### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Geometry of River Deposits and Its Relation to River Channel Geometry and Dynamics
- 3. Transition Probabilities for Characterization of River Deposits
- 4. Scaling and Solute Macrodispersion in Braided River Deposits
- 5. Discussion and Conclusion
- Acknowledgments
- References
- Supporting Information

[2] This paper explores the link between the heterogeneity of subsurface river deposits and river geometry and dynamics. The primary motivations for this study are to (1) better understand these links, (2) provide the tools needed for systematic investigations and intersite comparisons, and (3) alleviate the difficulties associated with hydrogeological characterization (i.e., mapping the spatial distribution of sediment types (facies) and their corresponding hydraulic properties such as permeability and porosity). The possibility of estimating subsurface flow and transport parameters from geometric features of rivers is appealing because it allows deriving prior (in a statistical sense) estimates of such parameters in ungauged basins.

[3] The geometry, water flow and sediment transport of rivers affect the character of deposits in channel belts, which in turn determines the spatial variability of hydrogeologic properties (e.g., permeability and porosity) of fluvial deposits. The hydrogeologic properties of fluvial sediments are determined by grain size variations within individual strata, and by the geometry of sets of strata of different scales [*Jordan and Pryor*, 1992; *Davis et al.*, 1993; *Webb and Anderson*, 1996; *Davis et al.*, 1997; *Ritzi et al.*, 2000; *Willis and White*, 2000; *Bridge*, 2003; *Lunt et al.*, 2004a, 2004b; *Bridge and Lunt*, 2006]. Knowledge of these different scales of strataset has been derived from investigation of modern rivers for many years [*Cant and Walker*, 1978; *Allen*, 1982; *Bristow*, 1993; *Willis*, 1993; *Collinson*, 1996; *Miall*, 1996; *Best et al.*, 2003] (summarized by *Bridge* [2003]). Recent advances in our ability to monitor river bed morphology using photogrammetric techniques [i.e., *Lane et al.*, 2001; *Westaway et al.*, 2000] and to observe the resulting deposits using ground-penetrating radar combined with trenches or cores [*Bridge et al.*, 1998; *Regli et al.*, 2002; *Jol and Bristow*, 2003; *Best et al.*, 2003; *Skelly et al.*, 2003; *Lunt et al.*, 2004a, 2004b; *Wooldridge and Hickin*, 2005; *Sambrook Smith et al.*, 2005; *Mumpy et al.*, 2006] have allowed quantitative, comprehensive assessment of the formation of fluvial deposits.

[4] The geometry of the deposits of braided rivers and its relation to subsurface solute transport properties are the focus of this paper. Research effort in this direction has been contemplated by several researchers. The National Center for Earth Dynamics (NCED), for example, has identified the understanding of subsurface architecture as one of its major research thrusts. In particular, NCED assigns high priority to understanding how surface channel properties, spatial patterns, and temporal evolution interact with net deposition to create the architecture of sedimentary deposits, and how these deposits are modified by subsequent erosion and biochemical alteration. The ability to address these issues depends on (1) development of databases suitable for empirical studies and hypothesis testing; and (2) the availability of tools for coherent analysis of these databases. Work in support of the first element has been going on for decades, as mentioned above. Scaled laboratory experiments [i.e., *Schumm et al.*, 1987; *Ashmore*, 1982, 1991; *Ashworth et al.*, 1999; *Paola*, 2000; *Moreton et al.*, 2002; *Sheets et al.*, 2002; *Doeschl-Wilson and Ashmore*, 2005] have also proved useful, but experimental studies cannot be expected to recreate the full range of bed forms seen in rivers, and consequently the smaller scales of heterogeneity in fluvial deposits. Both field and experimental data should be applied to constrain numerical simulations of fluvial processes and deposits [i.e., *Paola et al.*, 1992*; **Mackey and Bridge*, 1995; *Sun et al.*, 1996; *Karssenberg et al.*, 2001; *Hickson et al.*, 2005]. Work in support of the second element is at an early stage. It is proposed here that data analysis can be undertaken using geostatistical tools such as presented here and in earlier studies, including *Rubin* [2003], *Ritzi et al.* [2004], and *Dai et al.* [2005].

