model, which we denote by for the takes the p.d.f. to have the form Equation 1, where (**c**) is the direction from **c** to and can take any non-negative values. The concentration of the distribution around increases as increases: for near 0, the distribution of is almost uniform; for large, is approximately normally distributed with mean and standard deviation . In a simpler model, , the mean directions are (**c**) and . For example, one might expect concentrations (as measures of inverse within-site paleocurrent variability) to be more variable in the distal domain than in the proximal domain of a DFS. In this respect model permits concentrations to vary across a DFS, while model assumes equal concentrations for simplicity. To assess how well model fits the data, model is needed, in which and can take any appropriate values. Model does not assume a different apex for each observation station, it merely allows not to point directly towards the single apex. These models are listed in Table 1, in which the subscript on model denotes the number of parameters involved.

In any of the models, inference on the unknown parameters is carried out using the likelihood function. Given the observed directions for and , the likelihood function is defined by

where is defined by Equation 1. For the models in Table 1 the likelihood function is obtained from (2) by expressing in terms of the appropriate parameters.

For example, for ,

In any of the models, a suitable method (with particularly desirable mathematical properties) for estimating the unknown parameters is maximum likelihood (e.g., Section 11.1 of Blaesild and Granfeldt 2003; Section 7.7 of Lindgren 1993; Section 3.2.2 of Lindsey 1996), i.e., the estimates of the parameters are the values that maximize . Thus, for , the estimates of maximize Equation 3. In either model, the vector is the estimated position of the apex, **c**.

An idea of the precision of the estimate, can be obtained through confidence regions (see Section 7.5.2 of Lindsey 1996, for a detailed explanation of confidence regions). Intuitively speaking, a confidence region is a region in which the user can be, for example, 95% confident that the true apex (**c**) lies within the region. Confidence regions (99%, 95%, and 90%) are generated by the code provided.

Approximate confidence regions for **c** in models and can be obtained from the likelihood. If are large, then the regions

for and

for , are approximate % confidence regions for **c**. Here denotes the upper % quantile of the distribution (e.g., ). Various standard optimization routines are available and can be used to obtain the maximum likelihood estimates and the confidence regions for **c**. We find it convenient to use the code that we have written for this in the R language (R Core Team 2014) for statistical computation.

Once the parameters have been estimated, the questions arise: (i) whether fits the data nearly as well as , (ii) whether fits the data well. These can be answered using the likelihood. For *p*, let denote the maximum value of the likelihood for model . Since can be obtained by restricting the values of some parameters of and can be obtained by restricting the values of some parameters of we have The questions are whether is significantly bigger than and whether is significantly bigger than . Define

The deviances *D _{same}* and

*D*measure the differences between and , and between and , respectively. The method of likelihood ratio tests allows a choice to be made between nested models (e.g., Section 12.5 of Gilchrist 1984) and leads to the following tests. If

_{fit}*D*is large compared with the distribution, or equivalently the -value (the probability of getting a value of

_{same}*D*at least as large as that observed) for testing against is small, then we conclude that gives a significantly better fit to the data than does, i.e., it is worth the extra complexity of in order to obtain a better fit. If the

_{same}*P*-value given by

*D*is large, then we conclude that it is worth sacrificing the better fit of in order to describe the data more concisely by . Similarly, if

_{same}*D*is large compared with the distribution, or equivalently if the -value for testing against is small, then we conclude that it is worth the extra complexity of to get a better fit. If the

_{fit}*P*-value given by

*D*is large, then we conclude that it is worth sacrificing the better fit of in order to describe the data more simply by .

_{fit}In certain contexts, some of the observations are axial, e.g., a channel-edge reading, where it is not possible to distinguish between the angles and . In this case, one can use variants of the models above, as indicated in Jupp et al. (1987).

## Application To Modern DFSs

Both the Gilbert DFS, situated in northwestern Queensland, Australia, and the Taquari DFS, situated in southwestern Brazil, (Fig. 1) are covered by good satellite imagery in which the apex and channels, both active channels and paleochannels, are visible across the DFS surface. Both DFSs appear to have experienced a minor degree of human modification and thus provide suitable candidates for testing the methodology.

