### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Methods
- 3. Results
- 4. Discussion
- 5. Conclusion
- Acknowledgments
- References
- 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

- Top of page
- Abstract
- 1. Introduction
- 2. Methods
- 3. Results
- 4. Discussion
- 5. Conclusion
- Acknowledgments
- References
- 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?

### 5. Conclusion

- Top of page
- Abstract
- 1. Introduction
- 2. Methods
- 3. Results
- 4. Discussion
- 5. Conclusion
- Acknowledgments
- References
- 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.