Introduction

Through the U.S. Geological Survey (USGS) Water Availability and Use Science Program, the USGS is involved in a multiyear, multidiscipline regional water-availability study for the Mississippi River Alluvial Plain, south-central United States. For the Mississippi Alluvial Plain study, the USGS, in cooperation with more than 10 Federal, State, and local stakeholders, is comprehensively studying and modeling the Mississippi River Alluvial Plain area, the Mississippi River Valley alluvial aquifer, and associated hydrogeologic units (Clark and others, 2011).

The Mississippi River Alluvial Plain and nearby contributing river basins consist of an extensive flat alluvial plain extending beyond the historical floodplain of the Mississippi River and other proximal streams (Kleiss and others, 2000; Clark and others, 2011; U.S. Geological Survey, 2018). The Mississippi River Alluvial Plain is a major physiographic feature spanning parts of eight States (Alabama, Arkansas, Illinois, Kentucky, Louisiana, Mississippi, Missouri, and Tennessee). The northern and central sections of the Mississippi River Alluvial Plain area mostly are used for intensive production agriculture, and the primary crops include cotton, rice, and soybeans.

The Mississippi River Valley alluvial aquifer is the surficial aquifer in the Mississippi River Alluvial Plain area (fig. 1), extending southward from the head of the Mississippi embayment regional aquifer system (Clark and others, 2011) and merging with the Coastal Lowlands aquifer system (Martin and Whiteman, 1999; Barlow and Belitz, 2016). The Mississippi River Valley alluvial aquifer is critical for irrigation-based production agriculture (Reba and others, 2017), which contributes to the Mississippi River Alluvial Plain area being a premiere agricultural region of the Nation.

The hydrology and hydrogeology of the Mississippi River Alluvial Plain and Mississippi River Valley alluvial aquifer are greatly affected by many streams and rivers that incise it (fig. 1; Clark and others, 2011). For example, when a stream incises the Mississippi River Valley alluvial aquifer, the stream may (1) recharge the Mississippi River Valley alluvial aquifer, (2) receive streamflow from the Mississippi River Valley alluvial aquifer, or (3) temporally do either. In many parts of the Mississippi River Valley alluvial aquifer, large groundwater withdrawals have resulted in multidecadal declines of water levels in some areas and have diminished Mississippi River Valley alluvial aquifer streamflows to some of the streams (Barlow and Clark, 2011). Further, Reba and others (2017) documented contemporaneous water-level declines in areas of the Mississippi River Valley alluvial aquifer and detailed scenarios for mitigation such as tail-water recovery systems and managed aquifer recharge systems. The identification of distributed groundwater recharge zones in the Mississippi River Valley alluvial aquifer is complicated by surface water and by discontinuous low permeability units, which form confining or semiconfining units and affect recharge (Dyer and others, 2015).

Accurate assessments of water availability in the Mississippi River Alluvial Plain area require improved representation of surface-water and groundwater interactions (Feinstein and others, 2006). Simulations of streamflow for selected sites (stream reaches) within and adjacent to the Mississippi River Alluvial Plain aquifer system are important because surface-water flows have a substantial effect on groundwater storage in the Mississippi River Valley alluvial aquifer (Reba and others, 2017). The simulations of streamflow can provide boundary conditions and calibration targets for components of groundwater models such as the MODFLOW Streamflow-Routing (SFR2) Package (Niswonger and Prudic, 2005). Although streamgage records measured on orders of decades are available at sites on many streams in the Mississippi River Alluvial Plain system area, streamflow associated with those streamgages is applicable only to the reach near the streamgage and does not necessarily accurately represent streamflow on reaches between streamgages or on tributaries. Further, the streamgage network itself within the Mississippi River Valley alluvial aquifer area is not as spatially or temporally dense as in the region surrounding the Mississippi River Alluvial Plain area. Most of the sites are on the Mississippi River Alluvial Plain boundary, where monthly mean streamflow needed to constrain the groundwater modeling efforts is not coincident with USGS streamgages (fig. 1). Streamflow simulations derived from streamgage records or from statistical models of streamflow for selected stream reaches can be used with the SFR2 Package (Niswonger and Prudic, 2005) to provide inflows and model verification to the Mississippi River Alluvial Plain and Mississippi River Valley alluvial aquifer groundwater models. For this study, the R software package (R Core Team, 2016) was used to develop a statistical model of monthly streamflow and generate simulations of streamflow for selected stream reaches in the Mississippi River Alluvial Plain area and nearby stream basins. The model, the input data, and simulated streamflow and base-flow data generated during this study are available as a USGS software release (Westenbroek and others, 2021) and a USGS data release (Westenbroek and others, 2022).

