Quantitative metrics that describe river deltas and their channel networks

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi: /2010jf001955, 2011 Quantitative metrics that describe river deltas and their channel networks Douglas A. Edmonds, 1 Chris Paola, 2 David C. J. D. Hoyal, 3 and Ben A. Sheets 4 Received 22 December 2010; revised 25 May 2011; accepted 29 August 2011; published 16 November [1] Densely populated river deltas are losing land at an alarming rate and to successfully restore these environments we must understand the details of their morphology. Toward this end we present a set of five metrics that describe delta morphology: (1) the fractal dimension, (2) the distribution of island sizes, (3) the nearest edge distance, (4) a synthetic distribution of sediment fluxes at the shoreline, and (5) the nourishment area. The nearest edge distance is the shortest distance to channelized or unchannelized water from a given location on the delta and is analogous to the inverse of drainage density in tributary networks. The nourishment area is the downstream delta area supplied by the sediment coming through a given channel cross section and is analogous to catchment area in tributary networks. As a first step, we apply these metrics to four relatively simple, fluvially dominated delta networks. For all these deltas, the average nearest edge distances are remarkably constant moving down delta suggesting that the network organizes itself to maintain a consistent distance to the nearest channel. Nourishment area distributions can be predicted from a river mouth bar model of delta growth, and also scale with the width of the channel and with the length of the longest channel, analogous to Hack s law for drainage basins. The four delta channel networks are fractal, but power laws and scale invariance appear to be less pervasive than in tributary networks. Thus, deltas may occupy an advantageous middle ground between complete similarity and complete dissimilarity, where morphologic differences indicate different behavior. Citation: Edmonds, D. A., C. Paola, D. C. J. D. Hoyal, and B. A. Sheets (2011), Quantitative metrics that describe river deltas and their channel networks, J. Geophys. Res., 116,, doi: /2010jf Introduction 1 Earth and Environmental Sciences, Boston College, Chestnut Hill, Massachusetts, USA. 2 Department of Geology and Geophysics, University of Minnesota, Twin Cities, Minneapolis, Minnesota, USA. 3 ExxonMobil Upstream Research Company, Houston, Texas, USA. 4 Barr Engineering Co., Minneapolis, Minnesota, USA. Copyright 2011 by the American Geophysical Union /11/2010JF [2] How similar or different are the deltas of the world? One of the difficulties in answering this question is that there are few widely accepted metrics quantifying delta geometry and depositional patterns. In contrast, a variety of metrics applied to tributary networks reveal remarkable consistency and scale invariance across many orders of magnitude [Rodriguez Iturbe and Rinaldo, 1997]. Many delta distributary channel networks at an instance in time look crudely like inverted tributary networks, so a reasonable starting strategy is to adapt drainage basin metrics to delta distributary networks. In a particularly noteworthy effort in this regard, Fagherazzi et al. [1999] and Rinaldo et al. [1999a, 1999b] applied metrics from tributary networks to tidal delta networks and found that tidal networks exhibited little scale invariance. This strategy has also been applied to tributary submarine channel networks [Straub et al., 2007] where the authors found scaling similarities with tributary fluvial networks. [3] It is currently not clear to what extent river deltas and their networks exhibit scale invariance or other quantitatively consistent geometry. The need for quantitative measures is obvious given the recent proliferation of studies on delta evolution [Swenson, 2005; Olariu and Bhattacharya, 2006; Edmonds and Slingerland, 2007; Jerolmack and Swenson, 2007; Seybold et al., 2007; Syvitski and Saito, 2007; Edmonds and Slingerland, 2008; Fagherazzi, 2008; Edmonds et al., 2009; Hoyal and Sheets, 2009; Jerolmack, 2009; Kim et al., 2009a; Martin et al., 2009;Seybold et al., 2009; Edmonds and Slingerland, 2010; Falcini and Jerolmack, 2010; Geleynse et al., 2010; Reitz et al., 2010; Rowland et al., 2010; Wolinsky et al., 2010]. Quantification of observed and predicted geometry opens the way to measuring features of deltaic morphology and interpreting their signatures from a process perspective. Quantitative measures also provide a common language for comparison and prediction of delta morphology. Furthermore, delta metrics are needed to quantitatively test the extent to which numerical 1of15

2 [Overeem et al., 2005; Seybold et al., 2007; Dalman and Weltje, 2008; Seybold et al., 2009; Edmonds and Slingerland, 2010; Geleynse et al., 2010] and experimental [Edmonds et al., 2009; Hoyal and Sheets, 2009; Martin et al., 2009] deltas are similar to field scale deltas. Documenting and quantifying similarities among deltas is also important as we develop restoration strategies for deltas that are threatened by sea level rise and wetland loss [Ericson et al., 2006; Coleman et al., 2008; Blum and Roberts, 2009; Syvitski et al., 2009]. As such, our goal here is to present a set of metrics to quantify delta morphology and channel network organization, and demonstrate their use on different deltas. [4] In this paper we apply five metrics to four distributary delta networks (Figure 1) that are minimally affected by basin processes (e.g., waves, tides), and closely approximate inverse tributary networks in that they are strongly bifurcation dominated. We believe the metrics are applicable to deltas generally, but we start with simple distributary deltas for two reasons. First, the number of competing variables in these deltas (Figure 1) is minimized, which helps clarify relations between metrics and underlying processes. Second, the inverted tributary network structure of the deltas gives the best chance of showing both internal similarity (i.e., scale invariance) and external similarity [Paola et al., 2009] among the numerical, experimental, and field scale deltas. To facilitate applying the metrics to deltas generally, we comment on how they might change on deltas influenced by changes in accommodation as well as basinal properties like waves and tides. 