Velocity estimation using a Bayesian network in a critical-habitat reach of the Kootenai River, Idaho

Authors


Abstract

[1] Numerous numerical modeling studies have been completed in support of an extensive recovery program for the endangered white sturgeon (Acipenser transmontanus) on the Kootenai River near Bonner's Ferry, ID. A technical hurdle in the interpretation of these model results is the transfer of information from the specialist to nonspecialist such that practical decisions utilizing the numerical simulations can be made. To address this, we designed and trained a Bayesian network to provide probabilistic prediction of depth-averaged velocity. Prediction of this critical parameter governing suitable spawning habitat was obtained by exploiting the dynamic relationships between variables derived from model simulations with associated parameter uncertainties. Postdesign assessment indicates that the most influential environmental variables in order of importance are river discharge, depth, and width, and water surface slope. We demonstrate that the probabilistic network not only reproduces the training data with accuracy similar to the accuracy of a numerical model (root-mean-squared error of 0.10 m/s), but that it makes reliable predictions on the same river at times and locations other than where the network was trained (root mean squared error of 0.09 m/s). Additionally, the network showed similar skill (root mean square error of 0.04 m/s) when predicting velocity on the Apalachicola River, FL, a river of similar shape and size to the Kootenai River where a related sturgeon population is also threatened.

1. Introduction

[2] The only spawning ground for the Kootenai River white sturgeon (Acipenser transmontanus) is a reach near Bonner's Ferry, Idaho (ID), USA [Paragamian et al., 2001]. This population of white sturgeon has been listed as endangered since 1994 due to ongoing failure of juvenile sturgeon to survive and become viable members of the population [Duke et al., 1999]. Failure of juvenile survival has been shown to be the result of changes to the sediment characteristics of the spawning reach after the Libby Dam became fully operational on the Kootenai River in 1974 [Paragamian and Kruse, 2001; Paragamian et al., 2001, 2002, 2009]. Once the dam was operational, regions of gravel and cobbles, which are conducive to sturgeon spawning [Parsley and Beckman, 1994], were covered by a layer of sandy sediment at least 1 m thick [Barton et al., 2004]. Paragamian et al. [2009] and McDonald et al. [2010] demonstrated that although the post-dam discharge and velocity magnitude varied from natural conditions, the patterns of velocity and maximum depth were consistent with pre-dam conditions, indicating that spawning locations were most likely constant over time. As a result, sturgeon continued to spawn in historical locations based on hydrodynamic conditions although sediment characteristics became unfavorable for juvenile survival.

[3] Beginning in the early 1990s, a recovery program was undertaken [Duke et al., 1999] with the aim of creating a habitat conducive to a sustainable sturgeon population. An important component of the recovery process includes restoration of morphologic and hydrodynamic conditions in the spawning ground habitat to improve juvenile survival. In order to design and implement a recovery program, information about past, present, and future hydrologic conditions must be known. Since data are not available for all river restoration scenarios, hydrodynamic model simulations of actual and plausible river conditions must be implemented [Czuba and Barton, 2011].

[4] A range of numerical models have been applied in this critical-habitat reach of the Kootenai River. Berenbrock [2005] implemented the one-dimensional Hydraulic Engineering Centers River Analysis System (HEC-RAS) flow model with the objective of determining the backwater/free-flowing transition upstream of Lake Kootenay, British Columbia that was hypothesized to control the location of spawning. Berenbrock and Bennett [2005] also implemented a one-dimensional sediment transport model with the aim of relating changes in sedimentation rate to changes in white sturgeon spawning. Barton et al. [2005] extended this approach by simulating flow and sediment transport using the quasi three-dimensional FastMech flow model to enhance the understanding of biological data in the context of river hydrodynamics. Most recently, Czuba and Barton [2011] improved the Berenbrock [2005] flow model using higher resolution boundary conditions with the objective of using model results in river restoration decision making.

[5] Development of a reliable numerical model represents a significant investment of time, expertise, and finances for both observationalists and modelers. For example, boundary conditions for numerical models must be obtained by measuring bathymetry, water levels, and discharge, often at high resolution both spatially and temporally. The model must then be implemented, calibrated, and results validated against independent observations. Finally, the simulation results must be interpreted with respect to multiple restoration decisions. However, decision makers may not fully understand or control the simulations developed by the modelers.

[6] A significant challenge of river restoration is, therefore, integrating the understanding provided by these high-resolution model results into the decision making process [Stewart-Koster et al., 2010]. Bayesian networks are a tool that can be used to efficiently transfer the results of both conceptual and numerical models to nonmodelers [Stewart-Koster et al., 2010], while allowing nonmodelers to efficiently manipulate elements of the model. Bayesian networks make this possible by quantifying the probabilistic relationships between variables using a graphical format.

[7] As detailed by Plant and Holland [2011], this sort of Bayesian prediction approach has five main advantages. Primarily, this approach requires few computational resources once the network is trained (using detailed model results or observations) which streamlines end-user application over multiple queries. Also, computational complexity is minimized as predictions are focused on the locations and variables of most interest; so do not require solution over an entire model domain or under extensive boundary conditions. Additionally, the network is easily updated once it is formulated which allows straightforward modifications. The Bayesian approach also produces a forecast probability distribution resulting in estimation of the predictive uncertainty of the forecast. Lastly, predictions can be made when input data are uncertain or unavailable, and the approach can be applied in reverse to establish sensitivities to forcing conditions as simply as when it is applied as a forward model. In addition to the advantages described above, Bayesian networks can be trained to assimilate observations other than those explicitly included in physical model simulations, allowing development of a hybrid conceptual-physical model.