Purpose and Scope

The primary purpose of this report is to document the construction and assessment of a surface-water hydrologic statistical model built to simulate monthly streamflows in and near the Mississippi River Alluvial Plain model area from 1901 to 2018. The statistical model described is constructed to support concurrent and future groundwater investigations involving groundwater-withdrawal scenarios, optimization, and surface-water monitoring network analyses. For this study, random forest (RF) statistical models were developed from streamflow observations and used to make simulations for sites in the Mississippi River Alluvial Plain area. A secondary purpose of this report is to describe the general application of the methods herein for predictions of monthly streamflow using differing datasets as new information is acquired in and near the Mississippi River Alluvial Plain area. The scope of this report is limited to the description, application, and assessment of RF models to simulate monthly mean streamflow and base flow at ungaged rivers in and near the Mississippi River Alluvial Plain area (fig. 1).

Study Area Description and Site Selection

The Mississippi River Alluvial Plain area generally overlies the Mississippi embayment regional aquifer system area. The Mississippi embayment regional aquifer system area is about 78,000 square miles (mi²), is in the humid southern United States, and underlies parts of 8 States—Alabama, Arkansas, Illinois, Kentucky, Louisiana, Mississippi, Missouri, and Tennessee (Clark and Hart, 2009). The Mississippi River Alluvial Plain contains a low-lying plain in which the Mississippi River and associated tributaries have deposited and reworked sediments. Streams in the area have slopes generally flatter than streams in the bordering uplands.

For this study, the surface-water model included 247 gaged sites (fig. 1) within and adjacent to the Mississippi River Alluvial Plain groundwater model boundary. The study area is larger than the Mississippi River Alluvial Plain area because of the need to acquire sufficient spatial and temporal representation of streamflow information and to include areas needed as part of other hydrologic modeling within the Mississippi River Alluvial Plain groundwater modeling study. The basin boundaries for streamgages used in the study included parts of States: Alabama, Arkansas, Colorado, Florida, Illinois, Kansas, Kentucky, Louisiana, Mississippi, Missouri, New Mexico, Oklahoma, Tennessee, and Texas. The criteria used for selecting streamgages used in the study were based on the availability and completeness of streamflow record. Some examples of streamgages not used are streamgages with peak streamflows only, partial-record low-flow or flood-flow sites, springflow sites, and sites with less than 12 months of observations. The streamgages selected are listed in appendix 1, table 1.1.

The 162 ungaged sites were selected at locations where simulations of surface-water flow were needed for other components of the Mississippi River Alluvial Plain study. The ungaged sites were near the boundaries of the Mississippi River Alluvial Plain model area in eight States: Alabama, Arkansas, Kentucky, Louisiana, Mississippi, Missouri, Tennessee, and Texas (fig. 1).

Random Forest Prediction Model Construction

RF prediction modeling is a statistical method using classification trees built on many explanatory variables to predict a response variable (Breiman, 2001; Kuhn and Johnson, 2016). RF models are a powerful generalization of regression trees and rule-based models. Output from RF models can either be a discrete classification or a continuous numerical value (regression; Breiman, 1998). RFs begin with many bootstrapped random samples from the data, resulting in about 63 percent of the original observations occurring at least once (Cutler and others, 2007). RFs aggregate bootstrap samples by splitting the data into branches using the best among a subset of predictors randomly chosen at that branch point (Liaw and Wiener, 2002). This type of classification tree, which divides the data based on the explanatory variables, is repeated many times, and each successive tree does not depend on earlier trees—each tree is independently constructed using a bootstrap sample of the dataset and, in the end, a collection of many trees is used for classification or regression based on the explanatory variables (Liaw and Wiener, 2002). Kuhn and Johnson (2016) provide valuable information for RF modeling, providing additional conceptual detail, comparison to other predictive statistical approaches, and numerous references, many of which are cited in this report.

RF models are deemed useful to this study because of the ability of the methodology to identify important predictor variables with minimal human intervention. Exploration of many potential predictor variables that act in difficult-to-describe nonlinear ways to affect streamflow can be time consuming when developing traditional regression models. Further, the use of meteorological variables with varying degrees of lag time adds complexity that is difficult to exhaustively examine for use in linear models. The RF algorithm has a practical advantage over using traditional statistical models because the algorithm can handle large numbers of potential input variables and makes no prior assumptions regarding the type of distribution of the input variables. The RF algorithm can identify the relative importance of potential input variables in an automated way that can facilitate model refinement as new datasets become available. Although linear models may potentially exceed the performance of RF models, the RF algorithm was chosen for this study for the potential practical advantages of implementation and flexibility to adapt to new data.

