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Keywords:

  • uncertainty;
  • two-dimensional flow model;
  • topography;
  • spatial stochastic simulation;
  • rivers;
  • error propagation

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information

[1] Many applications in river research and management rely upon two-dimensional (2D) numerical models to characterize flow fields, assess habitat conditions, and evaluate channel stability. Predictions from such models are potentially highly uncertain due to the uncertainty associated with the topographic data provided as input. This study used a spatial stochastic simulation strategy to examine the effects of topographic uncertainty on flow modeling. Many, equally likely bed elevation realizations for a simple meander bend were generated and propagated through a typical 2D model to produce distributions of water-surface elevation, depth, velocity, and boundary shear stress at each node of the model's computational grid. Ensemble summary statistics were used to characterize the uncertainty associated with these predictions and to examine the spatial structure of this uncertainty in relation to channel morphology. Simulations conditioned to different data configurations indicated that model predictions became increasingly uncertain as the spacing between surveyed cross sections increased. Model sensitivity to topographic uncertainty was greater for base flow conditions than for a higher, subbankfull flow (75% of bankfull discharge). The degree of sensitivity also varied spatially throughout the bend, with the greatest uncertainty occurring over the point bar where the flow field was influenced by topographic steering effects. Uncertain topography can therefore introduce significant uncertainty to analyses of habitat suitability and bed mobility based on flow model output. In the presence of such uncertainty, the results of these studies are most appropriately represented in probabilistic terms using distributions of model predictions derived from a series of topographic realizations.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information

[2] Two-dimensional (2D) numerical models have become a powerful, widely used tool for examining flow patterns in river channels. Over the past decade, an increasing number of studies have applied depth-averaged models to investigate aquatic habitat conditions [e.g., Crowder and Diplas, 2000; Stewart et al., 2005; Brown and Pasternack, 2009], bed mobility and sediment transport [e.g., Lisle et al., 2000; Clayton and Pitlick, 2007; May et al., 2009], and morphologic change [e.g., Chen and Duan, 2008; Li et al., 2008; Rinaldi et al., 2008]. Lane [1998]. summarizes the underlying fluid mechanical principles involved in high-resolution flow modeling and reviews the development of this approach, which dates back to the early 1990s; the chapter by Nelson et al. [2003] and volume edited by Bates et al. [2005] also provide useful background on 2D flow models in the context of environmental hydraulics.

[3] Along with information on river discharge, flow resistance, eddy viscosity, and boundary conditions (e.g., water-surface elevation at the downstream end of the domain), accurate topographic data is a fundamental requirement for multidimensional flow models [e.g., Hardy et al., 1999; Marks and Bates, 2000; Horritt et al., 2006; Merwade et al., 2008]. The importance of these inputs depends on the size and type of river considered, and also might vary spatially within a reach as a function of the morphology [Lane and Richards, 1998]. In general, however, the ability of a 2D hydraulic model to resolve and reliably simulate the flow field within a particular channel depends strongly upon the way in which the shape of that channel is represented. Consequently, any uncertainty associated with the topography provided as input to the model will propagate through the model to produce uncertain model output, that is, uncertain predictions of water-surface elevation, depth, depth-averaged velocity, and boundary shear stress.

[4] To date, the effects of uncertain topographic data on 2D river flow modeling have received relatively little research attention, though a recent study by Casas et al. [2010] examined the effect of subgrid, grain-scale topographic variability on a three-dimensional flow model. These authors found that incorporating microtopography into the model's boundary conditions had a small effect on predicted mean velocities, whereas changing the resolution of the computational domain modified model output by up to 60%. More work on the influence of uncertain topographic input data has been done in the context of flood inundation models [e.g., Bates et al., 2003; Yu and Lane, 2006; Sanders, 2007]. For example, Casas et al. [2006] evaluated the sensitivity of a one-dimensional hydraulic model to the type and resolution of the input topographic data and found that LiDAR remote sensing reduced errors in predicted water levels and flooded area relative to less detailed, ground-based GPS surveys. For in-channel flows, Pasternack et al. [2006] recognized the uncertainty inherent to 2D modeling and assessed the sources and magnitudes of error in predicted depths and velocities via comparison with field measurements. An average depth prediction error of 21% was attributed to the resolution of the topographic survey used to parameterize the model. Similar errors could result from an inappropriate choice of roughness, but Pasternack et al. [2006] used a single, constant value of Manning's n and found no trend in depth prediction errors as a function of depth or local bed material grain size, implying that local bed roughness was not a systematic source of uncertainty. Errors in predicted depths, which also contributed to inaccurate predictions of velocity and shear stress, were thus primarily due to the topographic data input to the model; the survey was not sufficiently dense to resolve small bed undulations that influenced point measurements [Pasternack et al., 2006].

[5] The logistical challenges associated with acquiring high-resolution topographic data in the field have stimulated interest in alternative methods of characterizing channel form [e.g., Lane et al., 1994]. A wide variety of efficient survey techniques are now available and include total stations [Keim et al., 1999], GPS [Brasington et al., 2000], laser scanning [Milan et al., 2007], optical remote sensing [Marcus and Fonstad, 2008], and airborne LiDAR [French, 2003]. McKean et al. [2009] recently evaluated the ability of a water-penetrating, green-wavelength airborne LiDAR system to provide topographic data suitable for 2D flow modeling in small streams. To make this assessment, randomly distributed errors with a mean of zero and a standard deviation of 10 cm were introduced into the original LiDAR data and model runs performed with and without the simulated errors. For plausible values of the critical shear stress for entrainment of bed material, model predictions of bed mobility were relatively insensitive to the random errors added to the remotely sensed topography [McKean et al., 2009].

[6] Though insightful, these studies were also limited in several important respects. First, although the field data of Pasternack et al. [2006] allowed the error in flow model predictions to be quantified directly, this comparison was made for a single topographic data set. Model errors were attributed to inadequate survey point density, and although these authors recognized the effects of topographic sampling strategy on model errors, the issue was not explicitly addressed in their study. More data are presumably better, and some general guidelines for topographic data collection are employed in practice (e.g., cross sections spaced every half-channel width along single-thread, meandering rivers), but quantitative investigations of the effects of survey density on 2D flow modeling are lacking. Prior research on the modeling of fluvial topography from point elevation measurements has emphasized the value of distributed, terrain-sensitive sampling strategies [Lane et al., 1994; Brasington et al., 2000; Fuller et al., 2003; Valle and Pasternack, 2006; Wheaton et al., 2010], but surveys composed of regularly spaced cross sections are traditional and remain widely used in practice.

[7] Although advances in remote sensing technology might provide an appealing alternative to traditional, ground-based approaches, the system described by McKean et al. [2009], for example, remains experimental rather than operational. Even for established passive optical sensors, various physical constraints dictate that remote sensing techniques are only appropriate under certain conditions [Legleiter et al., 2009]. Nevertheless, Marcus and Fonstad [2008] argue that recent improvements in instrumentation and image processing make remote sensing a viable option for high-resolution, catchment-scale mapping of river systems. At present, then, a wide variety of survey methods, some well-established and some only now emerging, are being used to obtain the basic topographic information required for 2D flow modeling. Each of these techniques has strengths and weaknesses, and the associated uncertainties should be examined systematically. A more important, general question is how uncertain topographic input data influences flow model predictions, regardless of how the topography was measured.

[8] Another subtle but significant limitation of the work of McKean et al. [2009] is the manner in which error was introduced into the remotely sensed topographic data: as uncorrelated, randomly distributed noise with a uniform mean and standard deviation throughout the reach. Our work, in contrast, is predicated on the notion that a systematic analysis of the effects of uncertain topographic input data on flow model predictions must be spatially explicit because elevation differences (i.e., perturbations) at one node of a model's computational grid or mesh affect model predictions at nearby nodes due to the strong coupling between adjacent locations implied by the governing equations [Nelson et al., 2003]. Conservation of fluid mass and momentum dictate that a change in the flow field at one point is communicated to other, proximal points throughout some region of influence surrounding the initial change. An important implication of this principle is that the bed elevation perturbations used to assess model sensitivity must themselves be spatially correlated. The topographic surfaces used as input to 2D flow models are inevitably interpolated to the model grid nodes from a set of sampled elevation points. If the spatial distribution and density of the sample points differs significantly from the model grid, this interpolation step is likely to result in spatially correlated errors [Wheaton et al., 2010], particularly if the interpolator imparts some degree of smoothing and fails to reproduce the original data points at their locations. Assuming that errors in the input topography are independent, identically distributed random variables neglects the spatial structure of the error field. Because topographic errors depend on the morphology of the channel and the methods used to measure that morphology, any approach that fails to consider these spatial patterns cannot realistically capture the effects of topographic uncertainty on 2D flow model predictions.

[9] Spatial stochastic simulation, in contrast, provides an appropriate framework for examining these effects by generating many, equally likely, spatially correlated realizations of the bed topography, each of which is consistent with a geostatistical model that describes the variability and spatial structure of the channel morphology. Moreover, various sampling strategies can be evaluated using conditional simulation techniques that reproduce the available survey data but quantify the uncertainty associated with bed elevation estimates for other locations. In this manner, uncertainty in the topographic input data can be propagated through the flow model to characterize the resulting uncertainty in model predictions of various hydraulic quantities.

