We employ a model of necessary complexity to represent basin scale hydrologic processes that regulate the storage and movement of water on monthly timescales, as shown in Fig. 1. The model includes two soil layers, where the top layer represents the water that is available to plants via roots (PAW), and the bottom layer representing depths of the soil that plant roots cannot access (plant unavailable water, or PUW). The model uses monthly time steps to integrate the state variables and is driven with hydrologic flux variables such as ET and precipitation. The model also includes other processes such as infiltration into the soil, surface runoff, drainage from each layer, recharge into the lower soil layer, and various model parameters (listed in Table 1) that control the simulations.

The model includes 13 parameters that represent process-based hydrologic mechanisms, ones that are hypothesized to be influential on basin-scale monthly resolution model simulations of the hydrologic cycle. As depicted in Fig. 1, there are two soil layers representing the PAW and PUW pools. Each of the two separate soil layers has its own inferred physical properties, such as the depth of each layer, soil moisture initialization, porosity, field capacity, and retention capabilities. Various fluxes are represented in the model, such as precipitation (P), evapotranspiration (ET), infiltration, surface runoff, and drainage. The parameters of the model dictate the simulation of each process in the hydrologic cycle, and by adding the two water pools (PAW + PUW), an estimate of total water storage (TWS) can be generated, which can then ultimately be compared with the GRACE-based TWS.

We describe here the model equations that dictate how the TWS is calculated in the model. To start, we know from the water mass continuity that the changes in TWS in the soil is equivalent to the balance between input (precipitation, P) and outputs (ET, and the total loss through drainage and runoff Q). In effect, P and ET are prescribed boundary conditions for the model. In this version of the model, 1TWSt=MtPAW+MtPUW, where MtPAW represents the PAW and MtPUW is the PUW each month, t. The soil is represented this way in the model because plants cannot access all the water stored in the ground; therefore, two separate layers are used to represent the soil water in the rooting zone (PAW) and the soil water that is not accessible to plants (PUW).

The following model equations are used to represent the storage and flow of water in the model. The mass continuity equations for water stored in the MPAW and MPUW layers are: 2Mt+1PAW=MtPAW+It-Dt,PAW-Ft-ETt3Mt+1PUW=MtPUW-Dt,PUW+Ft, where It is the infiltration into the top soil layer, Dt,PAW and Dt,PUW are the drainage terms for each layer, Ft is the recharge in between layers, and ETt is the ET term each month, t. We assume that a fraction of precipitation cannot infiltrate the soil. This occurs because during rainy events, the precipitation rates often exceed the percolation rates of the near-surface soil, which may become temporarily saturated. These processes occur on sub-monthly scales and cannot be explicitly accounted for in the model; therefore, we use a phenomenological approach that assumes a maximum infiltration rate: 4It=Imax1-e-PtImax, where Pt represents the precipitation rate each month and Imax is the parameter that represents the maximum infiltration. The excess precipitation is lost as surface runoff (St) and never enters the soil storage: 5St=Pt-It. The recharge flux between the PAW and PUW layers (Ft, positive when the flow goes from PAW to PUW) can be defined by Darcy's law, relating the difference in potentials between the two layers: 6Ft=ρlKt,layer10-6ρlgΨt,PAW-Ψt,PUW12LPUW-LPAW+1, where Ψt,PAW and Ψt,PUW [MPa] are the soil matric potential of each layer each month, ρl=1000 kg m-3 is the water density, g=9.807 m s-2 is the gravity acceleration, Kt,layer [m s-1] is the hydraulic conductivity of the source layer (i.e., PAW if Fi is positive, and PUW if Fi is negative), and LPUW (rooting depth) and LPAW (remainder of soil depth) are the parameters that represent the thickness of each layer [m].

