The MWBM (Fig. 1) is a modular accounting system that provides monthly estimates of components of the hydrologic cycle by using concepts of water supply and demand (Wolock and McCabe, 1999; McCabe and Markstrom, 2007). Monthly temperature (T) is used to compute potential evapotranspiration (PET) and to partition monthly precipitation (P) into rain and snow (Fig. 1). Precipitation that occurs as snow is accumulated in a snow pack (snow storage as snow water equivalent, or SWE); rainfall is used to compute direct runoff (Rdirect) or overland flow, actual evapotranspiration (AET), soil-moisture storage recharge, and surplus water, which eventually becomes runoff (R) (Fig. 1). When rainfall for a month is less than PET, AET is equal to the sum of rainfall, snowmelt, and the amount of moisture that can be removed from the soil. The fraction of soil-moisture storage that can be removed as AET decreases linearly with decreasing soil-moisture storage; that is, water becomes more difficult to remove from the soil as the soil becomes drier and less moisture is available for AET. When rainfall (and snowmelt) exceeds PET in a given month, AET is equal to PET; water in excess of PET replenishes soil-moisture storage. When soil-moisture storage reaches capacity during a given month, the excess water becomes surplus and a fraction of the surplus (Rsurplus) becomes R, while the remainder of the surplus is temporarily held in storage. The MWBM has been previously used to examine variability in runoff over the CONUS (Wolock and McCabe, 1999; Hay and McCabe, 2002; McCabe and Wolock, 2011a) and the global extent (McCabe and Wolock, 2011b). Table 1 lists the MWBM parameters, with definitions and parameter ranges for calibration.

The Ppt_adj and Tav_adj parameters specify seasonal adjustments for precipitation and temperature, respectively. The seasonal adjustment parameters were included to account for errors in the precipitation and temperature data used in this analysis. Sources of systematic and non-systematic errors of climate forcing data are well documented from the precipitation gage-derived sources (Groisman and Legates, 1994; Adam and Lettenmaier, 2003). Interpolation of these systematic errors from point scale to gridded domains may propagate these biases, especially in complex terrain (Clark and Slater, 2006; Oyler et al., 2015). The use of adjustment factors allows uncertainty associated with forcing data and model parameter values to be treated separately (Vrught et al., 2008).

The MWBM was applied to the CONUS (Bock et al., 2016) with 109 951 hydrologic response units (HRUs) from the geospatial fabric (Viger and Bock, 2014), a national database of hydrologic features for national hydrologic modeling applications (Fig. 2). This HRU derivation is based on an aggregation of the NHDPlus data set (US Environmental Protection Agency and US Geological Survey, 2010), an integrated suite of geospatial data that incorporates features from the National Hydrography Dataset (http://nhd.usgs.gov/), the National Elevation Dataset (http://ned.usgs.gov/), and the Watershed Boundary Dataset (http://nhd.usgs.gov/wbd.html). The sizes of the HRUs range from less than 1 square kilometer (km2) up to 67 991 km2, with an average size of 74 km2.

Inputs to the MWBM by HRU are (1) monthly P (millimeters) and monthly mean T (degrees Celsius), (2) latitude of the site (decimal degrees), (3) soil moisture storage capacity (millimeters), and (4) monthly coefficients for the computation of PET (dimensionless). Monthly P and mean T were derived from the daily time step, 1/8∘ gridded meteorological data for the period of record from January 1949 through December 2011 (Maurer et al., 2002). Monthly P and T data were aggregated for each HRU using the USGS Geo Data Portal (http://cida.usgs.gov/climate/gdp/) (Blodgett et al., 2011). Latitude was computed from the centroid of each HRU. Soil moisture storage capacity was calculated using a 1 km2 grid derived from the Soils Data for the Conterminous United States (STATSGO) (Wolock, 1997). The monthly PET coefficients were calculated by calibrating the Hamon PET values to Farnsworth et al. (1982) mean monthly free-water surface evapotranspiration. McCabe et al. (2015) describes these PET coefficient calculations in detail.