[10] Due to the interaction between river morphology and hydraulics, the effects of topographic uncertainty on model predictions are likely to vary both spatially within a channel as a consequence of its form as well as over time in response to variations in discharge. In addition, if the channel experiences competent flows that cause the morphology to evolve, the effects of topographic uncertainty could also vary over time in response to the morphodynamics. Some portions of a channel thus might be more sensitive to topographic uncertainty than others. Similarly, the effects of uncertain topographic input data could be greater at lower flow stages for which the influence of the morphology on the flow field might be more pronounced due to stage-dependent topographic steering processes [e.g., Whiting, 1997; Legleiter et al., 2007, 2011]. In an applied context, the site-specific probability distributions of model output derived via this stochastic simulation approach can be used to assess the effects of uncertain predictions of depth, velocity, and boundary shear stress on analyses of aquatic habitat or bed mobility.

[11] The primary objective of this paper is to examine the influence of topographic uncertainty on 2D modeling of flow fields in river channels, focusing on spatial patterns of uncertainty in relation to channel morphology and on stage-dependent hydrodynamic effects. To achieve this objective, we have adopted a spatially explicit stochastic simulation framework and focused on a simple, gravel bed meander bend. More specifically, we address the following questions:

[12] 1. How and to what extent do uncertain topographic input data propagate through a two-dimensional flow model to produce uncertain predictions of water-surface elevation, depth, velocity, and boundary shear stress?

[13] 2. How do these effects vary as a function of the spacing between surveyed cross sections, a common sampling strategy for measuring channel geometry and parameterizing flow models?

[14] 3. Does the nature and degree of uncertainty associated with model predictions based on uncertain topographic input data differ between base flow and a higher, subbankfull discharge?

[15] 4. Where within a typical, gravel bed river meander bend are model predictions more or less sensitive to uncertain topographic input data?

[16] 5. How do uncertain model predictions of depth, velocity, and boundary shear stress associated with uncertain topographic input data impact analyses of habitat suitability and bed mobility?

2. Methods

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information

2.1. Study Area and Field Data

[17] To assess the effects of uncertain topographic input data on 2D flow modeling, we focused on the Robinson Reach of the Merced River, located in California's Central Valley. This single-thread, meandering gravel bed channel was reengineered in 2002 to improve spawning conditions for salmon and is the subject of ongoing investigations of point bar development and habitat dynamics; additional detail regarding the study area is available elsewhere [Legleiter et al., 2011; L. R. Harrison et al., Channel dynamics and habitat development in a meandering, gravel-bed river, submitted to Water Resources Research, 2010]. The simplicity of this recently restored channel made the Robinson Reach an ideal site for systematically examining the effects of topographic uncertainty on flow model predictions. The basic form of this simple channel could be characterized relatively well, giving us confidence that perturbations of the modeled flow field were associated with simulated perturbations of the topography and not more complex morphologic features. Another convenience afforded by this site was the homogeneity of the bed material introduced as part of the restoration project. Our field observations indicated that this material was not sorted spatially, with the median particle size of 52 mm distributed uniformly throughout the reach. Similarly, the bed had remained relatively unstructured, without well-defined pebble clusters, stone cells, or other bed forms that would add fine-scale complexity to the bed topography. For this simple reach, then, we were able to assume that the irregularities in the elevation of the channel bed were those introduced by our spatial stochastic simulations.

[18] We considered a single, 340 m long meander bend with a channel bed slope of 0.0021, a bankfull width of 32 m, and a well-developed point bar along the right bank. A more detailed description, including photographs, of the Robinson Reach is provided elsewhere [Legleiter et al., 2011]; the site is located at 37.480° N, 120.483° W. The basic field data used to describe the river's morphology and condition our topographic simulations were obtained in November 2006 as part of a larger-scale, long-term investigation of channel dynamics [Legleiter et al., 2011; L. R. Harrison et al., submitted manuscript, 2010]. This survey consisted of a series of 48 approximately regularly spaced cross sections that encompassed the channel bed, banks, and proximal floodplain with a mean spacing of 7 m along the channel and 2 m between points along a transect, on average. Elevation measurements were located to resolve important slope breaks, such as the top and base of the bank. A total of 662 points were surveyed, resulting in a mean point density of 0.06 points/m2; point density was consistent throughout the reach.

2.2. Spatial Stochastic Simulation

2.2.1. Overview

[19] Spatial stochastic simulation (hereafter referred to as simulation) is a powerful, theoretically grounded geostatistical tool for quantifying uncertainty. Primarily developed in the context of mineral exploration, simulation is increasingly applied to a variety of environmental problems [e.g., Rossi et al., 1993; Holmes et al., 2000]. For a thorough discussion of the theory behind simulation and a description of various algorithms, the interested reader is referred to Goovaerts [1997] and Deutsch and Journel [1998]. Here, we provide only a brief overview of simulation concepts in the specific context of our application.

[20] The basic principle of simulation is illustrated in Figure 1, where a histogram is used to indicate that the bed elevation at each location within the channel is treated as a random variable. The variability and spatial pattern of this random field is described in terms of a parametric model called a variogram, a general geostatistical tool that has been used to characterize the spatial structure of river attributes [e.g., Chappell et al., 2003; Legleiter et al., 2007; Clark et al., 2008]. Spatial stochastic simulation involves generating a series of plausible, equally likely realizations of the topographic surface. Each of these realizations consists of a spatially correlated set of samples drawn from site-specific distributions of bed elevation, conditional upon any nearby bed elevation measurements. The strength and characteristic length scale of this spatial correlation reflect the variogram model, which acts to define the texture of the bed elevation realizations.

image

Figure 1. Flowchart illustrating the simulation concept; see text for discussion.

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[21] Simulation is closely related to the more widely used class of spatial prediction techniques known as kriging [e.g., Legleiter and Kyriakidis, 2008], but simulation is more appropriate for evaluating the effects of uncertainty on model predictions. Typically, a single value is used to summarize the bed elevation distribution for each model grid node; for kriging, this single value is the mean of the local distribution [Goovaerts, 1997]. The resulting collection of predicted (i.e., interpolated) bed elevations is then considered the topography and used as input to the flow model. As indicated in Figure 1, the model in turn produces a single set of predictions, with no indication of the uncertainty incurred by replacing the site-specific bed elevation distributions with single values. Spatial stochastic simulation differs in a fundamental way: rather than summarizing each bed elevation distribution with some measure of its central tendency, samples are instead drawn from these distributions and used to produce a series of spatially correlated topographic realizations. By applying the flow model to each realization in turn, a distribution of model outputs is generated for each grid node; this important distinction is represented by the histogram on the right side of Figure 1. Whereas kriging provides a smoother, less variable representation of the topography defined by mean values of the local bed elevation distributions, the simulated realizations consist of samples drawn from these distributions and thus more nearly reproduce the full variability and rougher texture of the underlying random field.

[22] Figure 1 also shows the practical value of this kind of spatially explicit stochastic simulation strategy. Each member of a series, or ensemble, of bed elevation realizations can be provided as input to the flow model and used to produce a corresponding ensemble of model runs. In this manner, topographic uncertainty can be propagated through the flow model to produce, for each node of the computational grid, a distribution of predicted values that reflects the impact of this uncertainty on the modeled flow field. A single prediction of, say, velocity with no indication of the associated uncertainty can thus be replaced by a histogram of velocities that quantifies the uncertainty that arises as a consequence of the uncertain topographic input data. This information can then be evaluated within a sensitivity analysis framework to assess the reliability of the model output and evaluate its utility for particular applications.

2.2.2. Implementation

[23] Survey data were transformed to a channel-centered coordinate system defined by a streamwise s axis and a normal n axis perpendicular to that centerline at each location along the channel [Smith and McLean, 1984; Legleiter and Kyriakidis, 2006]. The trend due to the channel bed slope was then removed by linear regression of s coordinates against surveyed bed elevations. A variogram model was fit to the residuals from this downstream trend using an iterative, graphical procedure and weighted least squares routine [Zhang et al., 1995; Pelletier et al., 2004] (Figure 2). All of the original survey data were used for trend removal and variogram analysis, and all simulations were produced using the same residual variogram model.

image

Figure 2. Residual variogram calculated using topographic survey data from which the streamwise trend due to the channel bed slope was removed. Both the initial variogram model fit iteratively by eye and the final model fit via weighted least squares are indicated. Point colors for the sample variogram represent the number of pairs contributing to the sample variogram estimate for that lag distance class.

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[24] Spatial stochastic simulation was performed with a moving average technique based on a fast Fourier transform of the covariance matrix [Oliver, 1995]. Conditioning to the survey data was achieved by indirect simulation of the kriging error [Goovaerts, 1997]; to account for the trend due to the channel bed slope, s coordinates were used as an external drift variable [Legleiter and Kyriakidis, 2008]. This algorithm is highly computationally efficient but can only be applied to regular grids. Following transformation to the channel-centered (s, n) coordinate system, the survey points used to condition the simulations were translated to the closest nodes of the model's computational grid without modifying the elevation values. The data were thus shifted from their original locations prior to simulation, but the high resolution of the model grid (section 2.3) implied that any such shift would be no greater than 0.71 m, which is significantly less than the mean spacing between points in the streamwise or cross-stream directions. Moreover, given the gentle bed slopes within this reach, a horizontal displacement of this magnitude corresponds to a very small difference in elevation for most locations, except, perhaps, for steeper banks. Translating survey points to grid nodes allowed us to efficiently generate realizations via a Fourier transform-based technique, whereas simulation algorithms for irregular data are more computationally demanding and would have restricted us to a much smaller model domain.