2. Background [5] Traditional metrics describing river deltas focus on the external shape of the deposit and the internal configuration of the channel network. The best known metrics applied to external delta morphology are qualitative and focused on the relative contributions of rivers, waves, and tides. Wright and Coleman [1972] showed that the nature of the subaqueous profile of the delta determines the attenuation of wave power and thus influences the resulting shape of the delta. Those ideas were extended into a ternary diagram [Wright and Coleman, 1973; Galloway, 1975] that sorts delta morphology into different end members based on the relative contributions of fluvial, wave, and tidal effects. Later researchers added a fourth axis to account for the effect of sediment size and type on delta morphology [Postma, 1990; Orton and Reading, 1993], which recent theoretical work suggest could be a control on channel pattern and delta form [Edmonds and Slingerland, 2010]. These metrics seem to suggest that deltas of a given class (e.g., river dominated deltas) have similar shapes and network properties, and while this seems possible, it has not been quantitatively evaluated. Moreover, it is not clear how deltas of different end member classes (e.g., river dominated and wave dominated) differ quantitatively, or how to measure the effects of mixed influences. Applying metrics to different delta endmembers might elucidate subtle similarities and help refine the existing morphological classification. [6] Most deltaic metrics focus on the bulk shape and provide little information on the nature and morphology of the shoreline. In experiments, temporal shoreline position shows fluctuation because of the autogenic signal of sediment storage and release in the delta [Kim et al., 2006]. Shaw et al. [2008] have drawn attention to defining and measuring the shoreline as an important attribute of delta shape, a theme pursued by Wolinsky et al. [2010] who demonstrated nonfractal deltaic shoreline growth. Our focus here is on measuring the channel network as a complement to these shoreline analyses. [7] The internal configuration of the delta channel network is one of the most well described aspects of the delta landscape. Early work on delta network topology [Coleman and Wright, 1971; Smart and Moruzzi, 1972; Wright et al., 1974; Mukerji, 1976; Morisawa, 1985] revealed that the structure of delta networks is variable and cannot be predicted from a single controlling parameter. This is an important difference from tributary networks, which show remarkable internal consistency across different landscape regimes [Rodriguez Iturbe and Rinaldo, 1997]. Later work, however, showed some similarities among channel networks [Marciano et al., 2005; Syvitski et al., 2005; Olariu and Bhattacharya, 2006; Edmonds and Slingerland, 2007; Jerolmack and Swenson, 2007; Syvitski and Saito, 2007]. For instance, of the three main delta types, river dominated delta networks show a relatively high degree of similarity. Boxcounting metrics suggests that these networks have a fractal dimension of approximately 1.8 [Seybold et al., 2007, 2009; Wolinsky et al., 2010]. This fractality is likely an outcome of the tendency of river dominated delta networks to selfreplicate at progressively smaller scales due to the nonlinear decrease in size of distributary channel dimensions with successive bifurcations [Edmonds and Slingerland, 2007]. Furthermore, observations show that metrics describing the angle of bifurcation, the width ratio between bifurcate channels, and the average distance from bifurcation to bifurcation exhibit similar distributions for a subset of deltas [Olariu and Bhattacharya, 2006; Edmonds and Slingerland, 2007]. Jerolmack and Swenson [2007] also showed that channel length metrics from deltas indicate two scales of channels that reflect mouth bar bifurcation and channel avulsion, and that the smaller scale mouth bar channels are suppressed as wave influence increases. Wolinsky et al. [2010] used time series of delta imagery to show that a numerical, experimental, and field scale delta show similar growth laws. In particular, wetted area is a consistent fraction of delta area during delta growth, while channel edge length grows faster than delta area due to channel bifurcation at the delta shoreline. [8] The studies presented above represent important steps in quantifying morphological aspects of deltas. The next step is to develop a consistent set of metrics that can be applied across delta types. Here we present a set of five metrics that can be used to measure river delta morphology. We focus initially on images because these are far more abundant for deltas than high resolution topography. The metrics are intended for application to a single image and so do not use bathymetry, flow measurements, or temporal sequences. Our specific goals in this paper are (1) to relate the expression of the metrics to underlying processes, (2) to use the similarities and differences in the metrics to infer process differences among the four deltas, and (3) to use these metrics to determine the extent to which numerically 2of15

3 Figure 1. Deltas measured in this study. (a) A 2008 image of Wax Lake delta, Louisiana (latitude 29.52, longitude ). Letter a refers to long lived crevasse splays in the delta network; (b) 2009 image of Mossy delta, Saskatchewan, Canada (54.07, ). As the Mossy delta prograded basinward it encountered a bedrock island (black outline) and incorporated it into the delta. Image copyrighted by Digital Globe Inc., obtained through National Geospatial Intelligence Agency commercial imagery program. (c) Numerical delta created in Delft3D. Green and blue colors represent subaerial and subaqueous portions, respectively; (d) experimental delta created with a cohesive sediment mixture [Hoyal and Sheets, 2009]. Letter a refers to long lived crevasse splays in the delta network. Rhodamine dye is injected into the water to make the channel network visible. and experimentally generated deltas resemble one another and field scale deltas. 3. Methodology and Description of Metrics [9] Two ideas guided our choice of delta metrics: first, we wanted at least some metrics to have direct analogs to metrics for tributary networks for comparisons across systems; second, we adopted a sediment focused approach as opposed to the hydrologic focus adopted for tributary networks. Thus we focus on sediment distribution, whereas metrics for tributary networks traditionally focus on the spatial and temporal characteristics of the hydrograph [Rodriguez Iturbe and Rinaldo, 1997]. The sediment focused view is based on the idea that the primary driver of delta top morphodynamics is the need to distribute sediment over the delta top and maintain the surface transport slope in the face of progradation and relative sea level rise. Because sediment flow across deltas cannot be measured directly from overhead images, two of the metrics we propose are synthetic measures that rely on formulations and assumptions. [10] For each delta we measured the five metrics on an image that encompasses the delta and its channel network. We selected images of each delta that show the fully developed channel network. The Wax Lake delta image is high resolution orthoimagery taken on 1 October 2008 and obtained from the U.S. Geological Survey