[8] We hypothesize that a network developed for the Kootenai River can not only reproduce the information with which it was trained, but can also be used to predict velocities at other times and locations that are dynamically consistent with the training set, making the network a useful simulation tool in itself. Our objective is to test this hypothesis by developing a Bayesian network for predicting depth-averaged velocity on the habitat-critical reach of Kootenai River based on numerical model results and then to use the network predictions to assess the spatial distribution of velocity related to sturgeon spawning habitat. We also aim to demonstrate the potential for broader applicability of this approach by making velocity predictions for the Apalachicola River, Florida (FL) which has similar hydrologic and hydrodynamic characteristics to the Kootenai River and is a spawning ground for a threatened population of Gulf sturgeon (Acipenser oxyrhynchus desotoi).

[9] In section 2, we describe the study site and the characteristics of sturgeon spawning locations in the Kootenai River. The physical properties upon which our network is based are described in section 3. Also in section 3, we review the Bayesian approach and following the methodology outlined by Chen and Pollino [2012], we outline our network design, training, and performance criteria for a section of the Kootenai River, ID. Results are presented in section 4, with respect to velocity forecast comparisons with observations on the Kootenai River at locations and times independent of where the network was trained. In section 5, we discuss the sensitivity of the forecast input variables and resolution and the transferability of our Bayesian network to the Apalachicola River, FL.

2. Study Site and Sturgeon Spawning Habitat

[10] The Kootenai River originates in British Columbia, travels through Montana and Idaho, then returns to British Columbia before emptying into the Columbia River at the end of its 721 km course [Paragamian et al., 2009]. The study site for this work is focused downstream of Bonner's Ferry, ID, between river kilometer 232.8 and river kilometer 244.5 (Figure 1), which includes the white sturgeon spawning habitat. Barton [2004] described the river morphology and bed characteristics in this meandering section of the Kootenai River as follows. The study area has an average sinuosity of 1.53, and a reach-averaged water surface slope of 2 × 10−5. The river bed is covered by dunes typically less than 1.3 m in height composed predominately of medium-grained sands (0.5–1 mm). Sand deposits typically range from 5 to 10 m in thickness.

Figure 1.

The study area extends from USGS Gage 12309500 at Bonners Ferry (river kilometer 232.8) to USGS Gage 12314000 at Klockmann Ranch (river kilometer 244.5). The Sobek model domain used to train the Bayesian network for predicting velocity covered the study area. Imagery was acquired as part of the National Agriculture Imagery Program.

[11] Correlation between river velocity and white sturgeon spawning location (maximum correlation coefficient = 0.45, significant at the 99% level) suggests that spawning occurs near the peak velocity [McDonald et al., 2006; Paragamian et al., 2009]. Correlation between river depth and spawning location (maximum correlation coefficient = 0.4, significant at the 99% level) suggests that spawning coincides to some extent with the deepest depths along the Kootenai River [McDonald et al., 2006; Paragamian et al., 2009]. Modeling efforts demonstrate that those locations remain constant regardless of discharge [Paragamian et al., 2009]. As described above, previous studies have demonstrated that deep, fast moving regions are correlated to sturgeon spawning. In section 4.2, we will test whether a Bayesian network can be used to identify similar correlations.

3. Methods

[12] If the relationships between morphologic and hydrodynamic characteristics that affect sturgeon spawning can be captured using a Bayesian network, then the network can be used to evaluate the likelihood that a reach will be suitable for sturgeon spawning without having to directly manipulate detailed numerical model scenarios. The following section describes Bayes Theory and how it is applied to develop a network to predict depth-averaged velocity.

3.1. Bayes Theory

[13] Bayesian inference is the use of Bayes theorem to forecast the probability, p, of an unknown variable or variables, Fi, given the joint probabilities between the unknown variables and known variables, Oj, where Fi is the ith value in a discrete set of forecast possibilities and Oj is obtained from the jth observation location or time. Formally, Bayes theorem is

display math(1)

[14] The right-hand side of the equation describes the probability of the observations given a forecast probability (known as the likelihood) multiplied by the prior probability of the forecast. The prior probability is the climatological estimate of the forecast before new information is included in the model [Pearl, 1988]. These quantities are normalized by the total probability of the observations. The network is trained to learn the terms on the right-hand side given data or model results [Heckerman, 2008], so that equation (1) is not directly solved in this approach, but the terms in the equation are estimated using the joint probability distributions between known and unknown variables. The following two subsections are focused on the method used to develop and train a network so that depth-averaged velocity (the unknown) can be forecast given readily available riverine observables.

3.2. Network Development From a One-Dimensional Model for River Velocity

[15] In order to achieve our objective of estimating depth-averaged velocity, we designed our Bayesian network based on simulations from the depth-averaged equations describing the flow of water in a one-dimensional channel often referred to as the Saint Venant equations,

display math(2)

and

display math(3)

where h is water head elevation from the reference level, t is time, x is the distance along the channel, u is the flow velocity in the x direction, g is the gravitational constant, and n is the Manning coefficient describing bed roughness [Stelling et al., 1998]. Several previous studies [Berenbrock and Bennett, 2005; Czuba and Barton, 2011] demonstrate that one-dimensional approximations of flow produce errors less than 20% of depth-averaged and cross-section averaged velocity when compared with observations on the Kootenai River.

[16] We implemented the Sobek model [Dhondia and Stelling, 2004; Stelling et al., 1998] to model these dynamics, however, any of the previously implemented one-dimensional models for flow on the Kootenai River would have been equally valid for network training. The model was forced with measured discharge at the upstream boundary and water level at the downstream boundary between 12 May 2010 and 9 September 2010. Bottom boundary conditions were defined by cross sections collected by the United States Geological Survey (USGS) [Barton et al., 2004]. The bottom roughness in the model, n, was calibrated to produce the best match between modeled and observed water level and was assumed to be 0.1 on the banks, representative of a rough, vegetated region, and 0.022 in the main channel, representative of bed composed of medium sand.