[25] The topographic data sets used to condition the simulations included: (1) the full, original survey; (2) progressively thinned versions of this data set, created by retaining only every other, every third, every fourth, and every sixth cross section; and (3) 100 randomly selected subsets of the original survey, each consisting of one-third of the full data set. In the first case, 100 simulations conditioned to the full survey were used to characterize the uncertainty associated with flow model predictions based upon the most complete topographic data set available to us. For the second set of simulations, 100 realizations were generated for each of the five data configurations summarized in Table 1 and used to quantify the effect of cross-section spacing on the uncertainty of flow model predictions. For each sample configuration, all of the survey points on a given cross section were included; our analysis thus focused on the effect of the along-channel spacing of the survey data while holding the cross-stream point spacing fixed. For the third series of runs, a single topographic realization was produced for each of 100 random subsets of the data. For these simulations, survey density was held fixed while the locations of the conditioning data varied, so this ensemble provided a means of examining where within a typical meander bend the flow model was more or less sensitive to topographic uncertainty.

Table 1. Topographic Data Configurations Used to Assess the Effects of Survey Density on the Uncertainty Associated With Flow Model Predictionsa
Cross Sections RetainedMean Spacing (m)Baseflow WidthsSubbankfull Widths
  • a

    Mean spacing between cross sections is reported in units of m and as multiples of the wetted channel width (dimensionless) for the baseflow (6.4 m3/s) and subbankfull (32.6 m3/s) model runs.

All70.380.24
Every other140.750.47
Every third211.120.71
Every fourth271.450.90
Every sixth402.161.34

2.3. Flow Modeling

2.3.1. Description of the FaSTMECH Flow Model

[26] The Flow and Sediment Transport for Morphological Evolution of Channels (FaSTMECH) model used in this study was developed by the U.S. Geological Survey and is distributed as part of the public domain MultiDimensional Surface Water Modeling System (MD-SWMS). FaSTMECH is actually a quasi-3D model that includes vertical structure and secondary flows to yield full velocity and shear stress fields, but in this study we used only the 2D version of the model. Recent applications of FaSTMECH include the bed mobility study of May et al. [2009] and our own work on the Merced River [Legleiter et al., 2011; L. R. Harrison et al., submitted manuscript, 2010]. Briefly, FaSTMECH predicts water-surface elevation, flow depth and velocity, and boundary shear stress by obtaining numerical solutions to depth-averaged equations for conservation of mass and momentum. Inputs to the model include bed topography, river discharge, and downstream stage. The flow is assumed to be steady and hydrostatic, and turbulence is treated by relating Reynolds stress to shear via an isotropic eddy viscosity. A full description of FaSTMECH and a discussion of the underlying theory is given by Nelson et al. [2003] and in the original work of Nelson and Smith [1989a, 1989b].

[27] An important feature of FaSTMECH in the present context is the fact that the model's governing equations are expressed in the same channel-centered, orthogonal curvilinear (s, n) coordinate system adopted in our geostatistical framework. Similarly, part of our rationale for using FaSTMECH is the manner in which the model derives numerical solutions. Whereas many 2D flow models are based on finite elements and thus require construction of a computational mesh, FaSTMECH is formulated in terms of finite differences and is implemented on a regular (s, n) grid. In addition to being relatively easy to define, such a grid is also more compatible with the stochastic simulation algorithm described above. Moreover, unlike the various finite element models, FaSTMECH provides the computational efficiency needed for the large-scale batch processing involved in this type of simulation study. A final, related advantage of FaSTMECH is the standardized, transparent file input and output structure, specified by the Computational Fluid Dynamics General Notation System (CGNS) (available at http://cgns.sourceforge.net/). This framework allowed us to efficiently deliver simulated topographic realizations to the flow model and extract output from the corresponding FaSTMECH solutions.

2.3.2. Model Setup

[28] For this study, we considered two different discharges: (1) a low base flow of 6.4 m3/s of interest for spawning habitat assessment; and (2) a higher, subbankfull discharge of 32.6 m3/s (approximately 75% of the bankfull flow) more relevant to sediment transport studies. For the base flow model runs, we used a 32 m wide grid with 343 streamwise and 33 cross-stream nodes, whereas the high flow runs were performed using a 40 m wide 343 × 41-node grid. In both cases, the cross-stream grid node spacing was 1 m, and the streamwise node spacing was 1.0008 and 1.0266 m for the low- and high-flow grids, respectively.

2.3.3. Model Calibration

[29] Before the individual simulated topographic realizations were provided as input to FaSTMECH, the flow model first had to be calibrated appropriately. For each discharge, this process involved performing an initial model run for each data configuration (the original survey, five different cross-section spacings, and one random subset of the full survey) using bed elevations predicted via kriging with an external drift as the input topography [Legleiter and Kyriakidis, 2008]. Model boundary conditions (e.g., stage at the downstream limit of the computational domain) were specified using water-surface elevation profiles surveyed in the field at each discharge. FaSTMECH represents flow resistance in terms of drag coefficients, denoted by Cd, and we used a single, spatially uniform drag coefficient to represent both skin friction and form drag throughout the reach. This assumption was justified by the homogeneous bed material and lack of distinct bed forms along this simple, restored section of the Merced River. For each discharge and each data configuration, a numerical optimization routine was used to calibrate Cd by minimizing the disagreement between predicted and observed water-surface elevations, quantified in terms of the root-mean-square error (RMSE). The lateral eddy viscosity parameter used to represent momentum exchange due to turbulence not generated at the bed was calculated from the reach-averaged depth and velocity following Barton et al. [2005].

2.3.4. Coupling the Flow Model to Simulated Topographic Realizations

[30] The calibrated model runs based on the kriged topography and optimized Cd were then used as templates for applying FaSTMECH to realizations consisting of bed elevations simulated at the nodes of the two model grids, one for each discharge. At both the upstream and downstream ends of the reach, a buffer of constant, kriged topography 30 m (approximately one channel width) in length was used to ensure that the same boundary conditions could be applied for all model runs and to avoid numerical instabilities. Through the middle 280 m of the reach, the input topography provided to the flow model was defined using the simulated bed elevations. For each topographic realization, a FaSTMECH run was executed and the output stored to assemble, for each grid node, distributions of model predictions. In total, 1200 runs were performed, based on 100 topographic realizations for each of six sampling strategies and two discharges.

2.4. Quantifying the Effects of Topographic Uncertainty on Flow Model Predictions

2.4.1. Ensemble Summary Statistics

[31] To evaluate the influence that the increased topographic uncertainty associated with sparser survey data sets had on predictions from the 2D flow model, we calculated reach-averaged ensemble summary statistics for four key hydraulic quantities: water-surface elevation E, depth h, velocity u, and shear stress equation image. An ensemble mean equation image, standard deviation equation image, and range R was calculated for each location by pooling over the realizations; these statistics thus characterize the local distributions of model predictions at each node of the computational grid. To summarize the effect of sampling strategy on the uncertainty associated with model predictions, we then aggregated these ensemble summary statistics by averaging their values over the reach. Our notation is summarized in Table 2. For example, we use equation image to refer to the ensemble mean depth for a specific location within the channel and add angle brackets equation image to indicate aggregation of this quantity over the reach. A numerical subscript outside the brackets denotes a quantile of the distribution of that quantity within the reach; equation image thus refers to the reach-aggregated median of the ensemble mean depths. Other combinations of hydraulic attributes and ensemble summary statistics are represented similarly.

Table 2. Notation Used to Refer to Ensemble Summary Statistics Calculated for Individual Nodes of the Model's Computational Grid, or Aggregated Over the Reach
SymbolVariable and Description
EWater-surface elevation
hFlow depth
uFlow velocity
equation imageBoundary shear stress
equation imageEnsemble mean of the variable X for a single grid node
equation imageEnsemble SD of the variable X for a single grid node
RXEnsemble range of the variable X for a single grid node
equation imageReach-aggregated ensemble mean of the variable X
equation imageReach-aggregated ensemble SD of the variable X
equation imageReach-aggregated ensemble range of the variable X
equation imagepth quantile of ensemble means of the variable X within the reach
equation imagepth quantile of ensemble standard deviations of the variable X within the reach
equation imagepth quantile of ensemble ranges of the variable X within the reach

[32] Because some of the topographic realizations produced relatively high bed elevations that were not inundated by the flow model, the number of simulations included in an ensemble varied slightly from one grid node to the next along the shallow margins of the channel. The sample size upon which the ensemble statistics were based thus varied among nodes, but the reach-averaged summary statistics were computed using all grid nodes that were inundated in any of the realizations. Also, although the computational grid was regular in the channel-centered coordinate system, the size of the model grid cells varies in a Cartesian reference frame due to the channel curvature. To account for these differences in cell size, we weighted each cell in proportion to its area in calculating reach-averaged statistics.