Then, the soil matric potential of each layer is defined as a function of relative soil moisture (SMt,layer), following Brooks and Corey (1964): 7ψt,layer=ψporosity1SMt,layerb, where ψporosity=-0.117 MPa, and the parameter b corresponds to the inverse of the pore size distribution index (Marthews et al., 2014). The unsaturated hydraulic conductivity (Kt,layer) is defined following Campbell (1974): 8Kt,layer=K0SMt,layer2b+3, where K0 [m s-1] is the parameter that represents the saturated hydraulic conductivity, and the parameter b is the same as in Eq. (7). The drainage function is parameterized as the removal of water that exceeds the field capacity, to represent fast (sub-monthly) loss of water under near-saturated conditions: 9Dt,layer=max0,ψt,layer-ψfieldQexcessψporosity-ψfield, where the scaling term Qexcess is a free parameter from 0–1 that removes a fraction of SM excess above field capacity, ψfield.

Last, one thing to note is that precipitation and ET biases in the Amazon are known to be significant, and ET can even have an inverted seasonal cycle. The model is capable of substantially relaxing and constraining the simulated evapotranspiration (ETt) and precipitation (Pt) values each month, through the parameterization and inference of scale factors (Pscale and ETscale). The data set used for P each month, namely Pdata,t, is derived from precipitation measurements from the Tropical Rainfall Measuring Mission (TRMM) 3B42 (Huffman et al., 2007), provided at 0.25∘×0.25∘ and 3-hourly spatiotemporal resolutions. The data sets used for ET each month, namely ETdata,t, is derived following the approach in Swann and Koven (2017) and Shi et al. (2019). That is, monthly total ET is derived from satellite observations of precipitation and TWS and ground-based measurements of river runoff. Unlike the ET retrievals from the Moderate Resolution Imaging Spectroradiometer, which have been shown to be seasonally biased in the wet tropics (Maeda et al., 2017; Swann and Koven, 2017), this ET estimation is robust across seasons (Swann and Koven, 2017). Runoff data sets for each watershed are obtained from the Observation Service for the geodynamic, hydrologic, and biogeochemical control of erosion/alteration and material transport in the Amazon (SO-HYBAM) in situ river gauge discharge measurements (discharge measurements can be found at http://www.ore-hybam.org/, last access: 21 March 2017). With these three data sets, we estimate subbasin-based monthly ET.

To clarify this further, three different derivations are used for the TWS variable. These three estimates provide a sense of uncertainty for the TWS. The uncertainty from the GRACE product is used in the likelihood function of the MCMC algorithm when fitting the model-simulated TWS to the GRACE derived TWS. Next, three products are also used in the precipitation and the runoff driving variables that were used, to get a sense of the uncertainty in each variable. To estimate the ET driving variable in this work, we use the mean of the TWS, P, and Q products and create a water balance that will allow us to estimate a mean for the ET driving variable. Then, by application of the ET scaling parameter, we try to estimate whether our initial calculation of ET required any scaling to match the data. Therefore, even though the GRACE TWS is somehow used in the derivation of the ET data, the uncertainty that is applied throughout the work allows us to still estimate ET that is not dependent on the GRACE data. See Shi et al. (2019) for more details on this derivation. In essence, the simulated fluxes are represented as ETt=ETscale⋅ETdata,t for ET, and Pt=Pscale⋅Pdata,t for precipitation, where ETscale and Pscale are inferable parameters. Combining all these equations in the logical flow presented in Fig. 1 of the manuscript allows the model to simulate total water storage as TWSt=MtPUW+MtPAW. To our knowledge, this model has not been presented before in the literature, and this manuscript is the first to report on the model simulation results.

The parameters of the model will be inferred such that the TWS in the model simulations match the observed GRACE TWS data. As GRACE TWS is known to have the smallest uncertainties in the water budget (see Pascolini-Campbell et al., 2020), we use this information to infer and understand the more poorly constrained variables or processes in the model. In this case study, we use the model for the Gavião watershed, located in the western Amazon (for the location of the watershed refer to the map in Fig. 1). We chose the Gavião watershed for this study owing to sufficient data availability, and because there is a strong seasonal cycle for this watershed, which allows the model to capture hydrologic signals more efficiently during the parameter inference.