[17] The Sobek model was validated by comparing water level and velocity simulations to observations from USGS gage 12310100 located within the study area at the Tribal Hatchery near Bonner's Ferry over the entire period modeled. The root-mean-squared difference between modeled and measured water level was 0.02 m, and the root-mean-squared difference between modeled and measured velocity was 0.10 m/s. Simulated velocity was also compared with transect averaged velocity observations collected using multiple passes of Global Positioning System drifters deployed on the river surface in July 2010 and adjusted to depth-averaged velocity by multiplying the mean surface velocity by a standard scaling value of 0.85 [Rantz, 1982]. The root–mean-squared difference between the adjusted drifter velocity and the model was also 0.10 m/s.

3.3. A Bayes Model for River Velocity

[18] Reformulating equations (2) and (3) as a Bayesian model, the forecasts, Fi, are

display math(4)

and the observations, Oj, are

display math(5)

where V is the depth-averaged velocity, H is the mean hydraulic depth, Q is the discharge, W is the river width, S is the water surface slope equal to ∂h/∂x, and n is the Manning's coefficient. x is the streamwise location of a transect and the index of the transect is given by the subscript k. A circumflex above a variable represents an observation. Using the Bayesian approach, the terms in equation (1) are estimated by determining the conditional probabilities between observations and forecasts through network training which naturally include forecast uncertainty. Estimates of forecast uncertainty are unavailable from the deterministic model described in section 3.2. At that point, the original model results described in section 3.2 can be discarded. Our approach to network design and training continues in sections 3.4 and 3.5.

3.4. Bayes Network Construction

[19] Once the variables relevant to the network were identified, the network was designed according to the methods described by Chen and Pollino [2012]. Our Bayesian network was formulated in Netica (Norsys Software Corporation, 2011, Netica version 4.16), a software program that allows the user to create a directed acyclic graph (DAG) where the variables in the network are represented as conditional probability tables (CPTs) and links between the CPTs represent the joint probability distributions between variables [Pearl, 1988]. Our Bayesian network, represented as a DAG, is schematically diagramed in Figure 2.

Figure 2.

The Bayesian network used to predict river velocity was trained for the study area shown in Figure 1 using the data presented in Figure 3. The nodes in the network were selected based on equations (2) and (3). The arcs represent joint probability distributions between the nodes. The mean and standard deviation for each variable are given below the distribution.

[20] The links between variables were determined in an iterative manner, where links were chosen then Bayesian predictions were made in a hindcast mode to assess network skill. In the hindcast, velocity was withheld from the network and other variables from the training set were provided to the network. Hindcast predictions of velocity were compared with velocity values in the training set for each test case. These Bayesian hindcast comparisons were evaluated using two criteria, the regression skill and the log likelihood ratio. The regression skill is defined as

display math(6)

where math formula is the mean-square error between the velocity predictions and the observations and math formula is the variance of the velocity observations that were withheld from the prediction. This skill term expresses the degree of correlation between training set values and Bayesian predictions. The log likelihood ratio is defined as

display math(7)

where math formula are the observations supplied to the Bayesian network to make the hindcast and math formula are the observations withheld from the hindcast so that they can be compared with Fi. The first term on the right-hand side of equation (7) represents the log of the probability of the hindcast given new information and the second term represents the log of the climatological probability, before new information is given. A value of LR greater than 1 represents an improvement in hindcast relative to the climatological probability.

[21] In contrast to solving for floating-point values of variables as in a numerical model, variables in the CPTs are discrete. The level of discretization has a significant effect on the skill of the model predictions. A major concern in discretization is over fitting the model to the data (bins too narrow), reducing model skill in new situations. Similarly, under resolving the variables (bins too wide) can reduce model skill. Discretization intervals were determined as a balance between maximizing precision of the predictions while minimizing computational complexity [Pearl, 1988]. For example, the number of unknown joint probabilities scales as the number of bins for each parent variable times the number of bins in the predicted variable such that it is optimal to keep the number of variables and bins to a minimum as long as predictive resolution is preserved. Additionally, the CPTs were discretized to create a somewhat uniform distribution, assuring that the integrated probability in each bin contained between 5 and 15% of the training set to avoid over fitting the model to the data. In some cases, breaks between the bins were also set to match natural breaks in the training set. Like choosing the links between CPTs, discretization of variables was determined in an iterative manner, where a discretization scheme was chosen, then hindcasts were made and compared using regression skill, equation (6), and likelihood ratio, equation (7). Once the final configuration of the best performing network was determined, the sensitivity of the network to including new information about each variable was tested and then each variable was ranked in order of importance relative to correctly predicting velocity.

3.5. Network Training

[22] Before training, the 22,400 conditional probabilities governing the relationships between the network variables were unknown. In order for the Bayes network to make predictions, it was trained with prior information. We trained the network using numerical results from the Sobek model (described in subsection 3.2) applied to a meandering section of the Kootenai River downstream of Bonners Ferry, ID, USA.