[5] The paper is organized as follows. Section 2 provides background information on the geometry of river deposits. In section 3, the suitability of transition probability concepts for analyzing the spatial variability of river deposits is evaluated. Section 4 demonstrates an application of these statistical tools for estimating subsurface solute transport properties. In section 5, discussion and conclusions are presented.

### 2. Geometry of River Deposits and Its Relation to River Channel Geometry and Dynamics

- Top of page
- Abstract
- 1. Introduction
- 2. Geometry of River Deposits and Its Relation to River Channel Geometry and Dynamics
- 3. Transition Probabilities for Characterization of River Deposits
- 4. Scaling and Solute Macrodispersion in Braided River Deposits
- 5. Discussion and Conclusion
- Acknowledgments
- References
- Supporting Information

[6] This section examines the geometry of bed forms and cross sets in rivers. It also evaluates similarity and persistence of geometrical characteristics across scales and across rivers. If such similarity and persistence can be identified, then this would suggest the possibility for generalizing the findings from individual field studies. We start with a few general observations, and proceed to evaluate them against observations.

[7] It is well known that there is a hierarchy of different bed form sizes in rivers [*Vanoni*, 1974; *Allen*, 1982, *Jordan and Pryor*, 1992; *Bridge*, 2003; *Ritzi et al.*, 2004]. Small-scale bed forms such as sand ripples, dunes, and bed load sheets are superimposed on larger-scale features such as unit bars. Unit bars are commonly amalgamated to form compound bars such as point bars and braid bars (Figure 1). It is also well established that the dimensions of dunes and bars in rivers are related to river channel geometry [*Simons et al.*, 1965; *Allen*, 1976; *Parker*, 1976; *Nelson*, 1990; *Yalin*, 1992; *Julien and Klassen*, 1995; *Karim*, 1999; *Bridge*, 2003; *Wu and Yeh*, 2005]. Dune length and height are primarily proportional to flow depth, with variations in height and length due to the effects of unsteady flows [*Allen*, 1976; *Nasner*, 1978; *Nelson and Smith*, 1989; *Mohrig and Smith*, 1996; *Kleinhans*, 2002; *Wilbers and ten Brinke*, 2003], and the length of unit bars and compound bars are proportional to channel width [*Parker*, 1976; *Fredsoe*, 1978; *Engelund and Fredsoe*, 1982, *Fujita and Muramoto*, 1985; *Tubino et al.*, 1999; *Wu and Yeh*, 2005].

[8] Each of the different scales of bed form in rivers is associated with a different scale of strataset (Figure 2). For example, small-scale cross strata are associated with ripples, medium-scale cross strata are associated with dunes, and large-scale inclined strata are associated with unit and compound bars [*Rubin and Hunter*, 1982; *Jordan and Pryor*, 1992; *Bridge*, 2003; *Ritzi et al.*, 2004]. Furthermore, it is being recognized that the mean height and length of bed forms are proportional, respectively, to the mean thickness and length of associated stratasets [*Leclair and Bridge*, 2001; *Lunt et al.*, 2004a, 2004b]. This means that the geometry of stratasets formed by dunes and bars is related to the geometry and dynamics of the river.

#### 2.1. Length Scales and Ratios in the Sagavanirktok River

[10] The bed forms and deposits of the Sagavanirktok River, Alaska (Figures 1 and 2), are used as an example of how strataset geometry is related to bed form geometry and dynamics. The nature of channel evolution and deposits of the Sagavanirktok river were studied using aerial photographs taken since 1949, trenches, more than 100 km of ground-penetrating radar (GPR) data, and 17 cores [*Lunt et al.*, 2004a, 2004b; *Lunt and Bridge*, 2004]. The Sagavanirktok River is a sandy-gravelly braided river with a channel belt width of 2.4 km, maximum bankfull flow depth of 3.8 m, valley slope of 0.0013 m/m, bankfull discharge of around 600 m^{3} s^{−1}, and mean grain size of 4 mm in the reach studied.