Grids, composed of 10 km by 10 km cells, were placed onto each DFS. Where satellite imagery allowed, 10 directional measurements were obtained along channel, or paleochannel reaches, within each cell (Fig 2). The measurements within each cell were assigned to one central grid reference to help simulate variability of flow directions typically experienced in a vertical succession at geological outcrops. A 10 km by 10 km cell size was chosen, as it was deemed large enough to capture channel sinuosity, and therefore directional variability, within channel reaches. For each DFS, the directions at 30, 20, 15, and 10 independently selected locations were selected using the “random” function in Excel and then analyzed using the methodology. An analysis of the full database for each DFS was also run for comparison purposes. Further analyses were conducted in which the positions of 20 independently selected sites were restricted to proximal, medial, or distal portions of the DFS to see how important the spatial distribution of data is when trying to estimate the position of an apex.

### Taquari DFS

The Taquari DFS has an apex-to-toe length of approximately 250 km and an area of approximately 50,000 km^{2} (Assine 2005). The Taquari DFS is situated in the Pantanal Basin, which has been characterized as being a backbulge depozone by Horton and DeCelles (1997) or a small extensional basin of the eastward-migrating forebulge of the Andes system by Ussami et al. (1999). It is currently depositing sediment under a tropical subhumid climate, with warm (32°C) and wet conditions persisting during summer months, and slightly cooler (21°C) and drier conditions prevailing in winter months (Assine 2005).

#### Results.—

All the datasets obtained from the Taquari DFS, with the exception of the Taquari proximal dataset, have P-values of < 0.05 in the “*p.fit*” column of Table 2. The small P-values indicate that, in each case, model fits the dataset significantly better than model . Thus, statistically, none of these datasets supports the notion that almost all of the mean directions point directly away from a single apex, i.e., there is local variability in paleocurrent directions that deviate away from a perfect fan morphology. This is to be expected in natural systems and is illustrated in Figure 3 in several plots. However, taking a broad view of the datasets it is clear that, generally, the mean directions suggest the presence of a single apex. To obtain this conclusion statistically would require the use of more sophisticated models than we have considered here; the few simple models considered here have been chosen for their general applicability to a host of systems. Based on the assumption that a single apex is present, the small P-values (< 0.05) in the “*p.same*” column of Table 2 suggest that model fits the data significantly better than model does for Taquari full, 30, 15, and 10 datasets. The Taquari 20, medial and distal datasets however, are different in that a P-value of > 0.05 is observed, suggesting that model fits the data almost as well as model , indicating that the locations have approximately equal concentrations, and thus model can be used.

For the Taquari proximal dataset, the value of *p.fit* is > 0.05 (Table 2), indicating that model fits this dataset almost as well as model supporting statistically the notion that the mean directions point almost directly away from a single apex. Once model has been selected, the question arises of whether or not model would fit the data almost as well as model , i.e., whether or not the locations have approximately equal concentrations. The value of *p.same* is < 0.05, demonstrating that model fits the dataset significantly better than model , indicating the locations have significantly different concentrations, so that model fits the data better.

In order to determine how close the estimated apices were to the true apex, the estimated apex from the best-fitting model from each dataset was plotted onto the Taquari DFS. For clarity, the estimated apices for models that did not fit the data best were not plotted. However, as can be seen in Table 2, they are often situated close to the estimated apex given by the best-fitting model. As can be seen in Table 2 and Figures 3 and 4, and as would be expected, the Taquari full dataset estimated the apex closest to the true apex, placing it 13.1 km away, closely followed by datasets Taquari proximal (15.4 km away) and Taquari 30 (19.7 km away), all of which have confidence regions that contain or are very close to the true apex. The analysis of dataset Taquari 10 estimates the apex the farthest away, placing it outside of a realistic region, followed by Taquari medial (42.4 km away), Taquari distal (40.3 km away), Taquari 15 (34.2 km away), and Taquari 20 (30.6 km away), with confidence regions again containing or being close to the true apex. The confidence regions are larger for the datasets in which the estimated apices are farther from the true apex. Although none of the datasets produces an estimated apex which is exactly equal to the true apex, the estimated apices from the Taquari full, Taquari 30, Taquari proximal, and Taquari distal datasets come within a credible distance of the true apex, i.e., ranging from as close as 5.2% of the total DFS length away from the true apex to < 10%, with all other datasets lying within 17% of total DFS length.