The GRACE mission by NASA (Tapley et al., 2004) has proven to be an extremely valuable tool for regional to global scale water cycle studies (Famiglietti, 2014; Reager et al., 2015; Massoud et al., 2018a, 2020a). GRACE data have been widely used to diagnose patterns of hydrological variability (Seo et al., 2010; Rodell et al., 2009; Ramillien et al., 2006; Feng et al., 2013), to validate and improve model simulations (Döll et al., 2014; Güntner, 2008; Werth and Güntner, 2010; Chen et al., 2017; Eicker et al., 2014; Girotto et al., 2016; Schellekens et al., 2017), to constrain decadal predictions of groundwater storage (Massoud et al., 2018a), and to enhance our understanding of the water cycle on regional to global scales (Syed et al., 2009; Felfelani et al., 2017; Massoud et al., 2020a). TWS estimates from GRACE include all of the snow, ice, surface water, soil water, canopy water, and groundwater in a region, and when combined with auxiliary hydrologic datasets, TWS can be utilized to infer process information on model parameters or other model states.

Various recent studies have demonstrated that GRACE-derived estimates of variations of TWS can provide freshwater availability estimates with sufficient accuracy (Yeh et al., 2006; Zaitchik et al., 2008; Massoud et al., 2018a). These GRACE-based methods have been applied to regions such as Northern India (Rodell et al., 2009; Tiwari et al., 2009), the Middle East (Voss et al., 2013; Forootan et al., 2014; Massoud et al., 2021), Northern China (Moiwo et al., 2009; Feng et al., 2013), California (Famiglietti et al., 2011; Scanlon et al., 2012; Xiao et al., 2017; Massoud et al., 2018a, 2020a), northern mid- to high latitudes (Trautmann et al., 2018), and the Amazon (Swann and Koven, 2017). In this study, estimates of TWS are obtained from the GRACE retrievals of equivalent water thickness (Landerer and Swenson, 2012; Sakumura et al., 2014; Wiese et al., 2016). We use three GRACE TWS retrievals from the spherical harmonic data versions generated by the Center for Space Research (CSR), GeoforschungsZentrum Potsdam (GFZ), and Jet Propulsion Laboratory (JPL). These three GRACE TWS retrievals are 1-degree solutions of land field products (each was downloaded from ftp://podaac-ftp.jpl.nasa.gov/allData/tellus/L3/land_mass/RL05/, last access: 14 June 2017). We calculate the arithmetic mean of the three GRACE TWS retrievals to represent TWS used in Eq. (1). We used this GRACE product to constrain simulations of the hydrologic model described in Sect. 2.1 for the Gavião watershed from January 2003 through December 2012.

In this study, we aim to estimate parameters of a medium complexity model that simulates the hydrologic cycle using physics-based equations that capture large scale dynamics of the watershed. We showcase how the data-constrained, physically based model can simulate the hydrologic cycle by fusing the model with auxiliary observations. When simulated on its own, the model can represent a wide range of physical possibilities, but when calibrated and trained to fit some desired observed metric, the model simulations begin to represent the underlying physical system it is being trained to. Many tools exist to achieve model-data fusion, such as Bayesian parameter inference with MCMC algorithms (Schoups and Vrugt, 2010; Bloom et al., 2015; Vrugt, 2016; Vrugt and Massoud, 2018; Massoud et al., 2019c, 2020b) or data assimilation (Reichle et al., 2002; Vrugt et al., 2005; Girotto et al., 2016; Khaki et al., 2017, 2018; Massoud et al., 2018b). These state-of-the-art tools require enough computational cost but can ensure that the underlying system dynamics are being accurately replicated to an agreeable amount of uncertainty. The model parameters in this study are estimated using Bayesian inference with MCMC (Vrugt and Massoud, 2018), where the final estimated distributions are not required to follow any form, such as Gaussian or bimodal. The final estimates of the model parameters, shown later to be the posterior of θ in Eq. (12), are the posterior solutions and are utilized to constrain the spread of uncertainty in the simulations.