EarthExplorer Web site. The pixel size is 1 m 1 m, which resolves all significant morphological features on the delta. The Mossy delta image was obtained through National Geospatial intelligence Agency commercial imagery program from Digital Globe, Inc. It was taken on 22 February 2009 and has resolution of 1 m 1 m. For the numerical delta, the pixel resolution (25 m 25 m) was set by the cell size from the computational grid. In the image, the delta has evolved for 4.5 years. If flow intermittency were included the delta would represent evolution over centuries because in the simulation the channels are continuously at bankfull flow. The experimental delta image has mm resolution and was taken after ~140 run hours of evolution under constant water and sediment supply. We selected these time slices for the numerical and experimental deltas because the deltas are mature and show no influence of initial conditions. In the images the Wax Lake and Mossy deltas are at near bankfull flow, while the numerical and experimental deltas are at bankfull. 3 of 15

4 active channel, such as the Ebro Delta, that when skeletonized would be a line. A delta with D = 2 would have a network that is space filling, such as might happen in a high density of channels. The presence of fractality alone does not necessarily reveal any information about process because it only means the object is self similar. If the cause of fractality can be related to a process, then the fractal dimension could discriminate between deltas where that process is active. Figure 2. Example of nearest edge distance calculated for every subaerial point on Wax Lake delta image. Black parts of the images represent the nondeltaic initial condition prior to delta formation and are not included in the nearest edge distance calculation. [11] For analysis we classified each image into land and water pixels, making no distinction between in channel, lake, or ocean pixels. We mapped channels of all sizes that contained water at that flow stage and given the resolution of the images we are confident that our analysis captures all relevant channels. [12] The five metrics are (1) the fractal box counting dimension, D, (2) the distribution of island sizes, (3) the straight line distance to the nearest water ( nearest edge distance ), (4) the spatial distribution of synthetic sediment fluxes, and (5) the estimated nourishment area for any channel point, which we view as the deltaic equivalent of catchment area in a tributary network Fractal Dimension [13] The fractal box counting dimension is calculated using a standard box counting approach [Rodriguez Iturbe and Rinaldo, 1997]. We classify an image of a delta into channel pixels and everything else, creating a binary image of the channel network. We then skeletonize the network by finding the centerlines of the channels. This is done because lines are responsible for determining the value of the fractal dimension rather than two dimensionality of channel width [Foroutan Pour et al., 1999]. Determining the box counting dimension on a unskeletonized image can introduce error when the box size is smaller than the channel width because the shape of that channel segment, which is Euclidean with a D = 2, will also be measured. This box counting method was tested on a classic fractal object, the Koch snowflake, and reproduced the expected fractal dimension of approximately [14] For a skeletonized delta network, the fractal dimension, D, can vary from 1 to 2. First, a D value greater than 1 suggests the object is self similar. Second, more complex delta networks will have D values closer to 2. A D value of 1 would correspond to a straight line that is Euclidean, whereas a value of 2 a space filling curve that is fractal. An example of a delta with D close to 1 would have a single 3.2. Distribution of Island Sizes [15] Island sizes are mapped by tracing the edges of land polygons that are completely surrounded by water at bankfull stage. Edge lines are defined by the bankline/water contact for a channel of any depth. If there is water in the channel at a given stage we trace it and thus we include small channels that dissect the larger delta islands, as for instance on Wax Lake delta (Figure 1a). We make no distinction between islands formed by deposition in standing water at the river mouth and those formed by channels that carve into existing land. Mapping island sizes gives insight into the shapes that construct the delta and the variable scales of those shapes. Specifically this metric is useful for distinguishing between the dominance of bifurcation or avulsion in construction of the delta. A highly channelized and dissected delta with many bifurcations would have many small islands, while an avulsive delta that leaves large tracts of land unchannelized, should have a higher average island size and more importantly a greater probability of large islands Nearest Edge Distance [16] We define nearest edge distance as the shortest straight line distance from a given land point within a delta to water, making no distinction between channelized and unchannelized flow (Figure 2). This metric is the inverse of drainage density for tributary networks, which is the ratio of total channel length to basin area. The nearest edge distance measures channel density and is thus basic to nutrient and sediment distribution processes on the delta. For instance, areas of the delta with high nearest edge distances might receive less sediment over time and become susceptible to drowning as the delta topset aggrades either due to progradation or sea level rise. Nearest edge distance could also help predict floral and faunal assemblages on deltas. A delta with a high average nearest edge distance might be preferentially colonized with flora and fauna that need less access to water. Understanding how nearest edge distance distributions vary for different deltas will help determine if there is an ideal distribution that promotes healthy delta wetlands Synthetic Sediment Fluxes [17] The synthetic sediment fluxes are a measure of the spatial distribution of bed load and suspended load sediment fluxes delivered to the shoreline. To calculate the sediment fluxes we discretize the channel network into channel segments between bifurcation points (Figure 3). Only channel paths that reach the shoreline are included in the analysis. All shorelines in this paper are defined by the opening angle method with a threshold angle of 75 (see example shoreline in Figure 4) [Shaw et al., 2008]. The opening angle method defines a pixel as shoreline if the angular swath of visibility 4of15

