Introduction

The importance of lake morphometry (e.g. lake depth and lake volume) in understanding the ecology of lake systems has long been recognized [1]. Scientists and managers use this information to describe a lake's residence time, build predictive models of nutrients, pollutants, and ecological populations, and to understand lake productivity. For individual lakes that are the focus of research and management, bathymetry surveys are some of the first data collected. From these data, volume is usually calculated using bathymetric contour maps and planimeters. Lake volume can also be estimated with modern GIS methods if maximum depth is known [2]–[4]. Calculating depth and volume of lakes is a simple task provided bathymetry surveys exist; however, gaining access to these data is often difficult as they are frequently only available as unpublished tables or paper maps. This is especially true in regional studies that include a large number of lakes.

As part of the US Environmental Protection Agency's Ecosystem Services Research Program (ESRP), we are evaluating how changes in nutrient loads impact the delivery of ecosystem services in lakes in the Northeastern Region of the United States (Figure 1) [5]. Obtaining bathymetry data for even a small percentage of the lakes within this region has been difficult and has forced us to model lake depth and volume using existing, publically available datasets. One key source of information that provides insight into lake depth is the National Elevation Dataset (NED) [6]. With this information it is possible to calculate changes in elevation surrounding lakes, which is likely similar to the change in depth within lakes as the same processes formed the surrounding topography and the lake basin [1]. Thus, we assume that lake basins surrounded by steep topography are likely to have a steeper slope and greater changes in depth than do lake basins with lower topographic relief.

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Figure 1. Map of study area showing the Major River Basin 1 Boundary, Hydrologic Unit Code Boundaries, State Boundaries, and Lakes included in this study.

https://doi.org/10.1371/journal.pone.0025764.g001

There are two goals of our research. First we develop a method to estimate maximum lake depth for all National Hydrography Dataset Plus (NHDPlus) lake polygons in the Northeast U.S. (USGS Major River Basin 1) with sufficient accuracy for regional scale studies of nutrient cycling and ecosystem services. Lastly, we make the predicted depth predictions, as well as the scripts used to generate those predictions freely available.

Methods

Depth Prediction

The geophysical processes that shape a lake basin are the same as the geophysical processes that shape the topography directly surrounding that basin [1]. Thus, it is reasonable to expect that the slope of the surrounding topography approximates the slope of the lake bottom. We further assumed that the depth of any given point within a lake is partly a function of the distance from shore since points farthest from shore tend to be deeper than near shore areas [2]. If these two assumptions hold true we can combine distance from nearest shore (D) and the median percent slope of surrounding topography (S_median) to predict lake depth () for any pixel within a lake with the following equation:(1)

Surrounding topography was defined as the land area surrounding a lake that is both within the drainage basin of the lake and is within a specified distance of the lake (Figure 2). The drainage basin was defined to be all NHDPlus catchments that intersect a lake. The specified distance for each lake is the maximum of the in-lake distance to the shore.

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Figure 2. Example map of “surrounding topography” showing lake buffer, overlapping catchments, and the areas that are both within the buffer and overlapping catchment.

https://doi.org/10.1371/journal.pone.0025764.g002

To determine _max we used the maximum value calculated with Formula 1. To do this we first calculated the in-lake Euclidean distance from the shoreline for each pixel in the lake and determined the maximum in-lake distance. Second we calculated S_median in the surrounding topography using the National Elevation Dataset, re-sampled to 30 meter pixels. Finally, both values were entered into Formula 1 and an initial _max was recorded for each lake. All analyses were scripted in Python with the geospatial analyses conducted with ArcGIS 10.0, and other statistical and basic database operations conducted with the R programming language [9], [10]. All work can be reproduced with the Python scripts (Text S1) and example data (Dataset S1), assuming the user has access to an ArcGIS 10.0 license and Spatial Analyst. The example data represent a small subset of lakes and may be used to demonstrate the use of the script. The entire dataset is available upon request.

Results and Discussion

Our initial predictions, when compared to the NLA measured depths, were on average greater than measured values and thus required an adjustment to correct for bias (Figure 3). To correct maximum depths in HUC Region 01 we multiplied initial depth predictions by 0.553 and in HUC Region 02 we multiplied all initial depth predictions by 0.462. These constants are the median slope of the linear fit of the 10,000 cross-validation iterations of the NLA depths and predicted depths with an intercept constrained to 0 (Figure 3). Prior to correcting the estimates, the RMSE for HUC 01 was 6.32 m, and for HUC 02 was 6.14 m. The uncorrected correlation coefficient for HUC 01 was 0.82, and for HUC 02 was 0.64. After corrections, the mean RMSE from the 10,000 validation datasets for HUC 01 was 5.95 m and for HUC 02 it was 5.09 m. The mean correlation for HUC 01 was 0.82 and for HUC 02 it was 0.69. Comparing our predictions to the independent depths from the online sources resulted in an RMSE of 7.0 and a correlation of 0.72. Initial and corrected maximum lake depth predictions for all 27,942 lakes are available for download (Dataset S2). Additionally, the web reported depths and NLA measured depths used for the cross validation are included for the subset of lakes for which those data exist (Dataset S2).