[23] Simulation results for every hour between 12 May 2010 and 1 August 2010 at 58 cross sections [Barton et al., 2004] located between USGS gage 12314000 at Klockmann Ranch and USGS gage 12309500 at Bonner's Ferry (Figure 1), resulting in 56,376 simulated values of math formula, and math formula. math formula was not available directly from model results, so math formula was estimated as math formula. The time series of model results used to train the network are plotted in Figure 3, where math formula ranges between 0 and 2 m/s, math formula ranges between 0.6 and 14 m, math formula ranges between 191 and 1210 m3/s, math formula ranges between 103 and 255 m, and math formula ranges between 0 and 0.003. Since math formula was assumed to be constant over the model domain, it was not included in the final network. Outside of these ranges the network is unable to make reliable predictions without further training. The probability distributions depicted in Figure 2 were determined using the training set described in this section. The model simulations were input to Netica, where the maximum-likelihood joint probabilities of the variables were determined using the counting-learning algorithm, the optimal algorithm for training sets without missing data (Norsys Software Corporation, 2011).

Figure 3.

The Bayesian network presented in Figure 2 was trained using these time series of Sobek model results at the 58 transects along the meandering section of the Kootenai River between Bonners Ferry, ID (USGS gage 12309500) and the Klockmann Ranch, ID (USGS gage 12314000). The period spans 4.5 months and includes low flow as well as flood conditions. Each line represents a different transect.

[24] Once the network was trained (Figure 2), the prior probabilities in the network, p(Oj), are actually a Bayesian prediction of the climatology itself. In this case, the climatological likelihood of a particular velocity prediction is known. For example, without any new information, we know a velocity of 0.6–0.7 m/s had a 0.11 chance of occurring in this section of the Kootenai River for the training period. More interestingly, however, the network can be used to make forward predictions of Vk based on new observations of the variables math formula, and math formula available from other sources.

3.6. Network Testing

[25] Once the network was developed and trained, we made predictions at times and locations independent from the model result times and locations used to train the network. From this point forward, predictions are defined as Bayesian network results, in this case Vk. We compared Vk predictions with independent observations of velocity from drifters and acoustic Doppler current profilers for two scenarios on the Kootenai River summarized in Table 1. The first scenario consists of drifter velocity data collected 13–15 August 2010, approximately 2 weeks after the training period was completed. The surface velocities from the drifters were converted to a depth-averaged velocity at a transect by averaging all of the surface velocities within a ±20 m range of the transect, then multiplying by a correction factor of 0.85 [Rantz, 1982] to account for the relationship between surface and depth-averaged velocity. The velocity measurement uncertainty was defined as the standard deviation of the surface velocities within the 20 m range of the transect. Transects where the network predictions were made are independent of transects used to train the network. In the second scenario, depth-averaged velocity was predicted for a surveyed transect near gage 12310100 at the Tribal Hatchery near Bonners Ferry between 2003 and 2011 by the U.S. Geological Survey using an ADCP over a variety of discharges.

Table 1. Description of the Two Scenarios Used to Test the Bayesian Predictions of Velocity
Test SetDatesMethodDescription
113–15 August 2010Drifter64 transects under similar flow conditions
22 May 2003 to 27 July 2011ADCP56 observations at a single transect

[26] To make the comparisons between the Bayesian predictions and the observations, we selected several performance criteria. Predictive skills were computed on the basis of the bias,

display math(8)

and the root-mean-squared difference,

display math(9)

[27] To verify the forecasts in a probabilistic sense components of the Brier Score decomposition [Murphy, 1973] is used. The Brier Score is

display math(10)

where M is the number of probability bins. We chose to use five probability bins, ranging from 0 to 0.2, 0.2 to 0.4, and so on. The choice to use five probability bins is independent of the 10 velocity bins. Each of the 10 velocity bins from each Bayesian prediction is treated as an individual forecast with probability p, and all forecasts are binned by their probability of occurrence. The parameter nm is the number of forecasts of any velocity in a probability bin, p(F)m is the average forecast probability in bin m, math formula is the observed frequency of occurrence for each forecast in bin m, and math formula is the observed frequency of occurrence averaged over all bins, so it is the climatologically averaged probability. The first term in equation (10), the reliability, describes the sum of squared differences between the probability of a forecast and the frequency that the forecast in bin m was observed. A perfectly reliable forecast has a reliability of 0. The second term in equation (10), the resolution, describes how different the probability of the observations in bin m is relative to the average probability of all the observations. The larger the value of the second term, the better the resolution of the forecast. A forecast can be completely incorrect, but have good resolution, which will drive the B lower. The final term in equation (10), the Brier Score uncertainty, describes the variability of the data. A high value of Brier Score uncertainty (low variability) will increase B. When B from two different sets of verification data are compared, it is important to compare the Brier Score uncertainty of the data, as it can have a large influence on B.

[28] Τhe Brier Skill Score,

display math(11)

where Βc is the Brier Score of the climatology, describes the skill of the forecast relative to the climatology. A Βs of less than 0 indicates a forecast that is worse than the climatological prediction, whereas a Βs between 0 and 1 indicates that the forecast is an improvement over the climatology.

3.7. Error in Observations

[29] Observations input to the Bayesian network also require an error estimate to account for increased prediction uncertainty caused by measurement error. Given that measurement uncertainty in the USGS data on the Kootenai and Apalachicola rivers were not recorded, neglecting measurement uncertainty is equivalent to assuming that the observations had no error, and the assumption of no error in observations has an impact on the shape of the predicted probability distribution. USGS guidelines suggest that measurements with a variation of greater than 5% of the discharge be redone [Oberg et al., 2005]. Since boats used for measuring river cross sections cannot make measurements in very shallow water, USGS guidelines suggest that distance from the survey end to the edge of the river is estimated using another method, either visually or with a range finder [Oberg et al., 2005]. A standard deviation estimate of up to 5% of the width may account for the actual uncertainty in surveyed river width. Although uncertainty in depth measurements from an ADCP are on the order of millimeters [Teledyne RD Instruments, 2013], the mean hydraulic depth is the cross-sectional area estimated from the ADCP transect divided by the width. Since the cross-sectional area, width, and discharge include estimates in the shallowest parts of the river some measure of uncertainty should be included as input to the Bayesian network. To determine the appropriate uncertainty for the input data, the sensitivities of network predictions were determined for cases assuming no measurement error, and assuming measurement errors of 2.5% of the variable magnitude, and 5% of the variable magnitude.