In recent decades, Bayesian inference has emerged as a working paradigm for modern probability theory, parameter and state estimation, model selection, and hypothesis testing (Vrugt and Massoud, 2018). According to Bayes' theorem, the posterior parameter distributions, P(A|B), depend upon the prior distributions, P(A), which captures our initial beliefs about the values of the model parameters, and a likelihood function, L(θ), which quantifies the confidence in the model parameters, θ, considering the observed data, Y. The likelihood function is a critical property of this calculation. This section shows the derivation of the likelihood function used in this study. According to Bayes' Theorem, the probability of an event is estimated based on prior knowledge of conditions that might be related to the event. In equation form, this looks like: 10P(A|B)=P(B|A)⋅P(A)P(B). For the purposes of this study, we can express P(A) as the prior information of our calculation, which assumes that log-uniform distribution for all parameters and the probability outside the parameter bounds is equal to 0 (the minimum and maximum values for each parameter are reported in Table 1). P(B) is the evidence and is a normalizing constant and therefore taken out of the equation. This leaves us with: 11P(A|B)∝P(B|A);P(A|B)∝L(θ), where P(A|B) is the final distribution of the model parameters, or the posterior of θ in Eq. (12) described in the next paragraph, and P(B|A) is equivalent to the chosen likelihood function, L(θ), also described in the next paragraph. Therefore, the MCMC algorithm samples model parameter combinations (θ) that will maximize the fit to the GRACE data, and thus will maximize the value of the likelihood function, L(θ).

The observed data in this case study is the GRACE satellite observations, and our goal is to find the optimal set of model parameters, θ, that produces a model simulation, X(θ), which maximizes the fit, or the likelihood, relative to the observations. Our likelihood function is therefore set up as: 12L(θ)=-1σGRACE2∑tYGRACE,t-XModel,t(θ)2, where t refers to the time index (in months) of the simulations, YGRACE,t is the observed GRACE data at month t, XModel,t(θ) is the optimized model simulations at month t using the parameters θ, and σGRACE2 is the uncertainty associated with the GRACE data, which was chosen to be a homogeneous 50 mm per month for our applications. As GRACE data are represented as anomalies from climatology, we format the model simulations into anomalies as well to perform this model-data-fitting experiment. That is: 13YGRACE,t=TWSGRACE,t-meanTWSGRACE, indicating that the form of the GRACE observations is in climatological anomalies. Furthermore, we format the model simulations in this manner for the parameter inference, as follows: 14XModel,t=TWSModel,t-meanTWSModel. We apply Bayesian inference on the model parameters in an optimization framework and sample the likelihood function in Eq. (12). This allows for the inference of the model parameters, or θ. These inferred model parameters will be used to inform and constrain the spread of uncertainty in the model simulations.

Successful use of the MCMC application in a Bayesian framework depends on many input factors, such as the number of chains, the prior used for the parameters, the number of generations to sample, and the convergence criteria. For our application, we use the adaptive Metropolis-Hastings MCMC, as described in Bloom et al. (2020). We use C=4 chains, the prior was a log-uniform distribution for each parameter and the ranges shown are listed in Table 1, the number of generations was set at G=100000, and the convergence of the chains relied on the Gelman and Rubin (1992) diagnostic, where we applied the commonly used convergence threshold of R=1.2. Given the high efficiency of running this parsimonious model (compared with other high dimensional and expensive models), it was computationally feasible to obtain the set of G=100000 simulations for the MCMC algorithm (i.e., less than 1 h of CPU time to perform the parameter inference).

To better quantify the reduction of uncertainty for each parameter, we apply an averaging kernel (AK) calculation (Worden et al., 2004), which is typically a measure of how a modeled state (posterior) is sensitive to changes in the “true” state (prior) and is a method that is common for satellite retrievals. The AK matrix is calculated as follows: 15AK=I-cov(Posterior)cov(Prior), where AK is the diagonal vector of the averaging kernel matrix, I is the identity matrix, “Posterior” is the Bayesian parameter posteriors sampled with MCMC, “Prior” are samples randomly drawn from the prior distribution, and cov is the covariance function. We take the main diagonal of the AK matrix, which represents uncertainty reduction from the prior to the posterior parameter distributions. The AK diagonal values for each parameter are listed in Table 1 under “AK Diagonal”. A value of AK=1 represents a 100 % reduction in uncertainty, whereas a value of AK=0 represents no information gain and therefore no reduction in uncertainty.