A parameter SA for the CONUS was conducted for the MWBM using the Fourier Amplitude Sensitivity Test (FAST) to identify areas of hydrologic similarity. FAST is a variance-based global sensitivity algorithm that estimates the contribution to model output variance explained by each parameter (Cukier et al., 1973, 1975; Saltelli et al., 2000). Advantages of using FAST over other SA methods are that FAST can calculate sensitivities in non-linear systems, and is extremely computationally efficient. The seasonal adjustment factors were not incorporated into the FAST analysis. We viewed the seasonal adjustment factors as more related to the forcing data, and for this application only parameters associated with model structure were included (first five parameters in Table 1).

FAST transforms a model's multi-dimensional parameter space into a single dimension of mutually independent sine waves with varying frequencies for each parameter, while using the parameter ranges to define each wave's amplitude (Cukier et al., 1973, 1975; Reusser et al., 2011). This methodology creates an ensemble of parameter sets numbering from 1 to N, each of which is unique and non-correlated with the other sets. Parameter sets are derived using the corresponding y values along each parameter's sine wave given a value on the x axis. The model is executed for all parameter sets using identical climatic and geographic inputs for each simulation. The resulting series of model outputs are Fourier-transformed to a power spectrum of frequencies for each parameter. Parameter sensitivity is calculated as the sum of the powers of the output variance for each parameter, divided by the sum of the powers of all parameters (total variance). The parameter sensitivities are scaled so that the sensitivities for all parameters sum to 1. Thus, parameters that explain a large amount of variability in the model output have higher (i.e., closer to 1) parameter sensitivity values.

FAST was implemented with the MWBM using the “fast” library in the statistical software R (Reusser, 2012; R Core Team, 2013). Parameter ranges used by FAST for generating wave amplitudes of parameter ensembles across the CONUS were based on Table 1. The “fast” R package pre-determines the minimal number of runs necessary to estimate the sensitivities for the given number of parameters (Cukier et al., 1973). For our application we generated an ensemble of 1000 parameter sets (as compared to the minimally suggested number of 71 estimated by “fast”). The use of the minimal number of parameter sets should be a consideration for more complex models, but the relative computational efficiency and parallelization of the MWBM allowed the model to simulate this larger number of parameter sets quickly to help ensure a robust parameter sensitivity analysis.

Many applications of SA in hydrologic modeling have evaluated parameter sensitivity for measured streamflow using performance-based measures such as bias, root mean squared error (RMSE), and the Nash–Sutcliffe efficiency (NSE) (Nash and Sutcliffe, 1970; Moriasi et al., 2007). In this study, parameter sensitivity is examined using two hydroclimatic indices that account for the magnitude and variability of both climatic input and model output: the (1) runoff ratio (RR), a ratio of simulated runoff to precipitation, and (2) runoff variability (RV) index, the standard deviation of simulated runoff to the standard deviation of precipitation (Sankarasubramanian and Vogel, 2003).

The relative sensitivities derived from the FAST analysis using the RR and RV indices at each of the 109 951 HRUs across the CONUS were scaled so that the 5 MWBM parameter sensitivities derived for each HRU summed to 100 (Fig. 4). RR (Fig. 4a) is most sensitive to the parameter Drofac in regions where MWBM runoff is not dominated by snowmelt and orographic precipitation, such as arid and subtropical areas of the CONUS. MWBM parameters that control snowpack accumulation and melt (Meltcoef, Tsnow, and Train) are more important to the RR in the extensive mountain ranges in the western CONUS, and northerly latitudes around the Great Lakes and in the eastern CONUS. The RR indicates the highest sensitivity to the Rfactor parameter in mountainous areas of the CONUS and areas of the west coast, and moderate to high sensitivity in areas where the sensitivity of RR to Drofac is low. Tsnow, Train, and Meltcoef all share similar patterns across the CONUS. The spatial variability of the sensitivity of RR to Meltcoef indicates different physical mechanisms controlling Metlcoef parameter influence on RR in different areas of the CONUS. In the west CONUS, the sensitivity of RR to Meltcoef is greatest in mountainous areas that accumulate and hold snowpack through the late spring, such as the Rocky Mountains, Cascade, and Sierra Nevada mountain ranges. In the east and midwestern CONUS, the sensitivity of RR to Meltcoef is greatest for HRUs with more northerly latitudes.