2.4.2. Characterizing Spatial Patterns of Uncertainty

[33] We also sought to understand in more detail the spatial structure of the uncertainty in flow model predictions that arose as a consequence of uncertain topographic input data. To do so, we used 100 FaSTMECH runs based on topographic realizations conditioned to the original survey to characterize the uncertainty associated with a relatively complete elevation data set. This set of runs also served as a basis for comparison for sparser data configurations. A second set of FaSTMECH runs was performed for simulations based on 100 random subsets of the original survey. Because the locations of the conditioning data varied from one realization to the next, this ensemble allowed us to examine the spatial pattern of uncertainty in flow model predictions resulting from uncertain topographic input data.

[34] To gain further insight on the manner in which simulated perturbations of the bed induced perturbations of the modeled flow field, we compared local departures of bed elevation z and predicted velocity u from their respective ensemble means. More specifically, these site-specific fluctuations were calculated as equation image and equation image for 100 subbankfull model runs based on simulations conditioned to random subsets of the original survey. The correlation coefficient between the realizations of z′ and u′, denoted by rz′,u′, was then computed for each node of the model's computational grid.

2.5. Habitat Suitability Analysis

[35] Two-dimensional flow models have become a powerful and widely used tool for quantifying aquatic habitat (see review by Leclerc [2005]). Certain species of interest, fall-run Chinook salmon in the case of the Merced River, often prefer different types of hydraulic conditions at different stages of their life history, and empirical data regarding these preferences have been used to construct suitability indices. These preference curves relate hydraulic quantities, typically depth, velocity, and bed material grain size, to a ranking of habitat quality on a scale of 0 to 1. In this study, we used spawning habitat preference curves compiled by Gard [2006] for the Merced River to compute values of a composite habitat suitability index (HSI) based on model predictions of depth and velocity at a typical late fall, base flow discharge of 6.4 m3/s. A constant grain size of 52.5 mm, the median particle size along the Robinson Reach, was used to define the substrate component of the suitability index. For each node of the computational grid, a local HSI distribution was defined by calculating HSI values from two sets of flow solutions: one based on 100 topographic realizations conditioned to random subsets in which one of every three points was retained and the other based on 100 simulations conditioned to the full data set. The former set of runs thus indicates the level of uncertainty associated with a sparser data configuration and the latter set provides an indication of the uncertainty involved in habitat suitability analysis given the best available survey data set for our study area.

2.6. Bed Mobility Analysis

[36] Another common application of 2D flow models is the assessment of bed mobility. For example, recent studies have used FaSTMECH to examine bed load transport in a meander bend [Clayton and Pitlick, 2007] and to predict scour and fill at salmon spawning sites [May et al., 2009]. These types of analyses rely upon model predictions of boundary shear stress equation image and could thus be compromised by uncertainty in these predictions. To examine the potential implications of topographic uncertainty for studies of bed mobility, we considered two sets of model runs performed for a subbankfull discharge of 32.6 m3/s. The first set was based on 100 simulations conditioned to random subsets of the original survey, and the second set based on 100 realizations conditioned to all of the data. The shear stress predictions from these model runs were used to define a local distribution of equation image values for each grid node, and we used various quantiles of these distributions to quantify the range over which equation image predictions varied as a result of uncertain topographic input data. We also used model output to estimate the largest particle size one might expect to be mobilized by a flow of this magnitude. Grain sizes at the threshold for entrainment were calculated by rearranging the Shields equation to obtain

  • equation image

[37] In this expression, equation image is the critical grain size corresponding to the xth quantile of the local shear stress distribution equation image. The critical dimensionless shear stress equation image was assumed to be 0.045. equation image and equation image are the densities of sediment and water, and g is the acceleration due to gravity.

3. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information

3.1. Flow Model Calibration

[38] For the base flow model runs, the calibrated drag coefficient Cd for the full data set was 0.0176 and the RMSE between predicted and observed water-surface elevations was 0.027 m. For the subbankfull discharge, the corresponding values were 0.0112 and 0.035 m, respectively. For the subsampled data sets, Cd values were lower by up to 12% for the sparsest data configuration (i.e., greatest spacing between cross sections). Because flow resistance was calibrated separately for each data configuration, differences in roughness parameterization could have contributed to differences in flow model predictions, in addition to the effects of sampling strategy. The drag coefficient value changed only slightly across a range of cross-section spacings from 0.24 to 1.34 channel widths, however, and differences in predicted water-surface elevations, depths, velocities, and shear stresses were thus attributed to differences among data configurations. Similarly, sensitivity analysis indicated that model predictions were not affected by twofold variations in lateral eddy viscosity. Field measurements of flow velocity along six transects indicated that modeled velocities were accurate to within 20% of the measured values; additional detail on model calibration and validation is provided by Legleiter et al. [2011].

3.2. Effects of Topographic Sampling Strategy on Flow Model Predictions

3.2.1. Simulated Topographic Realizations

[39] Figure 3 illustrates a series of three (out of 100) topographic realizations for each of the five data configurations evaluated in this study, representing mean cross-section spacings (equation image = along-channel distance between measured elevation transects) from 7 to 40 m, or 0.24 to 1.34 wetted channel widths for a discharge of 32.6 m3/s (Table 1). For the full survey data set (first column of Figure 3), the simulations vary little from one realization to the next because they are tightly constrained by the relatively high density of conditioning data. The similarity of the realizations indicates that topographic uncertainty in this simple channel is minimal when cross sections are separated by less than a quarter of the wetted channel width. If only half of the original survey data (i.e., every other cross section) is retained, the realizations diverge from one another to a greater degree due to the reduced number of conditioning data. As survey density decreases, the simulated bed elevations are not as well-constrained by measurements and instead increasingly reflect the underlying geostatistical model. In this case, the zonal anisotropy [Goovaerts, 1997] evident in the variogram (Figure 2), with a higher sill variance in the cross stream than in the streamwise direction, is reflected in the simulations as along-channel streaks of similar values. This pattern is a direct consequence of an anisotropic variogram that implies that bed elevations tend to be less variable and to vary over greater characteristic length scales along the channel than across the channel [Chappell et al., 2003; Legleiter and Kyriakidis, 2008]. As the mean spacing between cross sections increases to a channel width or greater, the individual topographic realizations begin to differ more noticeably from one another. For example, in the fifth column of Figure 3 the zone of high elevations along the point bar is broader and extends farther upstream above the third surveyed cross section in realization #9 (labeled A) than in the other two simulations shown here. Similarly, realization #53 has a wider, more continuous swath of low elevations at the lower end of the pool (near the outer bank between the fourth and fifth surveyed cross sections, labeled B) than the other two realizations. These differences reflect the greater topographic uncertainty associated with a sparser survey.

image

Figure 3. Simulated topographic realizations for various cross-section spacings equation image, expressed in meters and as multiples of the mean wetted channel width for a subbankfull discharge of 32.6 m3/s. Points indicate the locations of survey measurements used as conditioning data for the simulations. The realizations depicted here (#53, #82, and #9) were selected at random from an ensemble of 100 simulations of the bed topography for each data configuration. The A and B labels highlight areas in which the realizations differ from one another, as described in the text.

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3.2.2. Base Flow Model Runs

[40] Reach-aggregated ensemble summary statistics for each hydraulic quantity are plotted against mean cross-section spacing for base flow (6.4 m3/s) conditions in Figure 4. The uncertainty associated with FaSTMECH predictions, as indexed by the reach-aggregated ensemble standard deviation and range, becomes greater as the distance between surveyed cross sections increases. Values of equation image and equation image increase steadily with mean cross-section spacing for all four hydraulic quantities. Moreover, the larger error bars, which represent the first and third quartiles of the reach-aggregated distributions of the ensemble summary statistics (i.e., equation image and equation image), for the less dense surveys indicate that the uncertainty associated with flow model predictions also varies more within the reach when only sparse topographic data are available.

image

Figure 4. Ensemble summary statistics for hydraulic model output derived from 100 topographic realizations, aggregated over the reach for base flow conditions (6.4 m3/s). The solid lines connect reach median values of the indicated ensemble summary statistic for each hydraulic quantity, and the vertical error bars represent the first and third quartiles of the distribution of that statistic within the reach.

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[41] The large increases in equation image and equation image and in the interquartile range of these quantities that occurred for the greatest cross-section spacing reflect an instability in the flow model that arose when the distance between cross sections was more than twice the wetted channel width. For such sparse data, the topographic simulations were not well conditioned, which resulted in large differences in bed elevation among realizations that translated into much different flow model solutions. Some of the simulations produced very high water-surface elevations for the upper portion of the reach, with the point bar producing a sort of backwater effect. For other realizations, the modeled water surface profile was more uniform and elevations upstream of the point bar were much lower. Because both of these scenarios were included within the same ensemble for the largest cross-section spacing, equation image, equation image, equation image, and equation image were all much greater. These results imply that if the elevation data delivered as input to a 2D flow model are not of sufficient density, the model might produce fundamentally different solutions depending on how the topography between widely spaced cross sections is represented.