As can be seen in Figures 3 and 4A, nearly all datasets place the apex north of the true apex. Presumably this is a consequence of recent incision on the DFS where the incised portion has a sharp bend northwards of the old apex (Fig. 4). The channels that radiate outwards from the currently more active lobe of the DFS is therefore causing a skew in results, whereby the estimated apex is farther north than the true apex. This bend is evident in satellite imagery (Fig. 4), and was not sampled in any of the datasets, with the exception of the Taquari full dataset. In the case of the Taquari full dataset, although the sharp bend is visible in Figures 3A and 4A, the methodology gives observations at sites near the bend the same weighting as all other sites, and thus these observations are eclipsed by the other readings. Despite placing the apex north of the true apex, the estimated apices from datasets Taquari full and Taquari 15 place the apex very close the basin margin edge (1.3 km and 14.5 km away), with all other datasets, with the exception of Taquari 10, placing the apex < 34.8 km away (an error < 14% of total DFS length).

When analyzing estimates for the Taquari 10 dataset, an unrealistic estimate is provided by the code. This is considered to be the result of nearly four of the ten arrows (located in the southwestern portion of Fig. 3E) being nearly parallel to one another. As the directions of nearly half the dataset point in a southward direction, the estimate of the apex is pushed to the northeast. Because the directions are almost parallel, they give little information about how far away the apex is. This results in uncertainty and an unrealistic position of the apex in a northeast–southwest transect. Interestingly, unrealistic estimates are also given for model for Taquari 30 and 15 datasets (Table 2). We speculate that, due to model giving credible results, the concentrations, which is what differentiates model from model , are very different, thus the simpler model provides an improbable result.

### Gilbert DFS

The Gilbert DFS is smaller than the Taquari DFS, possessing an apex-to-toe length of approximately 170 km and an area of approximately 12,500 km^{2}. The Gilbert DFS is currently depositing under a tropical savannah climate in which monsoonal climate conditions prevail in an intracratonic basin that has been slowly subsiding since the late Tertiary (Jones et al. 1993). This tectonic framework coupled with continued sediment supply has resulted in the progradation of the Gilbert DFS into the relatively shallow Gulf of Carpentaria (Nanson et al. 2013), where the toe of the DFS is currently being reworked by marine processes.

#### Results.—

The P-values shown in Table 3 indicate for all of the datasets that model fits the data significantly better than model , implying that assuming that the mean directions point away from a single apex is too simple an approach to the data, due to local variability in mean directions. However, when it is assumed that the mean directions point away from a single apex, a safe assumption to make when taking a generalized overview of the datasets (Fig. 5), gives a significantly better fit than model for all datasets, excluding Gilbert medial, where model fits the data best.

As can be seen in Table 3 and Figures 4 and 5, the Gilbert proximal dataset estimates the position of the apex most accurately, placing it only 2.7 km away from the true apex, followed by the datasets Gilbert full (9.4 km away), Gilbert 20 (9.5 km away), and Gilbert 30 (11.3 km away). The dataset Gilbert 15 places the apex farthest away, estimating it to be 39.8 km away from the true apex, followed by datasets Gilbert medial (39.2 km away), Gilbert 10 (35.8 km away), and Gilbert distal (27.7 km away). Broadly speaking, the size of the confidence regions increases with increasing distance between the estimated apex and the true apex, except for the Gilbert medial dataset. As can be seen in Figure 5G, the randomly selected sites of the Gilbert medial dataset display a clear radial morphology, and as a result of this the estimated apex position is “pulled” closer to the medial localities and away from the true apex. As with the Taquari simulations, the estimated apices come within a distance of the true apex, with proportional margin of error ranging from 1.6% (Gilbert proximal dataset) to 23.4% of the total DFS length (Gilbert medial dataset), the majority of simulations being within 20% of the total DFS length away from the apex (Table 3).

In contrast to the majority of datasets for the Taquari DFS, for the Gilbert DFS the true apex lies within the confidence regions for all datasets except the Gilbert medial dataset (Fig 5). We consider that this is due to the more elongated morphology that the Gilbert DFS has in comparison to the Taquari DFS (Fig. 4), and therefore the estimated apices broadly align on the northwest–southeast-trending axis of the DFS with little lateral variability from this line. Thus the confidence regions contain the true apex. The central axis of the DFS has therefore been estimated accurately by the methodology with the apices varying along this axis. This is opposite to the results of the Taquari DFS datasets, which have confidence regions that possess a more circular form with predicted apices that pinpoint the location of the basin margin more accurately, but to the north of the central axis of the system. We speculate that the difference is a function of the Taquari DFS possessing a more radial rather than elongated fan morphology and a consequence of the newly developed lobe and associated incision of the trunk channel.