The spatial patterns of sensitivities of RV to the five MWBM parameters (Fig. 4b) show both similarities and deviations from the patterns shown in the RR maps. For the central part of the CONUS, the relative sensitivity for the parameter Drofac is high for both indices, and low for the parameter Rfactor for both indices. Meltcoef, Tsnow, and Train share the same relations between higher sensitivity and higher elevation (primarily in the western part of the CONUS), and higher sensitivity and more northerly latitude (primarily in the eastern half of the CONUS) for both indices. However, Drofac and Rfactor show distinctly different patterns of relative sensitivities for the eastern part of the CONUS for RV as compared to RR. The other three parameters follow the same general spatial patterns for RV as compared to RR, but with greater fine-scale spatial variation and patchiness. The differences between the spatial distributions of the sensitivities between the two indices highlight that applying SA to different model outputs can generate different levels of sensitivities for each parameter. In addition, the choice of objective function or model output for which to measure parameter sensitivity is important, as parameter sensitivities will differ depending on whether a user is evaluating measures of magnitude, the variability of distribution, or timing (Krause et al., 2005; Kapangaziwiri et al., 2012).

Figure 5 illustrates the variability of parameter sensitivities between NHDPlus regions R08 (lower Mississippi) and R14 (upper Colorado) (see Fig. 2) for the RR and RV indices, and between the RR and RV within a single region. The lower Mississippi and upper Colorado NHDPlus regions have a similar number of HRUs (4449 and 3879, respectively) and cover a similar area (26 285 and 29 357 km2, respectively). The lower Mississippi region has homogenous topography, with humid, subtropical climate, while the upper Colorado region has highly variable topography, and thus highly variable climatic controls on hydrologic processes. For the lower Mississippi region, only one parameter dominates modeled RV variance (Rfactor, Fig. 5a) and modeled RR variance (Drofac, Fig. 5c). In contrast, for the upper Colorado River region several parameters influence RV variability (Drofac, Rfactor and Meltcoef, Fig. 5b) and RR variability (Drofac and Meltcoef, Fig. 5d). In the lower Mississippi region the amount of snowfall is negligible, so the three parameters that control snowfall and snowpack accumulation in the MWBM have a negligible effect on the volume and variability of simulated total runoff. The Rfactor parameter controls almost all of the variance for the RV in the lower Mississippi region. In humid, subtropical hydroclimatic regimes of the CONUS, peak runoff is coincident with peak precipitation, which is significant because these periods are when the surplus runoff is greatest. In the upper Colorado, peak runoff is not coincident with peak precipitation, and the MWBM snow parameters have more control in modulating the variability and timing of runoff from snowmelt in the higher elevation HRUs. The comparison of the parameter sensitivities for these two regions illustrates how variable parameter sensitivities differ by region (i.e., different climatic and physiographic regions) and components of model response (i.e., volume and variability).

The spatial patterns and magnitudes of parameter sensitivities across the CONUS were used as a basis for organizing HRUs into hydrologically similar regions for parameter regionalization through MWBM calibration. This idea is rooted in the hypothesis that geographically proximate HRUs share similar forcings and conditions, and thus will behave similarly. This application uses similarity in SA results as a basis for organization, rather than similarity in physiographic characteristics. The derived regions are subsequently used to simplify model calibration across the CONUS and provide a basis for the transfer and application of parameters to ungaged areas.