[30] Bias was relatively unaffected by the choice of uncertainty and varied by only 0.04 m for all combinations of measurement error. RMSD decreased by up to 0.05 m when measurement error was included, but the magnitudes of the errors was less important, as long as some uncertainty was included. Reliability improved by 84% when measurement error of 2.5% included on width, depth, discharge, and slope and by 70% when a measurement error of 5% was included on all input variables. Bias and RMSD were minimized and reliability most improved when variables input to the network included measurement errors of 5% of the width, 2.5% of the depth, 2.5% of the discharge, and 5% of the slope.

4. Results

4.1. Network Evaluation

[31] The network was evaluated on the Kootenai River with the two data sets introduced in section 3.6 and Table 1. In the following subsections, each data set is described and the network predictions are evaluated using equations (8)-(10).

4.1.1. Test Set 1

[32] Vk was predicted using the Bayesian network at 64 transects along a 20 km reach of the Kootenai River downstream of Bonner's Ferry, ID. Data used to make Test Set 1 predictions were collected during a field experiment 13–15 August 2010. Predictions were made in the same region as where the Sobek model was run, but not at the same time or the same location.

[33]  math formula was estimated by determining the area between elevation of the river bed from a multibeam survey [Barton et al., 2005] and the relatively constant water surface elevation measured over this period with a GPS and dividing by the width. math formula ranged between 3.5 and 8.33 m at the transects of interest (Figure 4). math formula was defined as the distance between river banks defined by the intersection of the river banks with the water surface. math formula on the Kootenai River ranged between 125 and 190 m (Figure 4). Water surface slope, 2 × 10−5, was estimated from the gradient in water surface elevation measured along a downstream transect using GPS. math formula for the period between 13 and 15 August 2010 on the Kootenai River was measured at 256 m3/s using an ADCP, and the standard deviation was 15 m3/s.

Figure 4.

math formula and math formula estimated from a USGS digital elevation map and measured water level for Test Set 1. Discharge during this period was relatively constant at 256 m3/s, and water level slope was 2 × 10−5. These observations were used as input to the Bayesian network for velocity predictions on the Kootenai River 13–15 August 2010.

[34] An example velocity prediction is plotted in Figure 5. The width of the predicted distribution clearly decreases relative to the climatological prediction when the input variables are included. The observed and predicted velocities for all transects in Test Set 1 are presented in Figure 6 (top). The gray bars in Figure 6 (top) represent the probability distributions at the 0.5 (darkest), 0.90, and 0.95 (lightest) levels. Variation in Vk follows variation in math formula and is also affected inversely by changes in math formula. Bias between the network predictions and observations is 0.06 m/s, and RMSD is 0.09 m/s. RMSD between Bayesian velocity predictions and observations is nearly the same as the RMSD between Sobek velocity model results and observations. In all but one of the 64 estimates of Vk, differences between the mean of the predicted distribution and the observation fell within the 95% prediction interval, indicating that uncertainty was correctly parameterized by the distribution (Figure 6, bottom). The Brier Score for velocity prediction (equation (10)) is 0.186. Reliability, the first component of the Brier Score, is 0.008. The second component of the Brier Score, resolution, is 0.044. The third component of the Brier Score, uncertainty, is 0.222. The Brier Skill Score is 0.33, indicating the relative improvement of the prediction over the climatological prediction.

Figure 5.

An example probability distribution for the observation, prior prediction, and updated prediction at river kilometer 232.97 for Test Set 1. A width of 170 m, depth of 5.15 m, discharge of 256 m3/s, and slope of 2 × 10−5 was passed to the Bayesian network. The width of the 95% prediction interval decreases between the climatological (prior) prediction and the updated prediction.

Figure 6.

Bayesian network velocity predictions for Test Set 1 compared to observations. In the upper plot, light gray represents the 95% prediction interval, medium gray represents the 90% prediction interval, and dark gray represents the 50% prediction interval. The red line represents the mean velocity from the predicted probability distribution. The thick black line represents the mean velocity measured with drifters and the error bars represent 1 standard deviation in the velocity measurements. The lower plot demonstrates that the 95% prediction interval in gray, accounts for the difference between predictions and observations.

4.1.2. Test Set 2

[35] In contrast to making predictions at many transects for a single time, we also made predictions at a single transect measured by the U.S. Geological Survey 56 times between 2003 and 2011, encompassing a wide range of discharges. Observations of velocity, mean hydraulic depth were collected with an Acoustic Doppler Current Profiler (ADCP) near the USGS gage 12310100 at the Tribal Hatchery near Bonners Ferry, ID. The original purpose of the data collection was to establish a rating curve for this section of the river. math formula ranged between 207 and 1180 m3/s (Figure 7). math formula ranged between 146 and 209 m (Figure 7). math formula ranged from 2.24 to 9.21 m (Figure 7). math formula was assumed to be 2 × 10−5 [Barton, 2004].

Figure 7.

math formula, math formula, and math formula measured with an ADCP on the Kootenai River near the Tribal Hatchery. Slope was assumed to be 2 × 10−5. These observations were used as input to the Bayesian network for Test Set 2.