## Discussion

The analyses of the modern datasets indicate that although it is commonly the case that more data will provide more accurate estimates, this is not always true. For example, dataset Gilbert 20 (Fig. 5C) provides a more accurate estimate of the apex than dataset Gilbert 30 (Fig. 5B), and Taquari proximal (Fig. 3F) provides a closer estimate than Taquari 30 (Fig. 3B). When the locations of the data are restricted, datasets that use only proximal data produce the most accurate estimates for both DFSs. However, when data are restricted to just medial or distal sections, there does not appear to be a clear trend of accuracy in estimates. For example, the estimated apex for the Gilbert distal dataset (Fig. 5H) is closer to the true apex than the predicted apex for the Gilbert medial dataset (Fig. 5G), while the estimated apex for the Taquari medial dataset (Fig. 3G) is closer to the true apex than is the estimated apex from the Taquari distal dataset (Fig. 3H). We consider that the analyses of the proximal datasets give the most accurate estimates, because the channel orientation readings have a radial pattern over a much smaller area than in the medial and distal datasets. Therefore, by limiting the area the accuracy of the estimate of the true apex increases. The most accurate results obtained are generally from datasets that express a consistent radial pattern over a good spatial area which includes proximal localities, such as the Taquari medial and 30 datasets (Fig. 3G, B) and the Gilbert proximal and 20 datasets (Fig. 5F, C). However, all other datasets give encouraging results, in that all estimated apices, with the exception of those from the Taquari 10 dataset, are < 17% of the total DFS length away from the true apex for the Taquari DFS, and within 24% of the total DFS length away from the true apex for the Gilbert DFS.

Based on the analyses of datasets from the Taquari and Gilbert DFSs, the methodology provides credible estimates of the position of an apex. Statistical models with more parameters may give even closer estimates of the apex by adding extra parameters and rules that will take local conditions into account. For example, marine reworking and its effect on paleocurrent directions at the toe of the Gilbert DFS could be incorporated into the models. Also, heavier weighting could be provided to readings that highlight the change in the main trunk channel direction from northwest to west on the Taquari DFS. However, being able to make such observations within the geological record is rare, and it is not the aim of this paper to pinpoint the exact location of the apex for the two systems, but rather to provide a general methodology that can be applied to a host of systems that were deposited under varying local conditions.

It is important to stress that the models and assume that the mean directions point away from a single apex, and thus if the estimated apex position is to be used, it is up to the user to decide that such an assumption can be made based on geological observations and the use of *p.fit*.

## Application To the Rock Record—The Salt Wash DFS

To apply the methodology outlined above to an ancient dataset, we conducted an analysis on data collected from the Salt Wash fluvial system. Craig et al. (1955) and Mullens and Freeman (1957) interpreted the Salt Wash Member of the Upper Jurassic Morrison Formation as the deposits of a DFS that extended over the southwestern portion of the Colorado Plateau, USA (Fig. 6). Mapping and facies analyses conducted by Peterson (1980, 1984a), Kjemperud et al. (2008), and Weissmann et al. (2013) more recently revealed that the underlying Tidwell Member (Fig. 6), composed of minor fluvial and lacustrine facies, represents the distal facies of the prograding Salt Wash DFS. Thus the Salt Wash DFS is lithostratigraphically composed of both the Salt Wash and Tidwell members.

The details of the tectonic configuration of the western USA during the Late Jurassic are contentious (Decelles 2004; Decelles and Burden 1991; Currie 1997; Currie 1998; Heller et al. 1986; Lawton 1994; Dickinson and Lawton 2001). These authors generally agree that a broadly northwest to southeast compressional regime existed, whereby an eastward-dipping subduction of the paleo-Pacific Farallon plate occurred below the northwestward-propagating North American plate (Decelles 2004; Turner and Peterson 2004). Compression was not uniform across the margin, inasmuch as the northern portion experienced an orthogonal compressional regime which resulted in a shortening of the crust and the formation of the Sevier Highlands (Decelles 2004). The southwestern margin experienced a compressional regime that was nearly parallel to the margin, resulting in a sinistral strike-slip intracontinental rifting phase and the formation of elevated rift shoulders, the Mogollon Highlands (Dickinson and Lawton 2001; Decelles 2004; Turner and Peterson 2004). Turner and Peterson (2004) interpreted the Mogollon Highlands, bounding the southern margin of the depositional basin, and the Sevier Highlands, bounding the western margin of the depositional basin, to be key sediment sources for the Morrison Formation.