Recently, Miller and others (2018) used RF modeling to simulate natural monthly mean streamflows for more than 2.5 million stream reaches in the conterminous United States with the RF simulated and observed streamflows corresponding to about 2,000 streamgages. However, these data were not available at the time the Mississippi River Alluvial Plain aquifer system study began and do not necessarily coincide with the sites in the periods of interest of the Mississippi River Alluvial Plain study; therefore, a similar methodology was used to develop RF models to simulate streamflows in the Mississippi River Alluvial Plain region.

In this study, an RF surface-water statistical model is used to simulate streamflow in the study area. RF models were developed from data at the gaged sites and were in turn used to make monthly mean streamflow and base-flow predictions at 162 ungaged sites in the study area and monthly mean streamflows for missing periods at gaged sites in the Mississippi River Alluvial Plain area. The modeling is based on the statistical relation between explanatory variables and response variables. The explanatory variables were based on the physical characteristics and climatic time series representative of the drainage basins above streamgages. Physical characteristics that were initially used as explanatory variables in the RF model included drainage area size and location, elevation and slope of land surface, surficial geologic age classification, and climatic data, which included monthly precipitation, evapotranspiration, and temperature. The response variable for the model was the time series of monthly mean streamflow and monthly mean base flow. The RF model treats each set of explanatory and corresponding response variables as independent observations; however, lagged time-series data are included in the explanatory variables for each observation.

Results of Random Forest Model Performance

This section of the report describes the RF model performance. The RF model results were compared against the observed streamflows and base flows to evaluate model performance.

Random Forest Performance

Measures of variable importance for an RF model of simulation of monthly mean streamflow indicated that drainage area (drainage_area_va), current month precipitation (precip), 1-month lagged evapotranspiration (ET0Sub1), 2-month lagged evapotranspiration (ET0Sub2), and preceding 6-month precipitation (preTot6) were the most important predictors for monthly mean streamflow within the study area (appendix 2, table 2.1). The inclusion of geologic basin characteristics during model testing did not improve the model performance, and therefore, geologic basin characteristics were not used as explanatory variables in the final models. Also, temperature was not determined to be an important variable, which is perhaps logically consistent with the relatively higher importance of precipitation and evapotranspiration, likely because precipitation and evapotranspiration have a direct effect on water mass (liquid or vapor) in streamflow whereas temperature has a high correlation with evapotranspiration as an indirect driver to the process.

Model Validation

The RF training used for this study has leave-one-out cross validation built into the model training procedure. This training procedure can exclude a certain percentage of observations in the input dataset used to train the model. This is to ensure that the model does not overfit the input dataset; however, observations from all gaged sites in the input dataset are available for random selection to be used in training the model unless the site is explicitly removed from the input dataset. For this study, to quantify the predictive accuracy of RF model results, and to quantify the overall predictive accuracy of the RF model, an additional cross-validation procedure was used wherein an RF model was trained with the dataset of streamflow and base flow for all gaged sites with the exception of one gaged site. In this procedure, for each of the gaged sites, a separate RF model was trained using predictor variables and monthly mean streamflows for all gaged sites excluding the “current site,” which was left out from the data available for training the model. This procedure ensured that no observations from the excluded site were available for random selection during the training process. The RF model trained on all the data except the excluded site was then used to predict monthly mean flows at the excluded site. Model performance metrics were computed for the observed monthly mean streamflows and simulated monthly mean streamflows for the excluded site for the period of available observations at the site, which varied depending on the length of operation of the site. This procedure was repeated for each of the 247 sites.

Seven cross-validation and goodness-of-fit performance metrics were calculated for the computed monthly mean base flows and simulated streamflows at 247 streamgages and summarized (table 1, fig. 2; appendix 3, table 3.1). The Nash-Sutcliffe efficiency (NSE; Nash and Sutcliffe, 1970) values were greater than 0.80 at 155 sites, and the Nash-Sutcliffe efficiency of the natural-logarithmically transformed streamflow prediction (NSEL) values were greater than 0.80 at 109 sites (appendix 3, table 3.1). The mean root mean square error (RMSE) was 871.8 cubic feet per second (ft³/s), and the root mean square error-observation standard deviation ratio (RSR; Moriasi and others, 2007) was less than 0.60 at 210 sites (appendix 3, table 3.1). The absolute value of the percentage of bias (PBIAS; Moriasi and others, 2007) was less than 10 percent at 128 sites. The Pearson correlation coefficient (Helsel and Hirsch, 1992) between observed and simulated streamflow ranged from 0.74 to 0.99, and the Spearman correlation ranged from 0.75 to 0.99 (appendix 3, table 3.1).