[36] Results of the Bayesian velocity predictions for Test Set 2 are presented in Figure 8. Bias between observations and predictions is 0.01 m/s and RMSD is 0.13 m/s, similar to results for Test Set 1. For all 56 times when velocity was predicted, observed differences between the predicted velocity and the measured velocity fell within the 95% prediction interval, indicating that modeled uncertainty correctly parameterized this difference. The Brier Score (equation (10)) is 0.077, and the components of the Brier Score are 0.006 for reliability, 0.027 for resolution, and 0.099 for uncertainty. Reliability for the predictions is 33% better than for Test Set 1. Resolution decreased by 39% in Test Set 2 relative to Test Set 1. Brier Score uncertainty, which is inversely related to the variability of the velocity, is lower in this example than in Test Set 1 because Test Set 2 encompassed a wider range of velocities. The Brier Skill Score was 0.22, indicating that the forecast prediction is an improvement over the climatological prediction but not as large an improvement as in Test Set 1.

Figure 8.

Bayesian network velocity predictions for Test Set 2 on the Kootenai River near the Tribal Hatchery compared to observations. In the upper plot, light gray represents the 95% prediction interval, medium gray represents the 90% prediction interval, and dark gray represents the 50% prediction interval. The red line represents the mean velocity from the predicted probability distribution. The thick black line represents the mean velocity measured with drifters and the bars represent 1 standard deviation in velocity measurement. The lower plot demonstrates that the 95% prediction interval in gray accounts for the difference between predictions and observations.

4.2. Velocity and Sturgeon Spawning

[37] As introduced in section 1, Kootenai River white sturgeon is listed on the endangered species list due to recruitment failure [Duke et al., 1999]. In order to characterize the relationship between spawning and hydrodynamics leading to the decline of the species, McDonald et al. [2010] used a three-dimensional numerical model to determine the effect of flow on white sturgeon spawning habitat. Spawning was quantified as spawning events per unit effort (SPUE) from 1992 to 2002. They observed a correlation, significant at the 99% confidence level between SPUE and maximum velocity, as well as between SPUE and maximum water depth. The correlation was consistent over a range of discharges between 535 and 1600 m3/s. They observed a similar correlation between SPUE and mean velocity, albeit with a lower correlation coefficient (ρ) than when maximum velocity was used.

[38] A central objective of developing this Bayesian network for the Kootenai River is to disseminate model results beyond numerical modelers for practical interpretation of the model results. In this section, we demonstrate that Bayesian predictions can be applied in the same manner as McDonald et al. [2010] applied results from quasi three-dimensional numerical model of velocity. However, the Bayesian approach does not require the user to have detailed knowledge of a numerical model.

[39] We predicted mean velocity at transects, where SPUE was measured using a river discharge of 256 m3/s, a slope of 2 × 10−5, and river depths and widths defined as described in section 4.1.1. While the discharge, depths, and widths provided as input to the Bayesian network were lower than during the spring spawning season, McDonald et al. [2010] demonstrated that the relative pattern of velocity is consistent over a range of velocity magnitudes and discharges. The spatial correlation between mean velocity and SPUE was computed for 10 lags of 100 m upstream and downstream. Similar to McDonald et al. [2010] results, we observed a broad peak in correlation between −200 and 400 m (Figure 9). The maximum correlation coefficient was located at a lag of 200 m and was of similar magnitude as the correlation coefficients in McDonald et al. [2010]. McDonald et al. [2010] note that they also observed a slightly lower correlation coefficient when using mean velocity from the numerical model.

Figure 9.

Velocity predicted from the Bayesian network at transects defined by Test Set 1 (top), spawning events per unit effort (SPUE) (middle), and correlation between predicted velocity and SPUE (bottom).

4.3. Hindcast Sensitivity Analyses

[40] A benefit of the Bayesian approach is the ability to quantify the sensitivity of the prediction to the input variables. The objective of the following sensitivity analysis is to understand the relative importance of each variable in the network to hindcasting Vk. To determine the sensitivity of the predictions to each variable, we withheld individual and groups of variables from the prediction and examined the performance metrics.

[41] As a benchmark, we hindcast Vk using math formula from the Sobek model as an input. This test quantifies the loss of skill due to the discretization of the Bayesian network. The regression skill for the benchmark test is 0.97, significant at the 95% level, between the model result and the Bayesian network indicating that the velocity prediction variable is highly resolved. Bias is between math formula and Vk, calculated with equation (8), is 0.0 m/s and root-mean-square difference (RMSD) between the predictions and observation, equation (9), is 0.1 m/s, indicating that there is no bias caused by discretization and a slight increase in scatter due to discretization. The likelihood ratio (LR, equation (7)) is 54,872, so predictions are a great improvement over the climatological estimate of velocity. Since math formula was provided directly to the Bayesian network, these statistics represent the best possible predictions the Bayesian network can make given that all the information in the network was available.

[42] Statistics from the rest of the sensitivity analysis were compared to the benchmark. We compared math formula from Sobek model results with hindcast Vk given inputs to the Bayesian network of math formula, and math formula from Sobek model results. This results in a regression skill of 0.93, significant at the 95% level, a decrease of only 2% from the maximum possible regression skill. A scatter plot of the hindcast velocity reveals that velocities less than 1.3 m/s are well reproduced by the Bayesian network, and higher velocities are underestimated leading to the slight decrease in skill (Figure 10). Bias is 0.00 m/s and RMSD is 0.1 m/s, the same as when Vk is provided directly to the network. The LR is 38,812, still a significant increase over the climatological prediction.

Figure 10.

In hindcast mode, the Bayesian network reproduces Sobek velocities less than 1.3 m/s with little scatter, while higher hindcast velocities are under estimated. The solid line represents the one- to-one relationship between hindcast and observed velocities.