Attempts in the past to constrain the source area and apex location for the Salt Wash DFS are of a general nature. For example, Robinson and McCabe (1997) noted, based on paleocurrent data and facies observations, that the Salt Wash fluvial system had its source in an area west and southwest of Shootering Canyon, Utah, located in Figure 6. Craig et al. (1955), based on analyses of paleocurrent data, facies and thickness patterns, postulated that the apex for the Salt Wash DFS was positioned somewhere within west-central to northwestern Arizona, southeastern California, and southern Nevada. Mullens and Freeman (1957) suggested that the apex was situated in south-central Utah, based on the thickest portion of the Salt Wash being preserved there. However, the interpretation given by Mullens and Freeman assumes that the deposits are the most proximal deposits of the original system, rather than being the most proximal deposits preserved and exposed.

Other authors, such as Peterson (1984b) and Dickinson and Gehrels (2008) attempted to identify a source area rather than apex location for the Salt Wash DFS. Based on petrology and mapping studies, Peterson (1984b) stated that most of the sediment found in the Salt Wash and Tidwell members was derived from a source area situated more than 160 km southwest and 100 km west of outcrops in northeastern Arizona, with only some of the material for the system coming from older Mesozoic formations in the southwestern and western parts of the region. Dickinson and Gehrels (2008) postulated from U-Pb detrital-zircon data that it is feasible that the Salt Wash DFS could have, either wholly or partly, been sourced from the Mogollon highlands, located in Figure 7. It is clear from a review of the literature that there is a high range of uncertainty about pinpointing the apex of the Salt Wash fluvial system, with current estimates spanning an area of approximately 50,000 km^{2}.

In an attempt to constrain the location of the Salt Wash DFS apex better, we applied 749 paleocurrent readings from 27 localities across the system (Fig. 6) using the methodology outlined above. At each locality, paleocurrent readings from sedimentary structures (trough and planar cross-bedding) in different sandstone bodies were combined and assigned to one grid-reference, to maintain a system-scale overview of paleoflow direction. All paleocurrent readings were, when applicable, corrected for later structural dip. It is noted that the measurements taken from bedforms from the Salt Wash DFS are not directly analogous to the channel orientations taken from the two presented modern datasets. As noted by Miall (1974), directional variance increases with decreasing structure scale. However, identifying comparable structure scales for the ancient and modern datasets is not possible, due to the difficulties with taking cross-bedding measurements from modern systems, and alternatively channel orientations from the ancient. We envisage that by taking several measurements along channel reaches within individual 10 km by 10 km cells, and centralizing all readings to a point in the center of the cell, the variability observed within smaller structures in the rock record is imitated.

## Results and Geological Implications

As can be seen in Table 4, model fits the data better than model . Based on the analyses of facies distributions and paleocurrent readings spatially across the system, we consider it geologically sound to assume that the Salt Wash DFS had a single apex. P-values generated by the code indicate that under the assumption that the mean directions point away from a single apex, model fits the data better than model (Table 4). The estimated apex position and associated confidence regions for model are shown in Figure 7.

The estimated apex for model is located in northwestern Arizona, 160 km northwest of Flagstaff, Arizona; 250 km northeast of Las Vegas, Nevada, and 125 km southeast of Saint George, Utah (Fig. 7), which is within the predicted area given by Craig et al. (1955). The results are consistent with the geological assumptions that the Salt Wash DFS had a single apex and imply that the Salt Wash DFS had an apex-to-toe length of approximately 550 km and that approximately 150 km of proximal material is either unexposed or removed by postdepositional erosion. The results complement observations made in the field, where it was noted the most proximal deposits exposed on the system do not represent true proximal deposits in that they possess a relatively low net to gross (74% sand), which is considered to be relatively low based on descriptions given by Nichols and Fisher (2007) and Weissmann et al. (2013) for proximal DFS deposits.