To characterize the sensitivity of the monthly TWS variability to model parameters, we perturb posterior parameters and generate corresponding TWS simulations. Figure 2 shows the sensitivity of the model-simulated TWS to minor perturbations in parameter values. In these plots, the green curves show changes in simulated TWS (dTWS) when each parameter is perturbed (dPar) by 1 % of its prior range, indicating the magnitude and the time steps of model sensitivity. Results in these plots show that sensitivity to initial conditions is higher for the first 12-month period but is diminished after that. Furthermore, the sensitivity of simulated TWS varies between wet and dry seasons.

The rooting depth parameter (Fig. 2a) is sensitive during initialization as well as during the wet periods, the maximum infiltration parameter (Fig. 2b) seems to only be sensitive during the wet periods, and the parameter representing the initialization of soil moisture in the top layer (Fig. 2c) is only sensitive during initialization. Figure S1 in the Supplement shows how the remaining parameters affect TWS sensitivity. To summarize these curves in a single value (i.e., [mm change in TWS per 1 %-unit change in parameter]), we show in Table 1 under “TWS sensitivity” the aggregated value for each parameter, calculated as the mean variance of all (dTWS/dPar) curves for each parameter.

After the model parameters and states variables are constrained by the GRACE data for the Gavião watershed, relationships between the model parameters and simulated states begin to emerge. We show in Fig. 8a the scatter plot between posterior solutions of model-simulated TWS and the excess runoff parameter. This figure shows that the region inside the black box, or the high-density region of the posterior, is the region within the posterior domain that has high information content (i.e., plausible solutions with a high likelihood). The true value provided by the GRACE data is marked with a red line in Fig. 8a. Other regions of this space, such as locations with excess runoff values below 0.2, produce unlikely model simulations, and similarly, locations with excess runoff values higher than 0.5 are also less likely. This can also be seen in Fig. 3, in the posterior histograms for the excess runoff parameter. Similar relationships between other parameters and state variables (including soil moisture of layer 1 and discharge from layer 1) are shown in Fig. S8. Overall, these plots not only show the emergent relationships between variables as informed by GRACE, but also indicate if and how they correlate in the Gavião watershed.

Similarly, the posterior parameter solutions can be used to infer relationships between the parameters themselves. To this end, we show in Fig. 8b a scatter plot depicting the GRACE-informed correlation of the posterior parameter values for the soil moisture initialization parameters in the Gavião watershed. We see that the initial soil moisture in the bottom layer is greater than 0.2 [m3 m-3], and in the top layer it is less than 0.3 [m3 m-3], which can be seen in Fig. 3, in the posterior histograms for the soil moisture initialization parameters. One property that also emerges in Fig. 8b is that the initial soil moisture in the bottom layer is larger than that of the top layer. This indicates that, in the initial time step of the simulations, the bottom layer should have greater soil moisture than the top layer. These relationships can be created for any pair of parameters in the posterior space, and Fig. S9 portrays these relationships for several combinations of parameters, indicating what combinations of parameters are possible for this hydrologic system, as inferred by GRACE.

We summarize the results reported in this subsection with the following points. First, we find considerable correlations between the posteriors of individual model parameters and model states in the Gavião watershed. We also find considerable correlations between the posteriors of individual model parameters and other parameters. This is important, because the correlations between parameters and states indicate that the choice of hydrologic constants can have a considerable impact on simulated TWS. The relationships found in the parameter posteriors imply that although several parameters exhibit considerable uncertainty, only a subset of parameter combinations provide GRACE-consistent model solutions. In essence, these GRACE-based relationships portray what parameter combinations are possible for accurately simulating the chosen watershed.