a. General context

The goal of this study is to introduce different methods able to combine various datasets of multiple variables, in such a way that a priori constraints among the state variables are imposed to the state estimate.

The major developments for the introduction of state constraints have been accomplished in the framework of Kalman filtering. A first approach to use such state constraint consists of projecting the state variables in a space where the constraint is satisfied. However, this approach has some limitations. For example, the interpretability of the results is more difficult since both the state variables and the physical equations are modified (Simon and Chia 2002). Another solution uses the particular constraint as a “perfect measurement,” meaning that there is no uncertainty on this constraint. The drawback of this method is that the introduction of such “hard” or “strong” constraints into the system increases the problem dimension and can introduce numerical instabilities (Hayward 1998; Porrill 1988; De Geeter et al. 1997; Kalman 1960). Simon and Chia (2002) proposed to postprocess the unconstrained Kalman solution to impose a linear state constraint: at each time step, the unconstrained solution is projected into a constrained state space. This avoids the numerical instabilities associated with the perfect measurement approach. Furthermore, it does not increase the dimension of the problem since the state variables are kept in the same space (this facilitates the interpretation of results). In addition, it is shown that this projected solution is better than the unconstrained one since the elements of the error covariance matrix are smaller in magnitude. This postprocessing can be defined using maximum a posteriori, least squares, or linear projection approaches.¹ An even more advanced solution is proposed by Ko and Bitmead (2007) that projects the full dynamical system into a constrained space. This system-projection approach is shown to be even better than the postprocessing solution.

However, these developments have been made in the framework of Kalman filtering where a state variable dynamical system is considered. This often means that a model is used so that the state variables at time t are dependent on the state variables at earlier times (i.e., the dynamical system). As previously mentioned, the goal of this paper is to produce state estimates that are independent from any model. Therefore, the Kalman filtering cannot be considered here; only direct state estimation will be proposed. However, the technical developments made for the Kalman filtering will be useful for the integration tools proposed in the following.

b. Notations

Let ^T = (P, E, Q, ΔS) be the state variable quantifying the four water cycle components of interest in this paper (a superscript T is the transpose symbol). The goal of this study is to estimate . The closure of the water cycle is obtained when ^T · G = 0, where G^T = (1, −1, −1, −1), from Eq. (1).

Multiple observations can be obtained for each component:

• (P₁, P₂, …, P_p), the p precipitation estimates;
• (E₁, E₂, …, E_q), the q sources of information for evapotranspiration;
• (Q₁, Q₂, …, Q_r), the r runoff estimates; and
• (ΔS₁, ΔS₂, …, ΔSs), the s sources of information for the groundwater storage change.

Let

be the vector of dimension n = p + q + r + s gathering all the initial water cycle observations. The observing system can be represented using²

no bias error is considered here. The observation operator has the following structure:

Off-diagonal terms could be introduced if this information was available.

In this matrix, row i includes a “1” in the first column if observation _i measures P (or E, Q, or ΔS for the second, third, and fourth columns, respectively). The n-vector ε represents the uncertainties of each observation in _ε. The observation uncertainty is characterized by the covariance matrix = cov(ε), often chosen to be diagonal:

In this study, the goal is to estimate a relationship (i.e., filter) that finds an estimate _a (a stands for analysis) of based on the observations _ε. Therefore, an inverse problem needs to be solved where Eq. (3) is inverted from observation _ε to _a (Tarantola 1987; Rodgers 2000).

A simple inversion of matrix is not good enough. Since is not square, a pseudoinverse or a generalized inversion of matrix could be a candidate, but these simple algebra techniques suffer from numerical instabilities. In addition, other sources of information are available, so more sophisticated methods should be considered to exploit them. For example, the information on the observation noise (i.e., matrix ) is very valuable information that allows for weighing the observations _ε based on their reliability. Furthermore, in addition to the observations _ε, a priori information _b on can also be used (b stands for “background,” the terminology used in numerical weather prediction centers). An important question concerns of course the origin of this first guess (FG) information since it is not always easy to obtain such a priori information, especially when models are excluded. The covariance matrix of the FG errors needs to be specified based on the source of information for _b.

c. Simple weighting

Let us first consider the simple case where two observations P₁ and P₂, with Gaussian uncertainties ε₁ ~ N(0, σ₁) and ε₂ ~ N(0, σ₂), are available for the estimation of precipitation P. In this case, the optimal linear estimator (Rodgers 2000) is given by

This type of equation is valid when there is no bias error in the sources of information P₁ and P₂. Note that to be an optimal solution, the errors in P₁ and P₂ must be independent from each other. This type of estimator is more frequently used when merging raw satellite observations because the bias correction is easier in this case (Thome 2004; Aires et al. 2010). However, the variables that are combined here are geophysical products originating from a remote sensing algorithm and, most of the time biases do exist in this case, unless a calibration procedure has been used to reduce them. When information on the bias is available (e.g., by comparison to in situ measurements), this information can be used to suppress the bias. However, in many cases, bias information is difficult to obtain, and a pragmatic strategy is to estimate the mean state for each variable as the average of each of its observations: (the overbar refers to the mean). Then all the sources of information P_i on this variable are bias corrected toward this average: , for i = 1, 2.