The parameter sensitivities derived from the RR were used to organize the HRUs into two independently derived calibration regions; the first derived by identifying HRUs with unique combinations of the order of parameter sensitivities to the RR (highest parameter sensitivities to lowest, i.e., 1-Drofac (78 %), 2-Rfactor (16 %), 3-Meltcoef (4 %), 4-Tsnow (1 %), 5-Train (1 %)), and the second classification based upon identifying HRUs with unique sets of parameters whose sensitivities exceeded a specified threshold of parameter sensitivity (i.e., only Drofac, Rfactor, Meltcoef using a 5 % threshold in the first classification example). The purpose of the first classification was to delineate regions of similar model response or behavior based on the order of importance of the MWBM parameters to the RR for each HRU. This classification identified 16 distinct regions of HRUs across the CONUS based on the order of the parameter sensitivities of the five parameters (derived using the RR index). Sizes of these regions ranged from 94 km2 to almost 2 million km2. The second classification delineated regions with an identical set of the most important parameters to the RR based on parameters whose sensitivities exceeded a 5 % threshold. This step identified 12 regions of HRUs with unique combinations of parameter sensitivities exceeding 5 %. There has been progress in providing quantitative thresholds for the identification of sensitive and non-sensitive parameters for hydrologic modelers (Tang et al., 2007), but no definitive consensus yet exists. Therefore, a 5 % threshold was used based on visual delineation of major physiographic features, such as mountain ranges across the CONUS. The sizes of this second group of regions ranged from 94 km2 to more than 15 million km2. Maps of the two groupings of HRUs were intersected to create a total of 49 regions across the CONUS. NHDPlus region and subregion boundaries, proximity, and significant topographic divides were used to further divide the groups into 159 geographically unique calibration regions across the CONUS. The lack of streamgages available in some regions, especially areas with arid and semi-arid climates, necessitated merging regions together. Calibration regions that contained less than 3 streamgages from the 8410 gages present in the geospatial fabric (see Sect. 3.3) were combined with the proximate and most similar group which shared the most similar parameter sensitivities (both order and magnitude), resulting in 110 calibration regions across the CONUS (Fig. 6). Within each region the FAST results for both the RR and RV indices were used to determine which parameters to calibrate. Within each region, parameters with a median parameter sensitivity of 5 % for the RR and RV among the region's HRUs were selected for group calibration. Parameters not shown as sensitive were kept at the default value for the group.

The initial set of streamgages used for testing in the MWBM calibration procedures was selected from 8410 streamgages identified in the geospatial fabric (Fig. 7). The geospatial fabric includes reference and non-reference streamgages from the Geospatial Attributes of Gages for Evaluating Streamflow data set (GAGES, Falcone et al., 2010). Of the 8410 streamgages in the geospatial fabric, 1864 were identified as having reference-quality data with at least 20 years of record. These reference quality streamgages were judged to be largely free of human alterations to flow (Falcone et al., 2010). In the current study, reference quality was not considered in the initial streamgage selection because the 20 years of record was considered too restrictive. Therefore, a subset of the 8410 streamgages was selected for initial testing in the MWBM calibration procedures based on the following criteria:

Remove streamgages with less than 10 years of total measured streamflow (120 months) within the time period 1950–2010.

Two automated calibration procedures were implemented to produce an optimal set of MWBM parameters for each calibration region. The first procedure, Individual Streamgage Calibration, calibrated each of the 5457 streamgages individually. Results from the individual calibrations were used to further filter the streamgages within the second procedure, Grouped Streamgage Calibration, which calibrated selected streamgages together by calibration region.

The individual streamgage calibrations provided information regarding (1) the potential suitability of a given streamgage for inclusion in a grouped calibration, and (2) a baseline measure for evaluation of the grouped calibration results. Reference and non-reference streamgages were considered in this application; if the runoff at a streamgage could not be calibrated individually to a satisfactory level (based on criterion outlined in Sect. 3.4.2), then it was felt that it would not provide useful information for the grouped streamgage calibration procedure. Figure 9 shows the NSE (Fig. 9a) and logNSE (Fig. 9b) coefficients from the individual streamgage calibrations for the CONUS. Scattered throughout the CONUS are NSE and logNSE values less than 0.0 (triangles in Fig. 9). These poor results are likely streamgages with poor streamflow records, either due to measurement error or anthropogenic effects (dams, water use, etc.).