3.2.3. Subbankfull Model Runs

[42] For the subbankfull discharge of 32.6 m3/s, the flow model was less sensitive to uncertainty in the topographic input data. Figure 5 indicates that reach-aggregated ensemble standard deviations and ranges of all four hydraulic quantities increased with the mean cross-section spacing, as was the case for the base flow model runs, but at a more gradual rate. For example, as cross-section spacing increased from 7 to 40 m, equation image increased from 0.03 to 0.21 m for base flow conditions but only changed from 0.11 to 0.20 m for the subbankfull discharge. Moreover, because the point bar presented less of an obstacle to the flow at the higher discharge, the oscillations between high and low water-surface elevations from one realization to the next that were observed for the base flow model runs were not evident; as a result, the spread of the reach-aggregated distributions was not significantly greater for the largest cross-section spacing. This result implies that the effects of topographic uncertainty on flow model predictions are less pronounced at higher discharges, when the influence of the river's morphology on the flow field is diminished [Whiting, 1997; Legleiter et al., 2011].

image

Figure 5. Ensemble summary statistics for hydraulic model output derived from 100 topographic realizations, aggregated over the reach for a subbankfull discharge of 32.6 m3/s. The solid lines connect reach median values of the indicated ensemble summary statistic for each hydraulic quantity, and the vertical error bars represent the first and third quartiles of the distribution of that statistic within the reach.

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3.2.4. Mean Response of Flow Model Predictions for Base Flow and Subbankfull Runs

[43] A more subtle observation stemming from this analysis was the mean response of the model's predictions of velocity and shear stress to differences in survey density. Both equation image and equation image decreased slightly as the spacing between cross sections increased. This effect was more pronounced under base flow conditions, when equation image and equation image decreased by 13% and 23%, respectively, over the range of cross-section spacings considered (Figure 4); for the subbankfull model runs, the corresponding changes in these values were −4 % and −17 % (Figure 5). With fewer data available to condition the simulations, the shape of the point bar was not resolved as well. As a result, flow through the bend was less constricted by the bar, reducing the strength of the high-velocity core along the outer bank. Because velocities and shear stresses in this region were lower, the reach-aggregated ensemble means for u and equation image decreased as a result. The greater reduction in equation image and equation image at base flow, when these topographic steering effects were more pronounced [Whiting, 1997; Legleiter et al., 2011], implies that sparser surveys resulted in a representation of channel morphology for which the influence of the point bar on the flow field was reduced, leading to lower predictions of velocity and shear stress.

3.3. Spatial Patterns of Model Uncertainty in Relation to Channel Morphology

[44] To examine the spatial structure of the uncertainty in flow model predictions that arises as a consequence of uncertain topographic input data, we generated 100 topographic realizations conditioned to different, random subsets of the full survey. One third of the original data were retained for each of these simulations, 15 of which are depicted in Figure 6. Because the conditioning data were selected at random from a fairly dense survey, the gross morphology of the channel was faithfully reproduced by each realization, but closer inspection of the examples in Figure 6 indicates that the topography did vary in detail from one simulated surface to the next, to a far greater degree than when all survey points were used to condition the realizations.

image

Figure 6. Topographic realizations conditioned to different random subsets of the survey data; one third of the original data were retained for each simulation. The individual realizations depicted here were selected at random from an ensemble of 100 simulations of the bed topography, each based on a different, randomly selected subset of the original survey data.

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[45] Ensemble means, standard deviations, and ranges for the four hydraulic quantities of interest for base flow and subbankfull model runs are shown in Figures 7 and 8, respectively. These maps illustrate the spatial structure of the uncertainty in flow model predictions that is attributable to uncertain topographic input data. Figures 7a and 8a are based on simulations conditioned to randomly selected thirds of the original data set, whereas Figures 7b and 8b are based on realizations conditioned to the full survey. The use of a common color scale for both sets of plots emphasizes the increased uncertainty associated with a sparser data set; thinning our survey by a factor of 3 caused ensemble standard deviations and ranges to increase by at least a factor of 2 and in some cases by an order of magnitude.

image

Figure 7. (continued)

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image

Figure 8. Ensemble summary statistics for model predictions at a base flow discharge (6.4 m3/s), based on topographic realizations conditioned to (a) different, random subsets of the original survey and (b) the full data set. Note that the color scale is the same for both sets of plots. Arrows and text labels highlight specific features described in the text.

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image

Figure 9. (continued)

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image

Figure 10. Ensemble summary statistics for model predictions at a subbankfull discharge (32.6 m3/s), based on topographic realizations conditioned to (a) different, random subsets of the original survey and (b) the full data set. Note that the color scale is the same for both sets of plots. Arrows and text labels highlight specific features described in the text.

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3.3.1. Base Flow Model Runs

[46] Figure 7a indicates that for all four hydraulic quantities, model uncertainty, as indexed by ensemble standard deviations and ranges, was least at the lower and upper ends of the reach, within the buffer of constant, kriged topography; the location of this buffer is labeled K in Figure 7a (bottom right). The relatively large values of RE observed at the entrance to the reach (labeled A), however, indicate that the effects of topographic uncertainty can be propagated spatially by the model to produce uncertain model predictions even in locations where the bed elevation was held constant. Because the model enforces a boundary condition at the downstream limit of the domain, the water-surface elevation at this location is fixed. As a result, uncertainty in predicted water-surface elevations and depths is minimal in the buffer of kriged topography at the lower end of the reach and greater near the upper boundary. Importantly, however, the presence of moderate values of equation image, Ru, equation image, and equation image within the kriged buffer at the lower end of the reach (labeled B) indicate that the effects of topographic uncertainty can be propagated downstream by the model.

[47] Figure 7a (top row) shows that the greatest uncertainty in the modeled water-surface elevation occurred over the point bar, with the largest values of equation image observed along the inner bank (labeled C). Ensemble ranges were greater toward the outer bank upstream of the bend apex, relatively small across the entire channel below the apex, and then increased abruptly entering the straight riffle downstream. The spatial pattern of uncertainty in flow depth was dictated by that of E, along with the gross morphology of the channel. Throughout most of the reach, equation image was on the order of 0.15 m, with somewhat higher values along the margin of the point bar near the bend apex and toward the right bank downstream of the bar. A narrow swath of high values of Rh, within which predicted depths varied by over 1 m among the realizations, was observed along the center of the channel beginning near the bend entrance, extending through the apex, and then widening over the downstream end of the bar (labeled D). The ensemble range was largest near the center of the channel because the cross-stream position of the steep point bar face shifted laterally from one realization to the next depending on which survey points were retained as conditioning data.

[48] The spatial organization of uncertainty in model predictions of flow velocity also corresponded closely with the river's morphology. Figure 7a (third row) indicates that equation image was greatest in three regions labeled E: (1) near the right bank just upstream of the point bar, where the flow first encountered the bar and was steered toward the opposite bank; (2) along the outer bank just past the bend apex, where the high-velocity core was best developed; and (3) at the exit from the bend, where the flow accelerates over a convexity in the bed topography and enters the riffle. The pattern of Ru is similar, with predicted flow velocities varying by over 2 m/s among the realizations near the bend apex as the narrow zone of high velocities shifted laterally as a consequence of the uncertain topographic input data. The spatial structure of the uncertainty associated with model predictions of boundary shear stress (Figure 7a, fourth row) closely matches that of velocity because we used a single drag coefficient that directly relates stress to the square of the velocity; note that the full, quasi-3D version of FaSTMECH considers vertical structure and secondary flows and would thus predict more complex stress fields. For the 2D model, the greatest values of equation image and equation image occur where the bar forces the flow toward the outer bank to produce a zone of high stresses along the channel thalweg (labeled F). These metrics of uncertainty took on much lower values throughout the remainder of the model domain, except for a zone of moderately high values of equation image over the riffle at the lower end of the reach.

3.3.2. Subbankfull Model Runs

[49] Figure 8 illustrates the spatial distribution of ensemble summary statistics for each modeled hydraulic quantity at a higher, subbankfull discharge of 32.6 m3/s. The patterns evident in these maps were similar to those observed at base flow in some respects, but also highlighted notable differences in the manner in which model runs for the two discharges were affected by uncertain topographic input data. Most notably, Figures 8a and 8b, which represent simulations conditioned to random subsets of the original survey and to the full data set, respectively, are more comparable for the subbankfull model runs in Figure 8 than for the base flow runs depicted in Figure 7. For the higher, subbankfull discharge, ensemble standard deviations and ranges again were greater when the conditioning data were more sparse but were of the same order of magnitude as when the full survey was used to condition the realizations; the spatial patterns of uncertainty depicted in Figure 8a were muted but still evident in Figure 8b. For the lower, base flow discharge, in contrast, the uncertainty associated with model predictions based on the full data set was negligible in comparison to the uncertainty associated with sparser conditioning data (contrast Figures 7a and 7b). These results imply that the flow model was somewhat less sensitive to uncertain topographic input data at a subbankfull discharge than under base flow conditions, a finding consistent with our analysis of cross-section spacing.

[50] Another salient distinction between the base flow and subbankfull model runs is the apparent narrowing of the channel at the upper and lower ends of the reach (labeled K in Figure 8a), which reflects the buffer of constant, kriged topography used to ensure model stability near the boundaries of the domain. Through the middle of the reach, where bed elevations varied among the simulations, the channel appeared wider because some realizations had lower elevations along the channel banks that allowed these areas to be inundated for at least a few of the 100 model runs. Whether or not a given portion of the bank became wet was thus one manifestation of the effects of uncertain topographic input data on predictions from the flow model.