[43] The results of withholding other variables are presented in Figure 11. For each instance where incomplete information is provided to the Bayesian network, there is a corresponding decrease in the hindcast regression skill and LR. Hindcasts were also made using a single variable, and results are less robust than when multiple variables are input (Figure 11). This test indicates the relative importance of variables to the prediction. Only results for skill are plotted in Figure 11, as the order of importance of variables for LR was identical to the results for skill. Differences in magnitude of the LR decrease due to withholding variables are reported in the text.

Figure 11.

The Bayesian network was tested to determine the sensitivity of the velocity hindcasts to each variable in the network. The hindcast velocities were most sensitive to removal of depth.

[44] Withholding the depth variable while providing the discharge, width, and slope variables to the Bayesian network (Figure 11) results in the largest reduction in regression skill (17% decrease) and LR (44% decrease). In contrast, neglecting to provide the slope variable to the network while passing the discharge, width, and depth to the network has the least effect on regression skill (5% decrease) and LR (33% decrease). While slope is a physically important variable, the range of slopes modeled on the Kootenai River was relatively small resulting in slope having little effect on the predictive capability of the network. When providing a single variable as input to the network, width is the most important variable using regression skill (R2 = 0.33) and discharge is the most important variable using likelihood ratio (LR = 7662) as the criterion. When providing the network with a combination of two variables as input, the combination of depth and discharge is most skillful for both regression skill (R2 = 0.84) and likelihood ratio (LR = 27,172), resulting in only a 10% drop in the regression skill and a 30% drop in the LR when compared with the case where math formula, and math formula are all provided as input to the network.

[45] As described in section 3.3, significant effort was expended designing a network that produced optimal results based on variable discretization. Here, we demonstrate the sensitivity of the network to discretization by halving the number of bins in the network. We applied the same sensitivity tests as described for the full network. Skill and LR for the lower resolution network net decreased relative to the full resolution case. For the case where all variables were provided to the network except velocity, skill is reduced by 15% due to decrease in bin resolution, reduction in skill ranged between 6% and 25% of all other combinations of variables. LR is reduced by 43% for the lower resolution network, and reduction in LR ranged between 20% and 45% for all other combinations of variables. Plant and Stockdon [2012] point out that with reduced resolution the prediction is less distinguishable from the climatology, so a reduction in LR is expected. Because there is a consistent decrease in skill and LR with lower resolution, the sensitivity of the network is nearly identical to the higher resolution case suggesting that the network design is robust.

4.4. Network Transferability: Apalachicola River, FL

[46] Given the quality of the comparisons for the long-term observation station on the Kootenai River, we decided to assess performance at a different location from where the network was developed and trained. Whether the network will make reliable predictions can be determined a priori using experience. Experience, or equivalent sample size, is an estimate of confidence in a prediction given the training data [Heckerman, 1995] and for the counting-learning algorithm used here, is the number of times a particular combination of variables appeared in the training set.

[47] We tested predictions on the Apalachicola River, FL, another river where the local species of sturgeon are at risk, and the river frequently falls within the range encompassed by the Sobek model runs on the Kootenai River. Velocities were predicted on the Apalachicola River near USGS gage 02358700 at Blountstown using 102 observations of math formula collected by the USGS between 1971 and 1999 (Figure 12). Sixty-seven observations fell within the range used to train the Bayesian network on the Kootenai River. Cases falling outside the range of the training data were excluded from predictions. Experience ranged between 10 and 207 (Figure 13), indicating that the network trained on the Kootenai River should be applicable to the Apalachicola River.

Figure 12.

Observations of math formula, math formula, and math formula used to make predictions on the Apalachicola River near Blountstown, FL. Based on three sources, slope was assumed to be between 5 × 10−5 and 2 × 10−3.

Figure 13.

A plot of experience for each measurement on the Apalachicola River demonstrates that the network trained on the Kootenai River is applicable to the Apalachicola River because experience is >0 for all measurements. Width of the predicted probability distribution is inversely proportional to experience.

[48] Data were collected following USGS methodology [Oberg et al., 2005] and uncertainty in observations was treated as they were on the Kootenai River. Observations of math formula on the Apalachicola River ranged between 194 and 1070 m3/s. math formula ranged between 168 and 255 m (Figure 12). math formula ranged between 1.6 and 5.58 m. Water surface slope was not measured at the time other data were collected, however, several observations of slope from in situ measurements in spring 2011, the Shuttle Radar Topography Mission [Farr et al., 2007] and Biedenharn (unpublished paper, 2007, Cursory geomorphologic evaluation of the Apalachicola River in support of the Jim Woodruff Dam Interim Operations Plan) suggest that slope likely falls in the two steepest sloping bins of the network. Therefore, slope was set to have equal probability between 0.0001 and 0.002.

[49] Bayesian predictions for the Apalachicola River test set are presented in Figure 14 (top). Bias is 0.01 m/s and RMSD is 0.04 m/s. For all transects, the 95% prediction interval (gray band in Figure 14, bottom) accounted for the difference between observations of velocity and predictions. The Brier Score is 0.112, composed of a reliability of 0.010, a resolution of 0.058, and an uncertainty of 0.16. The predictions were 33% less reliable on the Apalachicola River relative to Test Set 1 on the Kootenai River while resolution on the Apalachicola River improved by 32% relative to Test Set 1. The Brier Skill Score is 0.32, indicating an improvement in the predictions over the climatological prediction of similar magnitude to the improvements in prediction on the Kootenai River in sections 4.2.1 and 4.2.2.

Figure 14.

Bayesian predictions of velocity on the Apalachicola River. (top) Light gray represents the 95% prediction interval, medium gray represents the 90% prediction interval, and dark gray represents the 50% prediction interval. The red line represents the mean velocity from the predicted probability distribution. The thick black line represents the mean velocity observations. (bottom) The 95% prediction interval in gray accounts for the difference between predictions and observations.