Results from this study indicate that regionalized parameters can be used to produce satisfactory MWBM simulations in most parts of the CONUS (Fig. 13). Despite the differences between the individual streamgage calibration and grouped calibration, Fig. 13 illustrates that the grouped calibration strategy, which focused only on sensitive parameters, can provide just as much information as the individual streamgage calibration with no constraints on the parameter optimization other than the default ranges. The MWBM is a simple hydrologic model as it has minimal parameters, which are conceptual in nature (not physically based). It may be that this type of model is best for regionalization when parameter sensitivity can be identified and HRU behavior can be classified by a small number of clearly defined spatial groups. More complicated models with many more interactive parameters may not respond as well to this simple type of regionalization; more parameters may lead to more parameter interaction and situations of equifinality which might confuse the analysis.

The adjustments of precipitation and temperature parameters for the individual streamgage calibrations accounted for local errors such as rain gage undercatch of precipitation. In addition, these climate adjustments also account for local anthropogenic effects on streamflow (e.g., dams, diversions) since streamgages were not screened for these effects prior to individual streamgage calibration. In the grouped streamgage calibrations, the same precipitation and temperature adjustments are applied at every streamgage within the calibration region, making these climate adjustments more of a regional adjustment and producing more of a reference condition for each calibration region.

The MWBM parameter sensitivities varied by hydroclimatic index (RR and RV) and across the CONUS (Fig. 3). The parameter sensitivity patterns give an indication of dominant hydrologic processes based on MWBM. The dominant process can be seasonal and MWBM performance may be enhanced by extending the use of SA along the temporal domain to identify and temporally vary the parameters that are seasonally important to the MWBM. For example, error in peak flow months is the primary cause for poor model performance in the Platte Headwaters (Fig. 9). For the Platte Headwaters, the final parameter set performed well for simulated z scores for the regionalized low- and median-flow conditions (Fig. 9a, July through April), but was not able to replicate measured mean monthly flows for May and June. In this case, the dominant processes controlling hydrologic behavior change with season and the parameters controlling the dominant response may have to change accordingly (Gupta et al., 2008; Reusser et al., 2011).

The pattern of MWBM accuracies shown in Figs. 8 and 14 are similar to those shown by Newman et al. (2015; Fig. 5a) in which a daily time step hydrologic model was calibrated for 671 basins across the CONUS. Our study and the Newman et al. (2015) study both indicate the same problem areas with the poorest-performing basins generally being located in the high plains and desert southwest. Newman et al. (2015) attributed variation in model performance by region to spatial variations in aridity and precipitation intermittency, contribution of snowmelt, and runoff seasonality.

The inferior MWBM results in the problem areas can be attributed to multiple factors which likely include inadequate hydrologic process representation and errors in forcing data (e.g., climate data), and/or measured streamflow. Archfield et al. (2015) state that the performance of continental-domain hydrologic models is considerably constrained by inadequate model representation of dominant hydrologic processes. For example, the simplicity of the MWBM presents limitations on the representation of deeper groundwater reservoirs, gaining and losing stream reaches, simplistic AET, and the effects of surface processes (infiltration and overland flow) that need to be represented at finer time steps than monthly.

The dominant hydrologic processes in the problem areas appear to be poorly represented at the daily (Newman et al., 2015) and monthly time steps. This may be due to inadequate forcing data, the quality of which “is paramount in hydrologic modeling efforts” (Archfield et al., 2015) and/or the lack of good reference streamflow data for calibration and evaluation. Both surely play a role and emphasize the need for incorporation of additional data sets so that calibration and evaluation of intermediate states in the hydrologic cycle are examined.