Table 1.    

Cross-validation performance metrics from comparing the observed monthly mean streamflows with simulated monthly mean streamflows, Mississippi River Alluvial Plain area, 1901–2018.

Table 1.    Cross-validation performance metrics from comparing the observed monthly mean streamflows with simulated monthly mean streamflows, Mississippi River Alluvial Plain area, 1901–2018.

Metric	Mean	Median	Standard deviation
Nash-Sutcliffe efficiency of streamflows	0.11	0.83	5.57
Nash-Sutcliffe efficiency of the natural-logarithmically transformed streamflow predictions	0.67	0.77	0.33
Root mean square error of streamflow, in cubic feet per second	871.8	356.9	1,588.3
Root mean square error-observation standard deviation ratio	0.58	0.41	0.74
Percentage of bias	−18.6	−4.6	52.1
Pearson correlation coefficient between observed and simulated streamflow	0.92	0.93	0.04
Spearman correlation between observed and simulated streamflow	0.93	0.94	0.04

An example site with good performance metrics is the Obion River at Obion, Tennessee, site (USGS station 07026000; NSE=0.975). Performance metrics and graphical comparisons of the data (figs. 3, 4) indicate correlation between observed and simulated monthly streamflows and similarity in magnitude.

An example site with poor performance metrics is the Hemphill Creek near Hot Wells, Louisiana, site (USGS station 07355000; NSE=−83.5). The negative NSE value indicates that the observed mean is a better predictor than the model. Although the performance metrics and graphical comparisons of the data (figs. 5, 6) indicate correlation between observed and simulated monthly streamflows, the simulated flows are higher for almost all monthly mean flows. Because the most important variable in the RF model is drainage area, it is possible that difficulties related to accurately delineating low-slope basins such as the one associated with Hemphill Creek near Hot Wells, La., can bias the simulated flows.

Monthly Mean Streamflow Simulation

The model validation procedure was repeated without excluding any particular site from the model training dataset; therefore, this validation represents a model trained with observations from all sites and does not leave any specific site out from potentially being sampled for training. Seven cross-validation and goodness-of-fit performance metrics for the model trained with all gaged sites were computed for the observed monthly mean streamflows and simulated streamflows at 247 streamgages and summarized (table 2; appendix 3, table 3.2). The NSE values were greater than 0.80 at 244 sites, and the NSEL values were greater than 0.80 at 227 sites. The mean RMSE was 326.5 ft³/s, and the RSR was less than 0.60 at 244 sites. The absolute value of PBIAS was less than 10 percent at 242 sites. The Pearson correlation coefficient between observed and simulated streamflow ranged from 0.906 to 0.998, and the Spearman correlation ranged from 0.893 to 0.998 (appendix 3, table 3.2).

Table 2.    

Summary of performance metrics comparing the observed monthly mean streamflows with simulated streamflows from the trained random forest model for the model trained with all gaged sites, Mississippi River Alluvial Plain area, 1901–2018.

Table 2.    Summary of performance metrics comparing the observed monthly mean streamflows with simulated streamflows from the trained random forest model for the model trained with all gaged sites, Mississippi River Alluvial Plain area, 1901–2018.

Metric	Mean	Median	Standard deviation
Nash-Sutcliffe efficiency of streamflows	0.96	0.98	0.10
Nash-Sutcliffe efficiency of the natural-logarithmically transformed streamflow predictions	0.92	0.97	0.11
Root mean square error of streamflow, in cubic feet per second	326	129	563
Root mean square error-observation standard deviation ratio	0.17	0.16	0.09
Percentage of bias	−1.3	−0.40	5.9
Pearson correlation coefficient between observed and simulated streamflow	0.99	0.99	0.01
Spearman correlation between observed and simulated streamflow	0.99	0.99	0.01

Monthly Base-Flow Simulation

The model validation procedure was repeated for base flow. For this procedure, no particular site was excluded from the model training; therefore, this validation represents a model trained with observations from all sites and does not leave any specific site out from potentially being sampled for training. Seven cross-validation and goodness-of-fit performance metrics were calculated for the computed monthly mean base flows and simulated base flows at 247 streamgages and summarized (table 3). The NSE values were greater than 0.80 at 241 sites, and the NSEL values were greater than 0.80 at 213 sites. The mean RMSE was 234.4 ft³/s, and the RSR was less than 0.60 at 246 sites. The absolute value of PBIAS was less than 10 percent at 204 sites. The Pearson correlation coefficient between observed and simulated streamflow ranged from 0.958 to 0.995, and the Spearman correlation ranged from 0.796 to 0.995 (appendix 3, table 3.3).