[11] The dimensions of bed forms of different scale (e.g., dunes and bars) were measured using aerial photographs, and depth mapping of the river bed based on GPR profiles taken through the active channels (Figure 1). The dimensions of the various deposits were measured over rectilinear grids, superimposed on GPR lines and trenches. Medium-scale cross sets (formed by dunes) could only be measured in trenches, however, there is significant other information that was obtained from GPR images, such as shown in Figure 3. The majority of deposits comprise simple sets of large-scale strata, formed by unit bar deposition, which increase in thickness toward their downdip margins and contain cross strata that increase in dip in the same direction (Figure 2). Cross-bar channel fills have concave-upward erosive bases, length-to-thickness ratios of approximately 2:1, and may contain small unit bar deposits. Compound sets of large-scale strata (formed by compound bars) can be recognized primarily by their erosive boundaries that truncate numerous simple large-scale sets, and commonly contain concave upward cross-bar channel fills on their upper surfaces.

[12] The dimensions of bed forms reported by *Lunt et al.* [2004a, 2004b] are maximum length and width, because they were taken crest to crest or trough to trough in the horizontal direction, and trough to crest, vertically. Dunes were found to have a mean maximum length of 15 m and unit bars have a mean maximum length of 345 m (or 3.5 channel widths) and mean width of 66.2 m (or 0.7 channel widths). The length and thickness of the stratasets were used to determine the scaling relationships between bed forms and their deposits (Figures 4 and 5) .

[13] The ratios of maximum length to thickness of the deposits were found to increase from medium-scale to compound large-scale deposits (Figure 4), and suggest that simple relationships between the dimensions of morphological features in active rivers can be used to determine the dimensions of deposits formed by that river.

#### 2.2. Data From Other Rivers

[14] Data from other published studies of modern rivers show that the relationships between the geometry of bed forms and deposits observed in the Sagavanirktok River are not unique. These data sets are from modern braided fluvial deposits, and include the Jamuna River [*Best et al.*, 2003], the Niobrara River [*Skelly et al.*, 2003], the Fraser River [*Wooldridge and Hickin*, 2005], and the South Saskatchewan River [*Sambrook Smith et al.*, 2005]. These rivers range in channel belt width from around 600 m to over 15km, and the mean grain size varies from 0.04 mm to over 15 mm. A comparison of the length to thickness ratios of bed forms and deposits from all these rivers (Figure 5) shows that the dimensions of bed forms (and hence deposits) vary in different sized rivers, as would be expected. As bed forms such as bars and dunes scale with the channel dimensions, larger rivers give rise to deposits with larger lengths and heights. However, for rivers of all sizes and bed material types, the wavelength-to-height ratio of bed forms is very similar to the length-to-thickness ratio of deposits. This holds true over the entire range of bed forms for which data are available. Importantly, this suggests that the dimensions of subsurface deposits can be predicted from a limited amount of information about the formative river. Also, it may be possible to predict the relative dimensions of different scales of strataset. This will be expanded in further detail in section 4.

### 3. Transition Probabilities for Characterization of River Deposits

- Top of page
- Abstract
- 1. Introduction
- 2. Geometry of River Deposits and Its Relation to River Channel Geometry and Dynamics
- 3. Transition Probabilities for Characterization of River Deposits
- 4. Scaling and Solute Macrodispersion in Braided River Deposits
- 5. Discussion and Conclusion
- Acknowledgments
- References
- Supporting Information

[15] There is strong evidence indicating that the dimensions of fluvial deposits are determined primarily by the dimensions of their formative bed forms. It also appears, from our previous discussion, that these dimensional relationships are transferable between rivers. Additional work is needed in support of these ideas, and this is turn requires quantitative tools for analyzing the structures observed in river sediments. Such tools must (1) be able to represent complex, hierarchical structures, with different patterns of spatial variability at each hierarchy, (2) be formulated in statistical terms, to be able to correctly represent the uncertainty associated with spatial variability, on one hand, and be easily applied in data-sparse situations, on the other, (3) address asymmetries in the spatial correlation patterns, such as in the case of upward fining or coarsening grain size trends in fluvial deposits [*Carle and Fogg*, 1996], and (4) be able to represent site-specific features yet allowing comparison between and synthesis of multiple data sets.