5 variability of delta channel structure, we neglect sediment extraction over the delta top and treat the sediment routing as pertaining only to sediment that is delivered to the shoreline. The resulting distribution of sediment fluxes measures the planform stability and likely future evolution of the network. If most of the sediment is carried by one channel, a strong degree of asymmetry would develop as that channel progrades basinward, which creates a less stable network configuration that is likely to avulse. A more uniform distribution across the shoreline would be expected to be more stable because the delta would prograde relatively uniformly across its shoreline, reducing the slope imbalance that leads to avulsion. Figure 3. Delta channel network of Wax Lake discretized into channel reaches between bifurcation points for sediment flux calculations. See text for details about how these calculations are made. to open water from a given point on the land water interface in the delta is greater than a threshold angle. We chose 75 because that provides a reasonable shape of the shoreline without extending too far upstream in the distributary channels. To calculate how sediment is distributed among the channel reaches we use the water discharge for the given delta specified in section 4, and assume a unit delivered quantity of sediment, 20% of which is bed load. Results are insensitive to the choice of bed load percentage below 30%. At each bifurcation point the discharge and sediment are routed according to the nodal point condition developed by Bolla Pittaluga et al. [2003]. In the formulation of Bolla Pittaluga et al. water discharge is routed in proportion to widths of the downstream bifurcate channels. The bed load is routed in proportion to the depths of the channels, accounting for the topographic slope induced by bifurcation channels of unequal depth. Since we do not have depth information, we calculate depth for each channel segment from the reach averaged width by assuming the channel follows hydraulic geometry laws empirically derived from distributary networks [Mikhailov, 1970; Andrén, 1994; Tabata and Hickin, 2003]. We favor the laws of Andrén [1994], where W = aq b, D = cq k, and U = jq m and a = 9.91, b = 0.39, c = 0.358, k = 0.383, j = 0.238, and m = 0.227, because these relationships because they accurately predict changes in width and depth with bifurcation order on natural deltas [Edmonds and Slingerland, 2007]. Tests showed firstorder trends are not sensitive to other hydraulic geometry exponents [Leopold and Maddock, 1953]. [18] We assume the suspended load is divided in the same proportion as the water discharge [Kleinhans et al., 2008]. Because this is intended as a synthetic measure of the 3.5. Nourishment Area [19] Nourishment area is an estimate of the delta area nourished by sediment passing through a given channel cross section (Figure 4). For deltaic systems, we consider nourishment area to be the analog to drainage area in tributary networks: if the main effect of drainage (tributary) network is to collect water, the main effect of the deltaic network is to distribute sediment. Nourishment area is defined by tracing the largest area downstream of a given channel cross section that a particle could reach, until the path reaches the shoreline (Figure 4). The nourishment area includes all wet and dry pixels within the polygon. Determining nourishment area polygons requires knowing only the flow direction within each channel. Nourishment area estimated this way is only a first approximation; the true nourishment area encompasses some part of the bordering overbank areas at the boundary of the nourishment area polygon. We chose not to incorporate these areas because without a model of water and sediment routing there is no objective way of determining the fraction of the adjacent area that is linked to the bordering channel. To estimate the Figure 4. Image of Wax Lake delta showing examples of nourishment area for select positions (white polygons). Black dotted line represents shoreline calculated with an open angle of 75 degrees [Shaw et al., 2008]. 5of15

6 effect of ignoring this, we applied a rudimentary correction by dividing the area in equal proportions for each channel bordering it. For an island bordered by two channels, 50% of the island would be nourished by each channel. We applied this correction to Wax Lake delta and found that it does not change shape of the distribution appreciably; it does change the nourishment area by about 10 20% for smaller areas and as little as 1 2% for larger areas. For each delta, uncorrected nourishment areas are calculated only at bifurcation points (Figure 4) because it is at these points that the area changes appreciably. The nourishment area metric describes the shape of the delta and the topology of the channel network. Since it is analogous to a drainage area, it provides a potential basis for a Hack type law for deltas [Hack, 1957]. Hack s law for drainage basins states that the maximum channel length is a power law function of drainage area with a power law exponent greater than 0.5. An exponent greater than 0.5 implies that larger basins are relatively elongated. 4. Description of Deltas [20] We applied the metrics to four deltas (Figure 1): Wax Lake delta, Louisiana, USA; Mossy delta, Saskatchewan, Canada; a numerical delta developed using the Delft3D modeling package; and an experimental delta created using a moderately cohesive sediment mixture. These four cases allow us to compare an experimental delta and a numerical model with two field examples. All four examples approximate an inverse tributary network, that is, they are nearly purely distributary with few rejoining channels. Furthermore, there is minimal influence from waves, tides, and buoyancy forces. Atchafalaya Bay, location of Wax Lake Delta, is microtidal with minimal wave energy [van Heerden and Roberts, 1988] and at high discharge the salt water is pushed out into the bay. Mossy Delta is in a lake, eliminating tidal and buoyancy effects, and wave effects are minimal as well. Metrics for these relatively uncomplicated deltas provide a useful reference for deltas where additional forcing effects should produce deviations from the trends presented in this paper. [21] The history of Wax Lake Delta begins in 1941 when the Wax Lake outlet channel was dredged to reduce flood duration and magnitude in the Atchafalaya Basin [Wellner et al., 2005]. Atchafalaya Bay had a depth of 2 to 3 m below sea level. By 1973, the delta became subaerial at low tide [Roberts et al., 1980] and subsequent progradation and river mouth bar deposition created an intricate distributary delta network (Figure 1a). The Wax Lake delta has a spatially averaged vertical accretion rate of about 2.7 cm yr 1 from 1981 to 1997 with sand deposition causing the bulk of accretion. Wax Lake s sedimentary framework is 50 70% medium sand and significant amounts of suspended load bypass the delta into the bay [Roberts et al., 1997]. The channel forming discharge in the Wax Lake outlet is about 4800 m 3 s 1 [Kim et al., 2009b] and the Froude number of the flow