Paleogeographic maps published by Decelles nd Currie (1996) and Decelles (2004) show the thrust front to be located in western Utah close to the Nevada border during the Late Jurassic, which is much farther west than the estimated apex. We therefore postulate that the Salt Wash DFS was sourced from the Mogollon–Sevier highlands syntaxis or from the Mogollon Highlands proper to the south. This is in line with statements made by Dickinson and Gehrels (2008). The results suggest that the Mogollon highlands were farther north than suggested in the paleogeographic maps of Currie (1997) and Turner and Peterson (2004), but agree with the extent mapped by Spencer et al. (2011). The results also suggest that the western margin of the basin indicated by Turner and Peterson (2004) is too far to the east. The paleogeography of the basin, and in turn the tectonics of the basin, could be further constrained with the application of the methodology presented in this paper to other fluvial systems found in the basin.

At a smaller scale, understanding the location of the Salt Wash DFS apex provides scope to build a robust facies model for the system, because sites can be more accurately placed on the system, thus enabling a more quantitative understanding (i.e., distance downstream) of facies variability on the system.

## Wider Implications and Conclusions

We have tested the methodology presented here and applied it to data obtained from satellite imagery and an outcrop DFS. Clearly, however, it can be applied more widely to any geological system that has a dispersive pattern from a point source. This methodology therefore has great potential in helping to constrain better a host of sedimentary systems, such as deltaic, submarine-fan, and alluvial-fan environments. Through this, a greater understanding of the paleogeography of an area can be gained. For example, by estimating the point source of a trough mouth fan, a fan that forms at the ice-margin front (Vorren and Laberg 1997), it may be possible to constrain paleo-ice-sheet extent better. Likewise, through the analyses of paleocurrents collected from deltaic systems, the extent of ancient shorelines can potentially be gauged. We anticipate that this methodology will be best suited to systems that crop out over large areas such as the Cretaceous systems spanning across Utah. Subset analyses are also possible with large datasets, allowing the identification of different lobes or progradation events, or even the identification of separate systems in stratigraphic units that have been mapped as basin fills, i.e., accurately mapping the Recapture and Westwater Canyon systems in the Morrison Basin (Craig et al. 1955). The methodology can be used at a variety of scales, which is important to identify when applying the methodology; if data from a single lobe are used, the methodology will predict the apex of that particular lobe, if data across a system are used, it will find the apex of a system. The application of the methodology is not restricted to sedimentary systems, as is demonstrated by Jupp and Spurr (1989), who estimate the three-dimensional position of a shock center for a volcanic shatter cone situated on the Slate Islands, Lake Superior, Canada.

Because directional indicators, such as channel orientations in three-dimensional seismic or bedform readings from borehole formation micro-imaging (FMI) data, can now be more readily gained, the methodology presented here is readily applicable also to subsurface datasets, when data are available either at or near the system scale, or data points are located in the proximal or medial portions of the systems, as is demonstrated in the worked examples from the Gilbert and Taquari DFS datasets.

In conclusion, a methodology that estimates the position of an apex statistically is presented and is tested on two modern DFS datasets, for which the location of the apex is known. Although confidence regions may be large when data are limited or highly variable, the methodology described provides an improved objective predictive tool for the geologist to constrain better the extent of depositional systems, enabling data points to be put into a system-scale context. Data from the Late Jurassic Salt Wash DFS are also analyzed, with results complementing field observations and shedding light onto the paleogeography of both the sedimentary basin and the Salt Wash DFS. Due to the user-friendly nature of the methodology presented and provided, this methodology is an attractive predictive tool for use on data obtained from outcrop-based and potentially subsurface-based studies.

## Supplemental Material

The paleocurrent data used in these analyses is available from JSR's Data Archive: http://sepm.org/pages.aspx?pageid=229.

The authors would like to thank BG, Chevron, ConocoPhilips, and Total for sponsoring Phase 1 of the Fluvial Systems Research Group consortium which funded the PhD project researching the Salt Wash DFS. Initial discussions with David Waltham are greatly appreciated. We also appreciate constructive comments from reviewers Chris Paola and Mike Blum, and Associate Editor Paul McCarthy which improved the manuscript. The authors would also like to thank Anna Kulikova, Guy Prince, Kelsey McNamara and Karen Oud for assistance in the field.