[16] These requirements are generally met by various geostatistical tools, and as a result, the description of sedimentary deposits and their flow properties using geostatistical methods has become common in both the hydrogeological and reservoir characterization literature [*Johnson and Dreiss*, 1989; *Johnson*, 1995; *Gelhar*, 1993; *Jussel et al.*, 1994; *North*, 1996; *Deutsch and Journel*, 1998; *Weissmann and Fogg*, 1999; *Ritzi et al.*, 2000; *Strebelle*, 2002; *Rubin*, 2003].

[17] Indicator variograms and transition probabilities were found to be particularly useful. *Johnson and Dreiss* [1989], *Rubin and Journel* [1991], *Johnson* [1995], *Rubin* [2003], *Ritzi et al.* [2004], and *Dai et al.* [2004] have all employed indicator variograms to characterize such complex, hierarchical deposits. The variogram describes the degree of correlation between two points over a range of separation distances. The form of the resulting variogram can be interpreted to reveal information about the correlation length scales. Transition probabilities [e.g., *Carle and Fogg*, 1996] describe the spatial correlation structure of a region in a similar way. In this case, the probability of ‘transitioning’ from one class to another (e.g., from a unit bar deposit to a channel fill) is calculated over a range of separation distances. The advantage of transition probabilities is that they allow the direction of transition to be incorporated. For instance, the probability of passing from a unit bar deposit to a channel fill may be different from the probability of the reverse transition. This is particularly useful in characterizing fluvial sediments, in which the flow direction has a strong impact on the permeability distribution in the deposits.

[18] Geostatistical tools are efficient in reducing complex spatial data into a set of summary statistics. However, it can be shown that they are also related to the underlying depositional processes. *Ritzi* [2000] related the ranges and types of autotransition probability and variogram models used for stratigraphic characterization to the variability in facies lengths. He showed that as this variability increases, the indicator correlation range increases whereas the indicator variogram model evolves from a periodic linear structure to aperiodic spherical structure to an aperiodic exponential structure. These concepts are useful because they allow, as we will show below, translation of the length distributions, such as reviewed in section 2, into spatial characterization tools, and eventually into flow and transport parameters.

[19] Consider *I*_{k}(*x*) to be an indicator space random function (SRF) in a two-level hierarchical river deposit with *N* mutually exclusive sedimentary facies, with *k* varying from 1 to *N*. A two-level hierarchy implies an assemblage of different facies, with the permeability in each of the *N* types of facies being also spatially variable within the facies. This implies variability at two levels: at the lower level, there is variability within the facies, and at the higher level, there is variability in the organization of the various facies in space. *I*_{k} assumes the values of either 1 or 0 with probabilities *P*_{k} and (1−*P*_{k}), respectively: *I*_{k}(*x*) is equal to 1 if *x* is within a unit of type *k*, and 0 otherwise. *P*_{k}(*x*) = *Pr*[*I*_{k}(*x*) = 1], where *Pr* denotes probability. *P*_{k} is equal, in a stationary domain, to the volume fraction of geological unit *k*, and must satisfy the constraint:

The indicator is used here to describe the spatial arrangement of the geological units. Intrafacies variability for variables such as permeability can then be characterized using continuous space random functions (SRFs), with different types of SRFs attached to different facies [*Rubin*, 1995]. Higher-order hierarchies were investigated by *Dai et al.* [2004, 2005].