Table 3.    

Summary of performance metrics comparing the computed monthly mean base flows with simulated monthly mean base flows from the trained random forest model for the model trained with all gaged sites, Mississippi River Alluvial Plain area, 1901–2018.

Table 3.    Summary of performance metrics comparing the computed monthly mean base flows with simulated monthly mean base flows from the trained random forest model for the model trained with all gaged sites, Mississippi River Alluvial Plain area, 1901–2018.

Metric	Mean	Median	Standard deviation
Nash-Sutcliffe efficiency of base flows	0.94	0.96	0.07
Nash-Sutcliffe efficiency of the natural-logarithmically transformed streamflow predictions	0.9	0.95	0.14
Root mean square error of base flows, in cubic feet per second	234.4	77.2	470.2
Root mean square error-observation standard deviation ratio	0.23	0.21	0.08
Percentage of bias	−2.7	−0.5	8.8
Pearson correlation coefficient between observed and simulated base flows	0.98	0.98	0.01
Spearman correlation between observed and simulated base flows	0.98	0.98	0.02

The performance metrics indicate that the RF model can provide time-series simulations of streamflow and base flow at gaged sites that are responsive to physical basin characteristics and meteorologic data. The performance metrics indicate that simulated streamflow and base flow based on the model for ungaged locations would improve simulations of surface-water inputs to groundwater models, which are often based on long-term mean flows at streamgages that may not be near the areas of interest or may have missing data. Therefore, the RF model was used to simulate monthly mean streamflow and base flow for the 162 ungaged drainage basins. The outlets of the drainage basins corresponded to nodes in the domain of the groundwater model. The models, the input data, and simulated streamflow and base-flow data generated during this study are available as a USGS software release (Westenbroek and others, 2021) and a USGS data release (Westenbroek and others, 2022).

Summary

Improved simulations of streamflow and base flow for selected sites within and adjacent to the Mississippi River Alluvial Plain area are important for modeling groundwater flow because surface-water flows have a substantial effect on groundwater levels in the Mississippi River Valley Alluvial Plain area. For this study, 247 gaged and 162 ungaged sites within and adjacent to the Mississippi River Alluvial Plain model boundary were selected for inclusion in a random forest (RF) model to simulate streamflow and base flow. Daily streamflow observations and computed base flow from 247 streamgages were used as the basis for the development of these RF models. The basin boundaries for streamgages used in the study included parts of States: Alabama, Arkansas, Colorado, Florida, Illinois, Kansas, Kentucky, Louisiana, Mississippi, Missouri, New Mexico, Oklahoma, Tennessee, and Texas.

RF models were developed for gaged and ungaged sites in and near the Mississippi River Alluvial Plain area. RF models were developed from data at the gaged sites and were in turn used to make monthly mean streamflow and base-flow predictions at 162 ungaged sites in the study area and monthly mean streamflows for missing periods at gaged sites. The explanatory variables for the RF models were based on physical characteristics of the basin including drainage size, elevation, slope, and geologic and climatic data. The climatic data included precipitation, evapotranspiration, and temperature records representative of the drainage basins above streamgages or ungaged locations of interest. The response variables for the models were time series of monthly mean streamflow and monthly mean base flow.

Seven metrics of model performance and goodness of fit were calculated, using the results of model validation, on the observed total monthly mean streamflows at 247 streamgages compared to simulated streamflows. The Nash-Sutcliffe efficiency between observed and simulated monthly mean streamflow was greater than 0.80 for 155 of the 247 streamgages, with a median Nash-Sutcliffe efficiency value of 0.83. The performance metrics indicate that the RF models are responsive to physical basin characteristics and climatic meteorologic data and would improve simulations of surface-water inputs to groundwater models for predicting the monthly streamflow and base-flow values at ungaged sites and for missing periods at gaged sites; therefore, the RF model was used to simulate monthly time-series data for 162 ungaged drainage basins. The streamflow and base-flow simulations can be used to improve inflow values and to verify the Mississippi River Alluvial Plain groundwater flow model. The statistical model, the input data, and simulated monthly mean streamflow and base flow are available as a U.S. Geological Survey software release and a U.S. Geological Survey data release.