entering the delta is 0.25 during bankfull flows. [22] The Mossy delta (Figure 1b), located in east central Saskatchewan, started forming in 1927 when the prograding sediment wedge from the 1870s avulsion of the Saskatchewan River reached the shore of Lake Cumberland [Smith et al., 1998]. Lake Cumberland had a depth of 1.5 to 2 m. On the basis of aerial photography of the evolution of the Mossy delta, as the delta prograded it bifurcated around mouth bars deposited at the river mouth [Edmonds and Slingerland, 2007; Wolinsky et al., 2010]. The Mossy delta is finer grained than the Wax Lake Delta with roughly 50% fine grained sand. The approximate bankfull discharge is 300 m 3 s 1 and the Froude number of the flow entering the delta is 0.3 during bankfull conditions (D. A. Edmonds, unpublished data, 2006). [23] The numerical delta (Figure 1c) used for this study was simulated with Delft3D v. 3.28, a morphodynamic model that simulates fluid flow, sediment transport, and morphological changes at time scales from seconds to years. Delft3D has been tested for a wide range of hydrodynamic, sediment transport, and scour and deposition applications in rivers, estuaries, and tidal basins [Hibma et al., 2004; Lesser et al., 2004; Hu et al., 2009; van Maren et al., 2009]. Our simulation of delta growth used a steady river discharge of 1000 m 3 s 1 carrying equilibrium concentrations (no erosion or deposition at the inlet) of cohesive (30 µm) and noncohesive (125 µm) sediment into a standing body of water devoid of waves, tides, and buoyancy forces. The computational grid is 300 by 225 cells, each 625 m 2 in area, with an initial bed slope of in the mean transport direction (up in Figure 1c), and initial depths ranging from 1 to 3.5 m. Coriolis force is neglected. Initial depths are randomly adjusted from 0 to 0.05 m using a white noise model to simulate natural variations. Bed roughness is set to a spatially and temporally constant Chezy C value of 45 m 1/2 s 1. A rectangular river channel 250 m wide and 2.5 m deep extending 1000 m basinward is carved into a 500 m wide subaerial, sandy shoreline along the southern boundary of the grid (see initial conditions in the supplementary video provided by Edmonds and Slingerland [2010]). Western, northern and eastern boundaries are open, allowing water and sediment to exit. The water surface elevation at these boundaries is steady and uniform. The model domain initially has 5 m of well mixed sediment (50% cohesive and 50% noncohesive) on the bed available for erosion. The Froude number at the inlet in this channel is 0.24 during the evolution of the run. [24] The experimental delta (Figure 1d) used for this study was created with a cohesive sediment mixture in a 5 3 m tank at the ExxonMobil Upstream Research Laboratory [Hoyal and Sheets, 2009]. The delta was created with constant water and sediment discharges of 10 L min 1 and 0.01 L min 1, respectively, issuing into a standing body of water 0.04 m deep through a channel m wide. The basin depth is initially uniform and water level was steady through the use of a weir in the tank. The experiment had a Froude number at the inlet of 0.21 during the 150 h experiment. The sediment mixture ranges from bentonite clay to coarse quartz sand, combined with a small amount of polymer to provide cohesiveness (details on experimental methodology are available in the work of Hoyal and Sheets [2009]). 5. Results and Interpretation 5.1. Fractal Dimension [25] The channel networks of the four deltas in this study all show fractal characteristics with a box counting dimension, D, ranging from 1.24 to 1.3 (Figure 5a). Fractality of 6of15

7 Figure 5. (a) Box counting dimension, D, for the skeletonized delta networks in the study. D is calculated by finding the slope of the best fit linear regression. All delta networks are fractal and characterized by a D 1.3. (b) Skeletonized image of the theoretical reference delta network created using equation (1) for a Q = 1000 m 3 s 1. Fractal dimension of this network is calculated in Figure 5a. Flow arrow only represents hypothetical direction. deltaic channel networks has been demonstrated [Seybold et al., 2007; Wolinsky et al., 2010] and it is not surprising that it also exists in the distributary deltas in this study. D for the Lena delta network was calculated to be approximately 1.8 [Seybold et al., 2007]. It is not clear if this higher dimension represents something fundamentally different about the Lena or if it reflects the style of image processing. We calculated D on skeletonized channel networks, whereas [Seybold et al., 2007] did not comment on how they preprocessed the delta network image. [26] To understand what the fractal dimension means we created a reference delta network that grows only via bifurcation around river mouth bars at the delta front (Figure 5b). The following procedure is used to create reference distributions for island sizes and nourishment area metrics, but is only presented here in detail. In such a delta there is a decrease of channel length (L) from bifurcation to bifurcation with increasing bifurcation order (defined as the number of times a channel splits upstream of a given bifurcation) [Edmonds and Slingerland, 2007] where L ¼ 104 h w U 2 ð1þ ð ÞgD 50 w max where r and s are the water and sediment densities (kg m 3 ), respectively, b is the scaling exponent, g is the gravitational acceleration (m s 2 ), D 50 is the median grain size (m), U is depth averaged velocity (m s 1 ), w is the channel width (m), and w max is the maximum channel width on the delta (m). To transform equation (1) into a reference delta network we assume (1) at all bifurcations the downstream bifurcate channels have equal widths and split discharge equally (2) all bifurcations have the same bifurcation angle, (3) all bifurcate channels conform to hydraulic geometry scaling, and (4) the deltas have a D 50 of 200 µm and s of 2650 kg m 3.In all cases we allow the delta to grow to eight orders of bifurcation, consistent with observations on modern deltas [Edmonds and Slingerland, 2007]. [27] We tested the first three assumptions and found that the reference delta is not sensitive to them. We chose to have equal width bifurcations and uniform bifurcation angle because this allows us to solve for the network geometries analytically. We implemented asymmetric width ratios and spatially variable bifurcation angles but the resultant distributions contained the same first order trends, thus we use the simpler model for clarity. Assumption 3 allows us to recast equation (1) into an expression that is a function of the discharge at the head of the delta using Andrén s [1994] hydraulic geometry exponents presented earlier in the methodology. Tests also showed that these first order trends are not sensitive to other hydraulic geometry exponents. [28] The reference delta has D = 1.19, which is