[20] The transition probability tensor *t*_{jk}(*h*), with *j*, *k* = 1, *N*, and *h* the lag vector between *x* and *x*′, is defined by the conditional probability [cf. *Ritzi*, 2000, equation (2)]:

where *Pr*[*I*_{k} (*x* + *h*) = 1, *I*_{j}(*x*) = 1] is the joint (bivariate) probability for the facies at *x*+*h* and *x* to be of types *k* and *j*, respectively. The cross-transition probability is *t*_{jk}(*h*) with *j* ≠ *k*. Of particular interest is the autotransition probability *t*_{kk}(*h*):

because it can be related, following *Carle and Fogg* [1996] and *Ritzi* [2000], to the statistical distribution of the lengths of geological unit of type *k*.

[21] The autotransition probability can be related [see *Ritzi*, 2000, equation (3)] to the indicator variovariogram, γ_{kk}(*h*) = 〈[*I*_{k}(*x*) − *I*_{k}(*x* + *h*)]^{2}〉/2, through the identity:

[22] The indicator variogram model is defined through spatial variability models based on the variance of *I*_{k} and a vector of correlation ranges, *a*_{k,i}, *i* = 1,..., 3, where *k* = 1,..., *N*, defines the facies type, and the second subscript, *i*, denotes the Cartesian direction in space. The correlation range is defined as the lag distance *h*_{i} where the correlation between the indicators, in the *i*th direction, vanishes.

[23] *Carle and Fogg* [1996] defined another length scale, _{k,i}, the mean length of unit *k* in the *i*th direction (where *i* denotes a Cartesian coordinate, with *i* = 1,2 denoting the horizontal axes. *Ritzi* [2000] showed that if the unit types occur in equal numbers, then _{k,i} can be related to the correlation range *a*_{k,i} through the identity:

where ϕ = 1, 1.5, or 3 for linear, spherical, or exponential models, respectively. If the unit types do not occur in equal numbers, (5) needs to be modified by an embedding coefficient [*Ritzi*, 2000]. The choice of model depends on the coefficient of variation (C_{v}) of ℓ_{k,i}, as shown by *Ritzi* [2000]. A linear model is used if C_{v} < 0.2, a spherical model is used for C_{v} ∼ 0.7, and an exponential model used for C_{v} ∼ 1.

[24] *Ritzi* [2000] showed that for *N* = 2, the variograms of the two facies are equal, i.e., γ_{11} = γ_{22}. As a result, the ranges of the indicators *I*_{1} and *I*_{2} are identical, which indicates, following (5), that the ratio of average lengths is equal to the ratio of their volumetric fractions:

with *P*_{1} = 1 − *P*_{2}.

[25] To examine this relationship, let us consider the spatial distribution of compound sets of large-scale strata. *Lunt et al.* [2004a, p. 391] in their study of the Sagavanirktok River refer to compound sets of large-scale strata truncated laterally by major channel fills. Compound sets and channel fills were found to have average maximum lengths of 627.9 and 37.9 m, respectively. The average maximum lengths were found to be larger than the average lengths by a factor of 1.26, with coefficient of correlation of 0.82, leading to _{1,1} and _{1,2} values of equal approximately to 498.3 m and 30.0 meters, respectively. The coefficient of variation of the maximum lengths were found to be equal to 0.37 and 0.48, for compound stratasets and channel fills, respectively, suggesting that the coefficients of variation of the average lengths are much larger, possibly of the order of 1. This in turn indicates, according to *Ritzi* [2000], that γ_{11} and γ_{22} are exponential, and that ϕ of equation (5) is equal to 3. Combining (1) with (6), we obtain that the volumetric fractions of the compound sets and the channel fills are *P*_{1} = 0.943 and *P*_{2} = 0.057, respectively, which is in excellent agreement with GPR- and trench-based based estimates. We obtain from (5) that *a*_{1,1} = *a*_{2,1} = 85 meters.

[26] Once the lengths _{k,i} and the volumetric fractions, *P*_{k}, *k* = 1, ..,*N*, are defined, the indicator variogram and the autotransition probabilities become readily available. The lengths _{k,i} can be related to the correlation ranges. *P*_{k} can be related to the indicator variance, which is in fact equal to *P*_{k}(1 − *P*_{k}), and the variance of the lengths can be related to the shape of the variogram, based on the work of *Ritzi et al.* [2000], as discussed earlier. Since the _{k,i} generally have different values in different directions, a statistically anisotropic variogram for each *k* should be expected. With the variograms defined, the autotransition probabilities can also be derived, for example, using (4).