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Figure 3. Initial maximum depth predictions compared to National Lakes Assessment (NLA) field measured depths.

Black line is one-to-one line indicating perfect agreement. Green squares are values from HUC Region 01 and green line is linear fit with intercept of 0 and slope of 0.553 for HUC Region 01. Blue triangles are values from HUC Region 02 and blue line is linear fit with intercept of 0 and slope of 0.462 for HUC Region 02.

https://doi.org/10.1371/journal.pone.0025764.g003

Our primary goal for this research was to develop a method to predict maximum lake depth using publically available data that would be applicable at regional scales. We were successful in that regard and can now estimate lake depth for all lakes with surrounding elevation data, catchments, lake shoreline data (available for the whole United States at the 1∶100 000 scale or better in many cases), and a dataset used to empirically adjust the estimates (i.e. the NLA is available for ∼1000 lakes across the US). Furthermore, our assessment indicates that on average our predictions are accurate, evidenced by the average RMSE and strong average correlation between our predictions and the validation datasets. The methods described here represent a good first step in estimating maximum lake depth when detailed data are not available. There is variation between our estimates and the validation datasets (i.e., cross-validation and web reported depths) that we were not able to explain. There are many possible sources of error that might account for this. We have identified two that we feel to be most important.

First, our uncorrected depth predictions were greater than the measured depths. This suggests processes in addition to those that formed the topography are controlling the depth of the lakes. The most likely processes are erosion and sedimentation. Our method corrects for the over-prediction, but it does not explicitly account for the lake-to-lake variation in sediment loads or age of the lake. Including these loads may account for the over-prediction and should also improve the overall accuracy of the method. One possible addition to this method would be to use existing sediment load models (e.g. USDA's Soil and Water Assessment Tool - SWAT) to estimate sediment loads for each lake [12]–[14].

Second, while the median slope of the surrounding topography does a good job of predicting the average trend in lake depths across a region, the median is likely missing local variations in slope that are important in describing lake bathymetry. Other methods may provide additional information. For instance, interpolation methods (e.g. splines, kriging, polynomial regression) have a long history in generating digital elevation models [15]–[17]. Modifying these techniques so that they may be used to estimate depth in lakes might improve upon a method that uses median slopes.

Lastly, this method makes it possible to greatly expand and enhance modeling studies to include a much larger spatial extent. We can now include volume, via proven GIS methods, and residence time in modeling efforts in a large number of lakes across large regions [2]. In an unpublished study by Milstead and others, they found that including volumes and residence times based on the estimates of depth reported here have improved predictions of nutrient concentrations.

In summary, we developed a method to predict maximum lake depth and lake volume that, on average, had relatively low error (i.e. RMSE ∼5–6 m and strongly correlated with validation data). Unpublished work by Milstead and others also shows that the depth predictions from this method will improve regional modeling studies of lakes and allow for better predictions of nutrient concentration across all lakes in a region. Although our predictions are reasonable, there is room for improvement. Exploring other statistical methods and including estimates of sediment load may provide better estimates. This method is not meant to be a replacement for detailed bathymetric surveys when greater detail is needed. It does allow us to include important information over a much greater number of lakes than was possible previously. Lastly, given wide variation in the age, origin and sedimentary processes seen in lakes of the northeastern United States, that this simple model can explain 50–60% of the variation in maximum depth based solely on a 30 m Digital Elevation Model (DEM) and available in situ data is promising and should utlimately improve our ability to model various aspects of lakes across broad regions.

Supporting Information

Acknowledgments

We thank Kristen Hychka, John Kiddon, Anne Kuhn, and the anonymous reviewers for the comments and criticisms that greatly improved the manuscript. Special thanks to Jane L. Copeland of SRA International, Inc. for her assistance with acquiring the NED data. This paper has not been subjected to Agency review. Therefore, it does not necessary reflect the views of the Agency. Mention of trade names or commercial products does not constitute endorsement or recommendation for use. This is contribution number AED-11-013 of the Atlantic Ecology Division, Office of Research and Development, National Health and Environmental Effects Research Laboratory.