5. Discussion

5.1. Implications for Network Transferability

[50] Despite the disparate settings of the Kootenai River, surrounded by the northern Rocky Mountains, and the Apalachicola River, flowing through the Gulf Coast plains, the rivers share many geomorphic characteristics that allows for the successful application of our Bayesian network designed for the Kootenai River to be applied on the Apalachicola River. Both rivers have drainage basins, sinuosity, grain size, and slope of similar magnitude (Table 2) and banks of both rivers are well vegetated. The hydrodynamic characteristics of the two rivers also overlap in most cases. However, some of the observed flood conditions on the Apalachicola River were outside of the range that the network was trained with and these cases were not used in the comparison.

Table 2. Range of Observed Geomorphic Characteristics
CharacteristicKootenai RiverApalachicola River
Drainage area (km2)50,50550,289
Sinuosity1.531.3–1.8
Sediment typeMedium sandFine to medium sand
Slope1 × 10−6 to 2 × 10−31 × 10−4 to 2 × 10−3

[51] Relating the width of the 90% prediction interval to experience suggests that experience is a good indicator of transferability of a network to a new river (Figure 15). Bias and RMSD between observations and predictions of velocity on the Apalachicola River were similar in magnitude to results on the Kootenai River. Reliability on the Apalachicola River was within a factor of two of reliability on the Kootenai River, so the Bayesian network appears to be transferrable to another river falling within the similar parameters as the Kootenai River. These statistics support the concept that when the network is well trained by a numerical model, the predictions may be made outside of the location where the network was trained and demonstrate the transferability of the Bayesian approach.

Figure 15.

The relationship between experience and width of the 90% prediction interval on the Apalachicola River demonstrates the transferability of the Bayesian network to a river other than where it was trained.

[52] The similarities between the geomorphology and hydrodynamics of the Kootenai River and the Apalachicola River indicate that a Bayesian network trained on one river can successfully be applied at another river and retraining a net for individual rivers is unnecessary. Ability to transfer between rivers likely lies in the underlying physical relationships governing flow in rivers. While the Kootenai River has been extensively studied and modeled over the course of the last 20 years, numerical modeling efforts on the Apalachicola River have been far less extensive and application of a Bayesian network can help to fill the gap in knowledge. Further, we assert that our network should be transferable to other rivers with similar geomorphic to the Kootenai River and hydrodynamic characteristics that fall within the range of our network. Determining the experience of the network on input variables for a new river will provide quantitative insight into network transferability.

[53] Our results also hint at the idea that a universal Bayesian network for rivers of a particular type may be developed if enough rivers and conditions are included in the training set. In this case, transferability between rivers was relatively straightforward because the two rivers appear to fall into similar categories. If one were to expand the number of nodes in the network to include the additional geomorphic characteristics described in Table 2, the network would be transferable between relatively disparate river types. The initial investment to obtain a useful ensemble of model scenarios or observations covering a wide range of river types may be large, but once developed, it could be used on many rivers of the same type. In order to be applicable on a variety of rivers, we recommend these variables be included: drainage area, slope, sinuosity, bottom sediment type, and radius of curvature.

5.2. Sturgeon Management

[54] Paragamian et al. [2009] suggested that the most efficient way to improve sturgeon spawning habitat would be to increase discharge to above 1600 m3/s for more than 2 weeks, resulting in the scouring of sand that covers the underlying cobbles and gravels at high-velocity locations where sturgeon are observed to spawn. However, they note that such an approach is not acceptable to the public because the plan would cause extensive flooding and erosion of valuable land. As an alternative, river habitat in the critical reach section of the Kootenai River can be restored. Knowledge of where high-velocity regions and spawning sites coincide can be used to inform where restoration will be most effective for improving recruitment. Additionally, as suggested by Schweizer et al. [2007] our simple-to-use Bayesian network would facilitate preliminary tests to understand changes in the velocity caused by changes to the river width or depth caused by restoration. As demonstrated by Borsuk et al. [2012], a Bayesian network for river velocity can be incorporated as a subnetwork into a larger tool for stream rehabilitation, even going as far as weighing the economic, morphological, and ecological consequences of different rehabilitation options.

6. Conclusions

[55] We developed a Bayesian network for the Kootenai River, ID based on results of the solution of the one-dimensional shallow water equations in the Sobek numerical model. We demonstrated that for times and locations on the river other than where and when the network was trained the network produced skillful probabilistic predictions of velocity with bias of 0.01 m/s and 0.06 m/s and root mean squared error of 0.08 m/s and 0.09 m/s on the Kootenai River, ID. These errors are of the same magnitude as errors between velocity predicted by the Sobek model used to train the network and velocity observations. We demonstrated the practical use of the Bayesian network in ecological studies of the Kootenai River white sturgeon by comparing spawning events per unit effort with predicted mean velocity with similar success as using a more complicated numerical model. The significance of each variable in the network was tested using a hindcast and showed that discharge is the most important variable needed to make a prediction of velocity, followed by, depth, width, and slope. Finally, the network was also applied to the Apalachicola River, FL at times when input variables were within range of the network. The Bayesian velocity predictions improved relative to climatological predictions. Bias in predicted velocity is 0.01 m/s and root-mean-square error is 0.04 m/s. The Bayesian network does not replace detailed models, but can enhance the use of model results by making them more accessible to the end user, and making models more broadly applicable by extending their use beyond the times and locations where the model was originally run.

Acknowledgments

[56] This work was supported by the Office of Naval Research. Special thanks to Anke Becker for performing numerical simulations used in the training set and Jamie MacMahan and his team at Naval Postgraduate School for supporting the Kootenai River field effort.

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