similar to the other deltas in this study (Figure 5a). The similarity of D between the reference case and the deltas in this study suggests that the process causing fractality may be channel splitting because of mouth bar deposition, which occurs at progressively smaller scales with every bifurcation. This potential process connection implies that a global analysis of the fractal dimension of deltas might provide insight into their process regime. For example, a reasonable hypothesis is that delta networks not influenced by mouth bar bifurcation will have a D closer to 1 because that characteristic geometry is not repeated at all scales. On the other hand, delta networks will have a D closer to 2 if they create more channels than those created just at the channel mouth. This would create a higher density of channels that become more space filling over the delta top compared to the simple reference delta Island Sizes [29] All four deltas contain many small islands (those less than 3% of the total delta area) and few large ones (those greater than 10% of the area) (Figures 6a and 6b). To understand what the distributions mean we compare them to a synthetic reference distribution from a simple delta that grows via only mouth bar deposition and associated symmetrical bifurcation. Then, assuming a constant bifurcation angle of 60, the island for a given order is a simple geometric shape (Figure 5b). Only islands contained within the delta network are included. 7of15

8 Figure 6. Island size statistics for the four deltas in this study. All island sizes are normalized by the total delta area. (a) Probability distribution functions for the four deltas of island size normalized to total island area. Theory line refers to the distribution generated from a river mouth bar model of delta growth. Probability distribution functions are smoothed and have a bin size of (b) Cumulative distribution functions. On the y axis is the probability, P(x), that a given value in the distribution is greater than the given value of normalized island area. (c) Area weighted cumulative probability distribution functions. For three of the four deltas 50% of the area is composed of islands with normalized area 0.03 and smaller. [30] Although this model represents a rudimentary cascade process, the Mossy and experimental deltas have island size distributions similar to the model reference distribution and are dominated by small islands, while Wax Lake and the numerical delta have fewer small islands and more larger ones than the model distribution (Figure 6a). The deviations from the reference distribution in Wax Lake and the numerical delta are interesting, and suggest that processes other than bifurcations around mouth bars are creating the larger islands. The numerical delta has a nearly bimodal distribution of island area. This occurred because bifurcations were slowly abandoned leaving unchannelized land on the eastern and western flanks of the delta near the shoreline (Figure 1c). The same applies to Wax Lake; serial photographs [Wellner et al., 2005] show that bifurcations at the delta front often failed and channels prograded without bifurcating, creating anomalously long and large islands in the east central part of the delta (Figure 1a). The same bimodal signature would be expected if many channels were created by avulsion. For instance, the experimental delta, which has an island distribution consistent with a mouth bar bifurcation generated network, has an anomalously large island of 0.27 (normalized to delta area) that is an abandoned lobe created after an avulsion (Figure 1d). [31] Most island areas are relatively small when expressed as a fraction of total delta area. For instance, islands smaller than 5% of the total area account in aggregate for 50% of the delta area (Figure 6c). The numerical delta deviates from this because its smallest scale is set by the grid cell resolution. A linear trend in these distribution functions corresponds to a consistent cascade in the sizes of the islands, with excursions caused by anomalously large islands. These anomalies are created by either avulsions as described above or suppression of bifurcation (e.g., the normalized island of size 0.16 in the Mossy delta caused by suppression of bifurcation near a large bedrock island (black outline in Figure 1b)). [32] Island area distribution is a useful diagnostic tool to characterize processes occurring in deltas. For example, aggradation is thought to drive channel avulsion [Mohrig et al., 2000; Slingerland and Smith, 2004] and Jerolmack and Swenson [2007] suggest that older deltas should be more avulsive because they are dominated by aggradation. We propose that older deltas (in this sense) should tend toward a bimodal island size distribution with large and small islands, in contrast to younger deltas for which island sizes should be more continuously distributed (unimodal) Nearest Edge Distance [33] The calculated nearest edge distance distributions all have the same general shape (Figure 7a). But the medians (normalized to average channel width on the delta) vary, ranging from 0.8 to 1.6, and upper bounds (Figure 7b) range from 5 to 10. The lower bound for each delta is different because of the variation in image resolution. The channel networks appear to organize themselves spatially to maintain remarkably constant average nearest edge distances over much of the delta length (Figure 8). [34] Similarities in the shape of nearest edge distance distributions (Figure 7) and the spatial organization of nearest edge distances (Figure 8) suggest that channel networks are organized to maintain a consistent spatial channel density. To evaluate this hypothesis we compared the nearest edge distance statistics for the deltas in this study to nearest edge distances statistics of a random delta network. Differences in the nearest edge distance distributions of delta networks relative to randomly generated ones provide a measure of organization. Within the outline of each delta, 8of15