[27] In order to provide the complete geostatistical characterization that would be needed for simulating a sedimentary deposit with many facies, it is necessary to go beyond autotransition probabilities, and augment the previous identities and definitions with cross-transition probabilities. The cross-transition probabilities can be obtained from observations. However, it is useful to investigate the relationship between the cross-transition probabilities and the autotransition probabilities, in order to obtain accurate definitions of the cross-transition probabilities, particularly in cases where the quality and/or number of field data are insufficient for determination of correlation models. In this case, two relationships can be used as starting points. The first is the identity [see *Carle and Fogg*, 1996, equation (17b); *Dai et al.*, 2005, equation (B2)]

and the second one is the identity:

which again holds for any *i.* In these equations and subsequently, *i*, *j*, *k* = 1, *N*.

[28] These identities must be translated into relationships between the autotransition and cross-transition probabilities. Unique relationships between the autotransition and cross-transition probabilities exist for *N* = 2 (as will be shown below). In the more general case of *N* > 2, equations (7) and (8) can be used as constraints.

[29] Equation (8) can be simplified using several auxiliary relationships. Using equation (2), each of the summed up terms on the left-hand side of (8) can be expressed using cross-transition probabilities, as follows:

with no summation over repeated indices here or subsequently, unless where explicitly mentioned. The right hand-side of (8) can be expressed using autotransition probabilities, as follows (see *Rubin* [1995, equation [8]] for a similar derivation):

Substituting equations (9) and (10) into (8) yields the following identity:

[30] To demonstrate the usefulness of these relationships, consider the case of *N* = 2. In this case, equation (6) yields

whereas (11) yields, for *i* = 1:

An identical equation is obtained for *i* = 2. Introducing (4) into (13) confirms that γ_{11} = γ_{22}, as suggested earlier. This result can also be obtained by combining (7) with the definition of conditional probability [see also *Carle and Fogg*, 1997, equation (10)]:

[31] For the case of *N* = 2, (7) and (11) provide a complete bivariate statistical characterization once any of the autotransition probabilities, either *t*_{11} or *t*_{22}, is defined. For *N* > 2, (7) and (11) translate into a set of transition probability equations that can be transformed into variogram equations or covariance equations, using (4). These equations can be used as constraints when calculating transition probabilities (or variograms) from data.

[32] These results are significant because they provide a simple way to obtain a complete (for *N* = 2), or constrained (for *N* > 2), bivariate characterization, using the statistics of the facies' lengths as a starting point. As was shown in section 2 and originally by *Lunt et al.* [2004a], these statistics can be obtained in some cases using GPR [*Neal*, 2004; *Rubin and Hubbard*, 2005]. In other cases, as we will show later, they can be obtained from analysis of the channel and bar dimensions.

[33] Let us assume that an exponential indicator variogram was identified, given by γ = *P*_{k} (1 − *P*_{k})[1 − exp (−∣*h*∣)/λ_{I}], for both *k* = 1, 2. λ_{I} is the integral scale, which in the exponential case is equal to 1/3 of the correlation range. Other variogram models can also be employed, as appropriate (see *Rubin* [2003, chapter 2] for a list of models and relationships between integral scales and correlation ranges). The autotransition probability *t*_{kk}(*h*), *k* = 1,2, can be obtained from equation (4) and the cross-transitional ones, *t*_{12}(*h*) and *t*_{21}(*h*), can be obtained from (12). With these expressions defined, the complete bivariate statistical distribution of the indicators has been obtained, employing only a small number of parameters.