[51] Figure 8a (top row) indicates that for the subbankfull model runs, uncertainty in predicted water-surface elevations was greatest over the point bar, with equation image and RE values up to 0.1 and 0.4 m, respectively, in this region (labeled A). Because the shape and amplitude of the bar surface varied from one topographic realization to the next, some flow model solutions required higher water-surface elevations over the bar than others, resulting in large values of equation image. High values of RE were also observed near the lower end of the reach, just upstream of the buffer of constant, kriged topography, but equation image was not especially large in this area (labeled B). The combination of a large RE with a relatively small equation image implies that there were a few outlying realizations with unusually low or high water-surface elevations that created a large ensemble range, which could reflect a numerical instability or artifact associated with the transition from the constant, kriged elevations to the simulated topographies.

[52] Even at a higher discharge that inundated more of the bar and presumably reduced the bar's effect on the flow field, model predictions of depth, velocity, and shear stress exhibited a high degree of sensitivity to topographic uncertainty that varied spatially in relation to the river's morphology. Importantly, Figure 8b indicates that this sensitivity was evident not only when random subsets of the original survey were used to condition the realizations, but also when the simulations were conditioned to the full data set. For flow depth, Figure 8a (second row) shows that equation image and Rh were largest in the middle of the channel downstream of the bend apex where the point bar sloped down into the adjacent pool (labeled C). As the geometry of the bar surface varied from one realization to the next, predicted depths at a given location differed by over 1 m. This observation indicated that the model was most sensitive to topographic uncertainty where spatial gradients of the topography were strongest, such as along the steep face of the bar. In Figure 8b, equation image and Rh took on very low values at points aligned along a series of cross sections that correspond to the original survey data locations. Because these measurements were used to condition the simulations, the bed elevation was fixed and the depth, calculated by subtracting the bed elevation from the water-surface elevation predicted by the model, therefore varied little among the realizations. Away from the data locations, however, greater uncertainty was associated with depth predictions.

[53] For velocity, equation image and Ru were relatively low for most of the reach, with a small area of higher values concentrated over the point bar (labeled D), for both the realizations based on random subsets of the original survey and those conditioned to the full data set. The manner in which the bar form was represented in a given realization exerted a strong influence on the predicted velocities in this region due to the relatively shallow depths and low velocities over the top of the bar as most of the flow was steered across the channel into the pool. The spatial pattern of uncertainty associated with the shear stress field was similar, with the greatest values of equation image and equation image again concentrated near the point bar. The highest values of these uncertainty metrics occurred closer to the outer bank, just past the bend apex in the zone of high stress through the pool (labeled E). Moderately high values of equation image and equation image were also observed toward the lower end of the reach where the flow exited the pool and entered the straight riffle downstream.

3.3.3. Influence of Topographic Perturbations on the Local Flow Field

[54] To quantify the relationship between simulated topographic irregularities and the local flow field predicted by the model, we examined the spatial pattern of the correlation coefficient rz′,u′ between departures of the bed elevation and flow velocity from their respective ensemble means. Figure 9 indicates that moderately strong positive correlations between z′ and u′ occurred throughout most of the channel, implying that a positive perturbation of the bed (i.e., a bump rather than a hole) tended to be associated with a relatively high velocity for that location. Because the water-surface elevation field was more spatially persistent than the simulated topographic fields, a positive elevation perturbation at a given location was unlikely to produce a significant increase in E at that same location, though proximal values of E might have been affected. If the bed elevation increased but the water-surface elevation did not, the flow depth would have been reduced, and the velocity must have increased in order to convey the same mass of fluid. This pattern did not hold over the top of the point bar (labeled A in Figure 9), however. In this region rz′,u′ was either very small in absolute value or negative, implying that positive perturbations of the bed tended to be associated with reduced velocities. For this part of the channel, the bar presented an obstruction to the flow that caused shoaling and deceleration as fluid was steered laterally toward the outer bank by the topography; positive elevation perturbations acted to accentuate these effects and thus resulted in lower velocities.

image

Figure 11. Local correlation coefficient between z′ and u′ for simulations conditioned to different, randomly selected subsets of the original survey data for a subbankfull discharge of 32.6 m3/s. Arrows and text labels highlight specific features described in the text.

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3.4. Effects of Topographic Uncertainty on Habitat Suitability Analysis

[55] An increasingly common application of 2D flow models is the assessment of aquatic habitat. To characterize the effects of topographic uncertainty on studies of this kind, we created local distributions of the habitat suitability index (HSI) described in section 2.5. A series of model runs, each based on a different realization of the bed topography, was performed for a base flow discharge of 6.4 m3/s, and the resulting predictions of depth and velocity were used to compute HSI values for each node of the model's computational grid.

[56] The results of this analysis are shown in Figure 10, in which Figure 10a corresponds to realizations conditioned to random data subsets, and Figure 10b corresponds to simulations conditioned to the full survey. The first rows of Figures 10a and 10b show five different quantiles, denoted by numerical subscripts in parentheses, of local HSI distributions within the reach. These maps indicate that a wide range of HSI values might be calculated for a given location depending on the topographic information provided as input to the flow model. For example, in the straight riffle at the lower end of the reach, the map of median HSI values based on simulations conditioned to a random subset of the survey data features two zones of highly suitable habitat separated by an area for which the HSI is on the order of 0.5 (labeled A). The quantiles of the HSI distribution in this region, however, indicate that the HSI could range from 0.1 or less to nearly 1 solely as a consequence of uncertain topographic input data. The HSI quantiles in Figure 10b, in contrast, are more similar to one another because all survey points were retained for conditioning, resulting in a lesser degree of topographic, and thus flow model, uncertainty.

image

Figure 12. Effects of topographic uncertainty on habitat suitability analysis for simulations conditioned to (a) random subsets of the original survey and (b) the full data set. In the first rows of each set of plots, five quantiles of the local habitat suitability index (HSI) distributions are mapped. The second row illustrates the probability of exceeding the specified HSI values, and the third row indicates probabilities of class membership for a series of habitat quality categories. Arrows and text labels highlight specific features described in the text.

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[57] The local distributions of HSI values generated via a simulation approach could also be used to compute probabilities of exceeding certain HSI thresholds, given the uncertainty associated with uncertain topographic input data. This type of representation is illustrated in the second rows of Figures 10a and 10b and provides additional insight on the effects of topographic uncertainty on habitat assessment. For example, the map of HSI(50) in the first row, third column of Figure 10a indicates a fairly continuous swath of high-quality spawning habitat at the upper end of the reach above the point bar. The map in the second row, fourth column, however, indicates that along the right side of the channel upstream from the bar, there is a probability of 0.5 or greater that the HSI does not exceed 0.75 (labeled B). Only near the upper boundary of the reach on the right side of the channel is the probability of exceeding an HSI value of 0.9 consistently high (labeled C). In Figure 10b, reduced topographic uncertainty results in exceedance probability maps that are more binary in nature. Here, more extensive dark red and dark blue tones indicate very high or very low probabilities, respectively, and the areas with intermediate colors representing moderate probabilities of exceeding each HSI cutoff are noticeably smaller.

[58] Another strategy enabled by stochastic simulation involves designating HSI values that bracket a set of habitat suitability categories and using the ensemble of flow solutions to calculate probabilities of membership within each of these categories. For example, the third row of Figure 10a shows that the probability of poor quality habitat (HSI < 0.1) is low throughout the reach; low quality habitat (0.1 ≤ HSI < 0.4) is only consistently predicted to occur in the pool, where depths and velocities are too great (labeled D); medium quality habitat (0.4 ≤ HSI < 0.6) is concentrated in an elongated, narrow zone left of the centerline above the bend apex (labeled E); high quality habitat (0.6 ≤ HSI < 0.9) is located in two separate zones in the downstream riffle and along the left bank at the top of the reach (labeled F); and excellent habitat (HSI ≥ 0.9) is only highly likely on the right side of the channel near the upper boundary of the reach (labeled G). Again, when the simulations were conditioned to the full data set, the reduced topographic uncertainty resulted in habitat class probability maps that were more binary (Figure 10b).

3.5. Effects of Topographic Uncertainty on Bed Mobility Studies

[59] Two-dimensional flow models are also widely used to characterize sediment transport, and a common objective is to evaluate the mobility of different particle sizes under a range of flow conditions. These studies rely upon model predictions of boundary shear stress and are thus affected by the uncertainties inherent to these predictions, including those that might arise as a consequence of uncertain topographic input data. To examine the influence of topographic uncertainty on bed mobility analysis, we performed two sets of model runs for a subbankfull discharge of 32.6 m3/s: one based on simulations conditioned to random data subsets (Figure 11a) and the other based on realizations conditioned to the full survey (Figure 11b). These two ensembles were used to produce distributions of predicted equation image values for each grid node, and five quantiles of these distributions are shown in the first rows of Figures 11a and 11b. Each of these maps exhibits the same general spatial pattern, with low stresses at the upper end of the reach above the point bar and higher equation image values extending downstream from the bend apex, particularly along the outer bank. The maps also indicate, however, that the distribution of equation image predictions for a given location can be quite broad due to the uncertain topographic input data upon which these predictions were based. For example, the zone of high shear stress in the pool has an ensemble median shear stress on the order of 50 Pa (labeled A), but the 5th and 95th percentiles of the equation image distribution are as low as 35 Pa and as high as 60 Pa, respectively. In contrast, in Figure 11b, based on bed elevation realizations conditioned to the full data set, the maps for the different quantiles are more similar to one another, indicating that the predicted stresses varied to a lesser degree among the realizations when the level of topographic uncertainty was reduced.

image

Figure 13. Effects of topographic uncertainty on bed mobility analysis for simulations conditioned to (a) random subsets of the original survey and (b) the full data set. In the first row of each set of plots, five quantiles of the model's shear stress predictions are mapped. The second row shows the grain size at the threshold of entrainment corresponding to each of these quantiles. The third row illustrates the probability of motion for five different grain sizes. Arrows and text labels highlight specific features described in the text.