9 we created a random network by randomly placing the island shapes present in each delta until the same bulk channel density (defined as number of channel pixels relative to land pixels) was achieved. We allow the island shapes to overlap, which creates a delta composed of new island shapes. Results show that for all deltas the nearestedge distances for the randomly generated networks are not spatially consistent and vary significantly over delta length (Figure 8). Figure 8 represents one randomly generated network, but we produced multiple realizations of random networks and found that the above interpretation is robust. Furthermore, paired t tests show that the random distributions are statistically different from the corresponding real distributions at the 5% confidence level. [35] We suggest, therefore, that delta networks organize to maintain a spatially consistent average nearest edge distance, which is related to the tendency of delta channel networks to be space filling (Figure 5). Equally striking is that all deltas have average nearest edge distances of one to two multiples of the average channel width, which implies there is a fundamental length scale that sets how delta networks fill space. That length scale is consistent with the widths of fluvial levees [Adams et al., 2004], suggesting the channel density is set by the need to nourish areas adjacent to the channel with sediment. [36] We thus propose that an internal feedback drives the delta network toward a state of constant average nearestedge distance: areas with high nearest edge distance receive less sediment over time, and eventually become topographic depressions. Such depressions should tend to attract channels, thereby reducing the nearest edge distance. Organic processes may work against this feedback: if vegetation and peat growth is fast enough, low lying regions could produce enough organic sediment to keep up with aggradation elsewhere in the delta. Feedback among channel distance, sedimentation, and channel creation has not yet been directly observed in deltas, but may play an important role in regulating the distribution of nearest edge distances. [37] The argument advanced above applies to that part of the delta surface being actively maintained by sediment deposition. One clear exception to this is avulsion, which results in abandonment of parts of the delta plain. Of the deltas we studied, the experimental example has experienced more avulsions than the other three, and that is partly reflected in the statistical signature of its nearest edge distance. The experimental delta has long whiskers in the box plots in Figure 7c that correspond to the large abandoned delta lobe created after the most recent avulsion Synthetic Sediment Flux Distribution [38] As explained previously, the synthetic sediment fluxes at the shoreline were calculated using the nodal point relation of Bolla Pittaluga et al. [2003] to route bed load, suspended load, and water through the network assuming that the channels are at equilibrium hydraulic geometry. It is difficult to evaluate the accuracy of this simple sediment 9of15 Figure 7. Nearest edge distance statistics for the four deltas in this study. All nearest edge distances are normalized by the average channel width from that delta. (a and b) Probability and cumulative distribution functions, respectively. Probability distribution functions are smoothed and have a bin size of (c) Box and whisker plots of nearest edge distance distributions. Line in the box is the median, edges of the box represent the 25th and 75th percentile, whiskers correspond to the most extreme data points not considered statistical outliers, and pluses are statistical outliers. We did not remove the statistical outliers from any calculations of nearest edge distances.

10 Figure 8. Average nearest edge distance as a function of distance down delta measured on delta images in Figure 2. All nearest edge distances are normalized by the spatially averaged channel width for that delta. Distance down delta is normalized by the longest ray from the delta apex to the shoreline. Nearestedge distances are binned and averaged across 25 equally space radial swaths measured from the delta apex. Error bars are one standard deviation from the mean. Smaller gray dots and gray envelopes correspond to the mean and standard deviation, respectively, of nearest edge distance for a delta with randomly placed islands. Average nearest edge distance falls off at beginning and end because sample size is small. flux model for a network given the assumption made, but these sediment flux estimates do compare well with the actual fluxes at the shoreline recorded for the numerical delta (inset Figure 9a), the only case for which we have independent flux values. Applied to the four test cases this metric shows that at the shoreline the sediment fluxes exiting the river mouths are not spatially uniform (Figure 9a). In general, for each delta, a few channels transport the majority of the sediment. Quantitatively, about 20% of the channels deliver 50% of the total sediment load to the shoreline (Figure 9b). If the predicted sediment fluxes were constant in time, the deltas would become asymmetric because some channels would prograde farther into the basin than others. The planform shape of the deltas (Figure 1), however, suggests that this is not the case. [39] One way of eliminating shoreline asymmetry is channel avulsion. While relevant for larger lobe switching systems, like the Mississippi and Yellow River Deltas, deltas in this study do not frequently avulse. A less dramatic mechanism is that the path of maximum sediment flux shifts among active channels without channel abandonment. We term this process soft avulsion because no channels are abandoned or created yet the location of maximum sediment flux changes. This process of soft avulsion is related to a growing class of avulsions that exploit previously made channels rather than creating new ones [e.g., Jerolmack and Paola, 2007]. A plausible mechanism for soft avulsions is that large channels carry more sediment and prograde farther basinward before bifurcating [Edmonds and Slingerland, 2007; Hoyal and Sheets, 2009], which in turn reduces their fluvial slope, causing a reduction in transport capacity. At some point, this path is no longer an efficient space filling mechanism, and the path of maximum sediment discharge shifts to another channel in the delta network without abandoning the previous channel (Figure 10). We suggest that soft avulsion is likewise an important process that delta networks use to uniformly fill space without abandoning and creating channels. Flow shifting among channels in a network is potentially important to predicting the evolution of engineered sediment diversions [Kim et al., 2009b], and could help delta networks maintain stability to perturbations by quickly shifting sediment flux at the shoreline without altering their channel network Nourishment Area [40] The cumulative distributions of nourishment areas (normalized to delta area) show more linear regions in their distributions in log log space than the other metrics (Figure 11a). One interpretation is that nourishment area distribution is power law, but at best the linearity exists only over one decade, providing only weak support for scale free behavior. [41] On the other hand, a simple model for sediment delivery provides a good fit to the data. The shapes of the cumulative distribution functions of nourishment area are related to the rate of decay of nourishment area size over the length of the delta. Edmonds and Slingerland [2007] showed that for a delta network built purely by bifurcations around river mouth bars, the distance between channels (L) decays with an exponent of b = as given in equation (1). The rate of decay of L is equivalent to the decay of nourishment area under the assumptions that the delta network is built only by bifurcation via mouth bar deposition, each bifurcation splits flow equally, and the 10 of 15