### 5. Discussion and Conclusion

- Top of page
- Abstract
- 1. Introduction
- 2. Geometry of River Deposits and Its Relation to River Channel Geometry and Dynamics
- 3. Transition Probabilities for Characterization of River Deposits
- 4. Scaling and Solute Macrodispersion in Braided River Deposits
- 5. Discussion and Conclusion
- Acknowledgments
- References
- Supporting Information

[47] Depositional processes in river channels create complex heterogeneity, which is characterized by multiple length scales. One way to model such heterogeneity is through a hybrid geostatistical model that combines discrete space random functions, in the form of indicators, and continuous variables. The indicators in this hybrid model describe the juxtaposition of the various facies, whereas the continuous random variables describe the distribution, within the facies, of variables such as permeability. Models built on this concept were presented by *Rubin* [1995, 2003], *Rubin and Journel* [1991], *Dai et al.* [2004, 2005], and *Ritzi et al.* [2004].

[48] In this study, we employ the hybrid geostatistical models to characterize the spatial variability of the river deposits, on one hand, and to develop macrodispersion coefficients for solute transport, on the other. Thus we end up with a link between the river channel geometry and subsurface solute transport.

[49] On the basis of data collected at the Sagavanirktok river in Alaska [*Lunt et al.*, 2004a, 2004b], it is shown that a geostatistical model for the river deposits can be identified based on GPR surface surveys, augmented by measurements taken at trenches. The link between the hybrid model and the GPR survey is based on identifying the distribution of facies lengths using GPR, on one hand, and an analysis of the relationship between the statistical properties of these lengths and geostatistical models, on the other [*Carle and Fogg*, 1996, 1997; *Ritzi*, 2000].

[50] An analysis of the length to thickness ratios of bed forms and deposits from several rivers (Figure 5) shows that the dimensions of bed forms (and hence deposits) vary in different sized rivers, as would be expected. The data suggests that for rivers of all sizes and bed material types, the wavelength-to-height ratio of bed forms is very similar to the length-to-thickness ratio of deposits. This suggests that the dimensions of subsurface deposits and the relative dimensions of different scales of strataset can be predicted from a limited amount of information about the formative river. This has potentially wide-reaching implications, because the different scales of strataset in the subsurface determine important hydrogeologic parameters such as macrodispersion coefficients for subsurface solute transport. This idea is demonstrated using data from the Sagavanirktok river study, for the case of transport in a compound bar deposit composed of unit bar and channel fill deposits.

[51] The field examples used here are primarily from braided rivers and their deposits. However, the relationships between channel geometry and deposit dimensions presented here will hold for a wide variety of fluvial settings. Subsurface studies commonly take great pains to distinguish the deposits arising from different fluvial planforms, which must be interpreted from sparse data. Generally, it is difficult to distinguish meandering from braided river deposits from subsurface data [*Bridge and Tye*, 2000; *Bridge*, 2003; *Lunt and Bridge*, 2004a, 2004b] and the only true diagnostic is the recognition of a complete braid bar, which is impossible to do using core data alone. However, measurements of unit bar length and channel width from field, experimental and theoretical studies [e.g., *Lewin*, 1976; *Tubino et al.*, 1999; *Lanzoni*, 2000] indicate that the ratio between these lengths in meandering rivers is similar to those of braided rivers. Therefore, although our discussion is based on braided river deposits, we believe it applies equally to meandering river deposits.

[52] Our approach relies upon the assumption that the river channel geometry has not varied significantly over the recent past, so that the formative river channel geometry would have been similar to that observed today. Application of our methods to subsurface deposits that were formed by rivers that are significantly different from their present-day counterparts requires information about the width and depth of formative channels, which must be determined from geophysical or borehole data in representative locations. *Bridge and Tye* [2000] presented a method for deriving formative channel dimensions from subsurface data.

[53] To test our proposed approach further, a comprehensive database is needed to allow analysis of the relationship between river channel geometry and deposit geometry in a wide range of environments, using common tools and measures. With such a database, it could be possible to borrow correlation models from well-documented river valleys for use in ungauged, or minimally sampled basins, and to derive statistical priors for parameters of fluvial deposits such as macrodispersion. The development of such a database requires a unified approach for characterizing river basins and fluvial deposits, such as proposed here.