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[60] Another geomorphic question often addressed via 2D flow modeling is: what is the largest particle size that will be entrained by a given flow? In the second rows of Figures 11a and 11b we have plotted the grain size at the threshold of entrainment for each of the shear stress percentiles in the top row. Because equation (1) is a linear relationship between shear stress and grain size, the spatial patterns in the first two rows of Figures 11a and 11b are identical. The critical grain size maps in the second row indicate that, for a given location, the grain size predicted to be mobile in the pool, where the stress is greatest, ranges from less than 50 mm (labeled B) to more than 80 mm (labeled C) over the ensemble of topographic realizations. In the riffle near the lower end of the reach, the 5th and 95th percentiles of the shear stress distribution correspond to mobile grain sizes of 30 and 70 mm, respectively (labeled D and E). As before, the range of predicted mobile grain sizes is less in Figure 11b, based on elevation realizations that were conditioned to the full, original survey and thus reflecting a lesser amount of topographic uncertainty.

[61] An important implication of these results is that an evaluation of whether a particular grain size will be mobilized by some discharge of interest is perhaps best expressed in terms of probabilities, due to the uncertainty associated with model predictions of boundary shear stress. For example, the third rows of Figures 11a and 11b show the probability of motion for five different grain sizes. In this case, these probabilities do not represent the effects of turbulent fluctuations of the boundary shear stress nor the presence of a range of grain sizes on the streambed [e.g., Grass, 1971], but rather the proportion of the topographic realizations for which the predicted equation image value exceeded the critical shear stress for that grain size. Even for the simulations conditioned to the full data set, the entrainment probability maps are not strictly binary but instead indicate the presence of zones where certain particle sizes might or might not be predicted to be mobile, depending on the topographic representation used to parameterize the flow model. For example, for a grain size of 45 mm, the probability of entrainment is very high through the pool (labeled F), but farther downstream is a zone of moderate entrainment probabilities on the order of 0.65 (labeled G), indicating that the shear stresses in this region were capable of mobilizing particles of this size for some of the topographic realizations but not for others.

4. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information

[62] One of the primary advantages of a spatial stochastic simulation approach is the ability to generate not just a single prediction that lacks any indication of the uncertainty associated with that prediction, but rather a full probability distribution of predicted values that reflects the uncertainty inherent to model outputs derived from uncertain model inputs (Figure 1). Information of this kind is potentially quite valuable in a management context because quantitative indices of model uncertainty can help to guide decisions that must be made in spite of such uncertainty. The simulation framework employed in this study quantifies, in a spatially explicit manner, the effects of topographic uncertainty on 2D flow models. The approach is also computationally demanding, however, and might not be practical to implement on a case-by-case basis. Nevertheless, our results provide some general insight regarding the influence of topographic uncertainty on various applications of 2D flow models. In this section, we first discuss additional sources of uncertainty not examined in this investigation and then highlight some of the key implications of uncertain topographic input data for studies of habitat suitability and bed mobility.

4.1. Additional Sources of Uncertainty

[63] This study has shown that predictions of water-surface elevation, depth, flow velocity, and boundary shear stress from a typical 2D flow model are subject to a significant degree of uncertainty, even for a single bend of a relatively simple gravel bed river and solely as a consequence of uncertain topographic input data. We wish to point out, however, that uncertain model output can arise for a number of other reasons we have not considered herein. Additional sources of uncertainty include: (1) parametric uncertainty related to the characterization of flow resistance and eddy viscosity, to name but two examples; and (2) structural uncertainty associated with the assumptions regarding process representation and numerical approximation that are incorporated into a particular flow model [Cao and Carling, 2002]. This study focused on a third source of uncertainty, the model's boundary conditions. Although we examined only the effects of uncertain topographic input data, one must bear in mind that other parametric and structural factors also introduce significant uncertainty to predictions from 2D flow models.

[64] An important limitation of this study, given our focus on the effects of uncertain topographic input data, was the relatively low resolution of our original survey. These data were collected to support ongoing, larger-scale investigations of channel morphodynamics and habitat development [Legleiter et al., 2011; L. R. Harrison et al., submitted manuscript, 2010] and consisted of a number of regularly spaced cross sections. Although this sampling strategy is widely used in practice, several recent studies have demonstrated that distributed, terrain-sensitive elevation measurements, along with appropriate use of interpolation methods, can yield improved representations of fluvial topography [e.g., Lane et al., 1994; Fuller et al., 2003; Valle and Pasternack, 2006; Wheaton et al., 2010]. Similarly, newly developed remote sensing techniques provide essentially continuous, high-resolution topographic information over greater spatial extents [e.g., Marcus and Fonstad, 2008; Legleiter et al., 2009; McKean et al., 2009]. Due to the low density and simple configuration of our survey data set, we were not able to fully explore the effect of survey density and sampling strategy on topographic and hence flow model uncertainty. Because our results were derived from cross section-based, relatively sparse elevation data, the magnitude and spatial pattern of the uncertainties we observed might not reflect the uncertainties that would be associated with a terrain-sensitive survey coupled with more advanced interpolation techniques (e.g., triangulated irregular networks, or TIN's). Future research could examine these issues by starting from a high density, distributed survey, progressively thinning the original data set to mimic lower resolution and/or cross section-based surveys, and using a stochastic simulation approach to quantify the influence of topographic sampling strategy on the uncertainty associated with flow model predictions. Similarly, because the influence of uncertain topographic input on 2D model output depends on channel morphology, comparative evaluations of relatively simple channels, such as the restored reach of the Merced River we considered, and more complex, natural streams could also yield more general insight.

[65] A common application of 2D flow models is the assessment of habitat suitability for various species of concern, across a series of life history stages. Because HSI values depend on 2D model output, the effects of uncertain topographic input data are propagated through the model to introduce uncertainty to habitat assessment, as illustrated in section 3.4. A number of other sources of uncertainty are also relevant, however, including the biological uncertainty associated with the preference curves used to relate flow conditions to habitat use. The empirical data used to construct these curves are often rather limited and pertain only to the species, life stage, and field site for which they were collected, although such curves are commonly extrapolated to other locations as well [e.g., Moir et al., 2005]. We have demonstrated how the uncertainty attributable to uncertain topographic input data can be summarized by expressing HSI values in probabilistic terms, and Leclerc [2005] describes probabilistic habitat indices that account for the biological uncertainty inherent to habitat evaluation. The limitations of HSI-based assessments are increasingly recognized, however, and a number of alternatives have been proposed, including individual-based, bioenergetic models [e.g., Railsback et al., 2003] and/or process-oriented ecological models that consider dynamics across a range of scales and levels of biological organization [e.g., Anderson et al., 2006]. Stochastic descriptions of hydraulic conditions in stream reaches [e.g., Schweizer et al., 2007] provide another means of accommodating various sources of uncertainty, and Mouton et al. [2009] discuss an approach based on fuzzy logic that can improve the reliability of habitat modeling. Van der Lee et al. [2006] used a Monte Carlo simulation technique similar to that employed in this study to quantify the effects of both habitat preference curves and input data on HSI values, but further research is needed to improve our understanding of the interactions among topographic, biological, and other sources of uncertainty in habitat assessment.

[66] Two-dimensional flow models are also widely used to examine bed mobility, predict sediment transport, and forecast channel change. In this context, a number of other factors, in addition to the effects of uncertain topographic input data, contribute to the uncertainty inherent to studies of this kind. In addition to the general issues associated with 2D flow modeling described above, uncertainty can also arise from the characterization of the bed material grain size distribution, the specification of a critical boundary shear stress for entrainment, and the effects of bed structure, armoring, and interparticle arrangement. In this study, attention was focused on the influence of topographic uncertainty on model predictions of boundary shear stress, but we acknowledge the potential significance of these other factors as well.

4.2. Implications of Topographic Uncertainty

4.2.1. Assessment of Habitat Suitability

[67] The sequence of maps presented in Figure 10 illustrate how a simulation-based approach can be used to quantify the uncertainty inherent to habitat suitability analyses based on flow model predictions that are derived from uncertain topographic input data. Typically, a single model run based on a single topographic data set is used to assess habitat suitability and each location is assigned a single HSI value, with no indication of the uncertainty associated with that value. Under the simulation framework depicted in Figure 1, however, each location is instead characterized by a distribution of HSI values. These distributions can be used to summarize, in a quantitative, spatially explicit manner, the uncertainty inherent to the assessment of habitat.

[68] In general, a comparison of Figures 10a and 10b, which differ in the number of data retained to condition bed elevation realizations, implies that the habitat suitability analysis based on the more complete survey data set (Figure 10b) was less ambiguous due to a lesser degree of topographic, and hence flow model, uncertainty. In the presence of such uncertainty, reporting a single HSI value that fails to acknowledge this uncertainty is inappropriate, and habitat conditions are more realistically described in probabilistic terms. To reiterate, the distributions of HSI values illustrated in various ways in Figure 10 arise solely as a result of the uncertainty associated with the topographic data delivered as input to a 2D flow model. Our results clearly indicate that topographic uncertainty can propagate through flow models to produce highly uncertain evaluations of habitat quality.