11 Figure 9. Statistics of sediment flux for the channels at the delta shoreline for the four deltas in this study. (a) Probability distribution functions of sediment fluxes for channels at the shoreline. Sediment flux at the shoreline is normalized by the total sediment flux. Inset plot is a comparison of the width based sediment flux prediction employed here and the actual sediment flux at the shoreline measured in Delft3D for the numerical delta. Probability distribution functions are smoothed and have a bin size of (b) Cumulative distribution functions for the sediment flux at the shoreline. Approximately 20% of channels carry close to 50% of the total sediment flux for each delta. bifurcation angle is uniform across the delta. A reference distribution is generated using the same methods as presented for fractal dimension, and island areas. For each reference distribution we specify the maximum bifurcation order, discharge at the delta head, average grain size (assumed to be 200 µm), and the location and number of distinct nourishment areas. [42] We create a unique reference distribution for each delta by building a theoretical delta network out to the same observed bifurcation order using equation (1). We take the same number of measured nourishment areas for a given delta, and randomly place those points within the theoretical delta network. The theoretical nourishment area is calculated for each point and the total number of points in a given order is not allowed to exceed 2 n, since this is the maximum number of bifurcations for a given order. For example, a randomly placed point in the area occupied by bifurcation order 2 would correspond to a nourishment area measurement at a second order bifurcation. The resulting theoretical distributions were fitted to the observed distributions by finding the b in equation (1) that produced the minimum chi square statistic between theoretical and observed distributions. [43] In a delta built only by mouth bar bifurcations nourishment areas will decay downstream with an exponent of b = as shown in equation (1) [Edmonds and Slingerland, 2007]. However, when equation (1) from their model is applied to these four deltas, better fits are achieved with much smaller values of b (Figure 11a). The smaller values of b indicate that the rate of decay of nourishment area downstream, and also of the spacing of bifurcations (Figure 11b), is slower than for a delta network built purely by mouth bar bifurcations. Wax Lake and the experimental delta have the smallest values of b and deviate most from the mouth bar bifurcation model. In the experimental delta this is related to the tendency of the system to create long lived crevasse splay channels in addition to channels that bifurcate around mouth bars [Hoyal and Sheets, 2009]. This is obvious from watching the delta evolve, but it can also be inferred from the cumulative distribution function of nourishment area. The experimental delta nourishment areas deviate significantly from the mouth bar bifurcation model at a normalized nourishment area of 0.3. This is because crevasse splay channels at the head of the delta ( a in Figure 1d) created more large nourishment areas than predicted by the simple bifurcation model. The small b in Wax Lake delta is also a result of the frequent crevasse splay channels that cut through levees and feed intradistributary lows ( a in Figure 1a), which effectively dissect the delta network and produce a slower decay of nourishment areas downstream. [44] Classical studies of drainage area have revealed consistent scaling relations between drainage area and, for example, channel length [Hack, 1957]. In the deltas we studied, nourishment area (A N ) scales poorly with the channel width (w) at the head of the A N (Figure 12, top), whereas it scales well with the length (l) of the longest stream within the A N (Figure 12, bottom). Channel width is a poor predictor of nourishment area because, as indicated earlier, only a few channels carry the bulk of the sediment load (Figure 9b) and the set of most active channels varies over time through the process of soft avulsion (Figure 10). This creates A N that is often out of balance with the associated channel widths. Channel length is a better predictor of nourishment area because it grows directly as A N grows and therefore the nourishment area is rarely misfit to length. Interestingly, A N l 2.1±0.35 (Figure 12b), but more studies are needed to determine whether the value of this exponent remains near 2 for a broader sampling of deltas. When the exponent is 2, then delta nourishment areas behave as simple Euclidean objects that grow via dilation of a fixed shape. This would be in contrast to drainage basins which are characterized by Hack s Law, A d l 1.6 a relationship 11 of 15

12 Figure 10. The process of soft avulsion in Delft3D. (a) The location of maximum sediment flux at the shoreline for the numerical delta varies through time. Theta is the angle from the delta apex to the position of maximum sediment flux. An angle of 180 degrees corresponds to a sediment flux directly to the east of the apex. (b) The path of maximum sediment distribution shifts to other channels without channel abandonment. Both channels persist once created. that holds over many orders of magnitude of drainage area (A d ). 6. Discussion 6.1. How Similar Are Delta Networks? [45] Taken together, our results suggest a mix of similarity and intrinsic variability in the four deltas we studied. For instance, all the channel networks are fractal with similar exponents (Figure 5) and they self organize to provide relatively consistent average nearest edge distance within each network and among different networks (Figures 7 and 8). Additionally all the networks are organized such that 20% of the channels carry 50% of the sediment (Figure 9b). All the networks have well defined power law relationships, with similar exponents, relating nourishment area to channel length or width (Figure 12). On the other hand, all four deltas do not show scale invariant power law Figure 11. (a) Cumulative distribution functions of nourishment area normalized by total delta area. We fit the distributions by transforming the mouth bar bifurcation model of Edmonds and Slingerland [2007] into distributions of nourishment area (solid line), as explained in the text. b is the scaling exponent (from the work of Edmonds and Slingerland [2007]) that provides the best fit to the observed data. (b) Decay of bifurcation length with bifurcation order for the best fit exponents calculated for each delta in Figure 11a. 12 of 15