[69] Stochastic simulation can thus yield insight regarding the magnitude and spatial pattern of not only the HSI values but also the uncertainties associated with those values. This kind of information could be quite valuable in an applied context. For example, if overstating the quality of the available habitat could have undesirable management consequences, a conservative approach in light of the uncertainty that might result from relatively sparse topographic input data would be to base habitat evaluations on a smaller quantile of the HSI distributions. Using HSI(25) values rather than the median HSI for each location would ensure that, for 100 equally likely realizations of the bed topography, no more than 25 of the corresponding flow model solutions produced HSI scores less than the HSI(25) value for that location.

[70] A similar approach would be to report probabilities of exceeding certain HSI thresholds, as shown in the second rows of Figures 10a and 10b. This strategy explicitly acknowledges the possibility that habitat conditions might not be as favorable as the nominal HSI (or some measure of the central tendency of the local distribution of HSI values, such as the median) indicates due to uncertainty in the underlying flow model predictions. Another alternative would be to define a series of habitat quality classes and use the local distributions of HSI values to determine the proportion of the realizations for which each node of the model's computational grid had an HSI value within each of these categories. This approach is illustrated in the third rows of Figures 10a and 10b, which makes the important point that every location within the channel has some probability of membership within each habitat suitability class, not just the one for which the probability is greatest. Again, because this kind of information quantifies the uncertainty involved in habitat assessment, a simulation approach can help guide management actions.

4.2.2. Evaluation of Bed Mobility

[71] In a geomorphic or engineering context, two-dimensional flow models are often used to assess bed mobility [e.g., Clayton and Pitlick, 2007; May et al., 2009], predict sediment transport [e.g., Lisle et al., 2000; Li et al., 2008], and evaluate channel restoration designs [e.g., Pasternack et al., 2004; Brown and Pasternack, 2009]. These studies rely upon model predictions of boundary shear stress and could thus be compromised by the uncertainty associated with these predictions. Under the stochastic simulation framework adopted herein, rather than a single estimate of the boundary shear stress for each grid node, a distribution of predicted equation image values was produced from ensembles of model runs based on a series of topographic realizations. Quantiles were used to summarize these local shear stress distributions and indicate the range over which model predictions of equation image varied as a result of uncertain topographic input data. In general, the results described in section 3.5 and illustrated in Figure 11 clearly indicate that, in addition to all of the other factors that influence bed mobility and complicate its assessment, uncertain topographic input data can propagate through a flow model to produce highly uncertain predictions of boundary shear stress and hence bed mobility.

[72] For example, the quantiles of the local equation image distributions mapped in the first rows of Figures 11a and 11b indicate that the greatest uncertainty in model predictions of boundary shear stress occurred near the outer bank, where the magnitude of the stress was greatest. This pool region would thus be a critical area to characterize accurately for modeling particle entrainment, bed material transport, and channel change. Such an effort could be undermined by the uncertainty associated with the predicted shear stresses, however. For the reach considered herein, the median shear stress through the pool was on the order of 50 Pa, but the interquartile range of predicted equation image values was on the order of 10 Pa. Given the binary nature of typical bed mobility assessments based on Shields criterion, a 10 Pa range in equation image could easily cause some particle sizes to switch from mobile to immobile depending on whether the fixed critical shear stress for entrainment is exceeded. Similarly, the second rows of Figures 11a and 11b clearly show that uncertain topographic information can lead to highly uncertain predictions of which particle sizes are likely to be entrained by a given flow. If not only bed mobility but also bed material transport rates are of interest, the consequence of uncertain flow model predictions are even more severe: the strongly nonlinear relationship between boundary shear stress and bed material flux implies that a broad range of predicted equation image values can translate into an even wider range of bed material transport estimates, solely as a consequence of uncertain topographic input data.

[73] Given the importance of accurate predictions of boundary shear stress for bed mobility studies, and the sensitivity of these predictions to topographic uncertainty, a stochastic simulation approach provides an effective means of characterizing the uncertainty associated with predicted equation image values and allows sediment movement to be described in terms of probabilities. For example, if we had not adopted a simulation framework and instead used a single predicted shear stress, the entrainment probability maps depicted in the third rows of Figures 11a and 11b would be binary; a given particle size either would or would not be mobile at each location within the channel. This type of analysis also yields insight regarding the effects of topographic sampling strategy on predictions of shear stress and mobile particle size. For example, inspection of Figures 11a and 11b indicates that the entrainment probability maps are somewhat more binary in appearance for the simulations conditioned to the full survey data set (Figure 11b) than for the simulations conditioned to the random subsets (Figure 11a). This comparison implies that obtaining more reliable predictions of mobile particle size requires more detailed topographic data to define model boundary conditions.

5. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information

[74] Two-dimensional flow modeling is widely used to support a growing range of applications, from habitat assessment to prediction of sediment transport and channel change. The ability of such models to support these analyses is potentially compromised, however, by the inherent uncertainty associated with model predictions of depth, velocity, and boundary shear stress. In this study, we examined an important but often overlooked factor contributing to uncertain model output: uncertain topographic input. We adopted a spatial stochastic simulation framework and generated many, equally likely realizations of the bed topography. Because these simulations were spatially correlated and conditioned to the available survey data, this approach allowed us to investigate the magnitude and spatial pattern of topographic uncertainty associated with different data configurations. By delivering each realization in turn as input to a typical 2D flow model, we obtained distributions of predicted water-surface elevations, depths, velocities, and stresses for each node of the model's computational grid. This strategy had the distinct advantage of providing not just single estimates with no indication of the associated uncertainty, but rather full distributions of values that quantify the uncertainties that result from uncertain topographic input data. This kind of quantitative, spatially explicit information regarding the uncertainty inherent to 2D flow models can facilitate the informed use of such models for various applications in river research and management.

[75] The goal of this study was to quantify the effects of uncertain topographic input data on 2D modeling of flow through a simple, gravel bed meander bend. To pursue this goal, we (1) considered various survey sampling strategies; (2) evaluated stage-dependent hydrodynamic effects; (3) examined spatial patterns of uncertainty in relation to channel morphology; and (4) assessed the impacts of topographic uncertainty on analyses of habitat suitability and bed mobility. For each node of the model's computational grid, the uncertainty in model predictions was characterized by calculating ensemble summary statistics (mean, standard deviation, and range) for bed elevation realizations and corresponding model runs conditioned to several different data configurations. The resulting metrics of uncertainty were (1) aggregated over the reach to summarize the effects of cross-section spacing and (2) mapped to reveal the spatial organization of model uncertainty within the context of the meander bend. Ensembles of model runs were also used to demonstrate how topographic uncertainty propagates through a flow model to influence assessments of habitat quality and bed mobility.

[76] The principal conclusions of this study, which correspond to the research questions posed in section 1, include the following:

[77] 1. Our results clearly indicate that uncertain topographic input data can propagate through a 2D flow model to exert a strong influence on predictions of several key hydraulic quantities. Uncertain topographic boundary conditions alone can thus lead to model solutions that are potentially highly uncertain, and uncertainties related to model structure and parameterization could further exacerbate the inherent uncertainty associated with flow model predictions.

[78] 2. The uncertainty associated with model predictions of water-surface elevation, depth, velocity, and boundary shear stress increased steadily, in direct proportion to the uncertainty in the topographic input data, as the spacing between surveyed cross sections increased.

[79] 3. Model runs performed for a higher, subbankfull flow (75% of bankfull discharge) were somewhat less sensitive to topographic uncertainty than model runs for lower, base flow conditions, when the bed topography had a more pronounced influence on the flow field.

[80] 4. The spatial structure of the uncertainty associated with model predictions was strongly related to channel morphology. The greatest uncertainty occurred over and adjacent to the point bar, where bed slopes were greater and flow patterns were dictated by topographic steering effects.

[81] 5. Habitat suitability index values computed from model predictions of depth and velocity varied considerably among the ensemble of bed elevation realizations and corresponding model runs, implying that habitat assessments can be affected in a significant way by uncertain topographic input data. In the presence of such uncertainty, habitat conditions are better described in terms of quantiles of the distribution of HSI values or probabilities of exceeding certain HSI thresholds. Similarly, because uncertain topographic input data propagate through the model to produce uncertain predictions of boundary shear stress, analyses of bed mobility are also subject to a high degree of uncertainty. Determining the critical grain sizes corresponding to various quantiles of the shear stress distribution or specifying probabilities of entrainment for a given particle size are thus more appropriate methods of representing these results.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information

[82] The California Department of Water Resources provided logistical support for field data collection along the Merced River and Chris Robinson granted access to his property. Xiangmin Jiao developed and made available computer code for efficiently reading and writing data in the CGNS format. This study was supported by a grant from the Office of Naval Research (N0001409IP20057) to Jonathan Nelson. Three reviewers provided helpful comments on an earlier version of this paper.

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  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Methods
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information
FilenameFormatSizeDescription
wrcr12804-sup-0001-t01.txtplain text document1KTab-delimited Table 1.
wrcr12804-sup-0002-t02.txtplain text document1KTab-delimited Table 2.

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