It is possible to extend the result of Eq. (6) to the case where more than two observations are available for a variable and where more than one variable is retrieved. Let us find the optimal linear estimator, such that

where is a 4 × n matrix. To be optimal, needs to take into account the observation uncertainties of the observation _ε described by .

Let us consider the first line of used to retrieve the first water cycle component in , that is, P. As stated earlier, there are p available observations: (P₁, P₂, …, P_p). The first line of is composed of zeroes, except for the first p columns that are given by

for j = 1, …, p. This expression extends easily to the second (for E), third (for Q), and fourth (for ΔS) rows. Again, this expression is the optimal solution only if all the errors are independent from each other. If a covariance matrix describing the dependencies of the input datasets errors was available, an eigendecomposition of this covariance matrix would create a new input dataset with uncorrelated errors. Unfortunately, this type of covariance matrix for the input dataset errors is generally not available.

d. A constrained linear technique

This approach is based on a simple linear model for the weighting of the satellite observations:

with additional constraints on gain matrix and on solution .

We note that is the (4, n = p + q + r + s) matrix that linearly links the new water cycle components to the satellite estimates:

Note that the simple weighting (SW) filter follows such structural constraint.

The coefficients in Eq. (10) are constrained by

(i.e., each line L in adds up to 1). This ensures that estimated water components in are a weighted sum of the observations for that particular component. However, if the estimation of is a well-posed problem, it might be possible to use simpler constraints, for example, the coordinates in can be constrained to be 0 ≤ k_ij ≤ 1, or their amplitude can be limited. This later and simple procedure is a Tikhonov regularization often used in regression problems. For instance, in the neural network (NN) theory, it is called “weight decay” (Bishop 1996).

Note that if only constraints of Eqs. (11) were applied to the filter of Eq. (10), the constrained linear (CL) solution would be very close to the SW one. The SW exploits the uncertainty characteristics to obtain the good coefficients in filter . The specificity of the CL approach is that, in addition to the structural constraint in , a water closure constraint is also added. This closure can be written as a constraint on the solution :

where M is the number of samples (i.e., observations ⁱ) used in this criterion. The term υ is a scalar; it represents the variance of closure errors. This number can be defined a priori from previous experiments or as an empirical regularization term. In this case, the smallest value for υ that still ensures water budget closure in the estimate _a is chosen.

To find the optimal solution , an optimization algorithm needs to be used to minimize _closure under constraints of Eqs. (11):

This optimization needs to be performed simultaneously on all the N available samples. The samples can be pooled into a general dataset, and the optimization be used to obtain a general filtering matrix . As previously mentioned, matrix has a similar structure to the SW solution. As a consequence, SW can be a good first guess solution for CL, and this can speed up the optimization process and the quality of the final results.

It should be noted that in this CL solution, and are not used, though they are also very valuable constraints. It is an intermediate solution between the SW and the following optimal interpolation (OI). This method will not be tested in the following experiments because it is more difficult to implement and the application that will be presented in this paper does not require such complexity. However, it is a very interesting approach, especially for problems where complex physical constraints need to be imposed on the solution.

e. The optimal interpolation method

The goal of this approach is to simultaneously exploit all the available a priori information and to find the best compromise for _a. For this purpose, a cost function (i.e., quality criterion) is first defined as

The first right-hand term is related to the information provided by an FG variable _b. Since the focus of this paper is to obtain a model-independent estimate of the water cycle components, this FG needs to originate from observations such as available climatology, simple averaging of the observations for each component, or the result of an SW filtering. The second term is related to the observations _ε. The third term represents the water budget closure constraint. This OI filter can be used without the FG information _b, but, of course, this would be detrimental to the quality of the solution.

The optimal solution is chosen to minimize the quadratic cost function J(). This is obtained when equating to zero the gradient ∇J() = 0. By posing _ε − · = _ε − · _b − · ( − ^b), and neglecting the constant terms, we search for the gradient of

This gradient is given by

Therefore, the solution follows:

where

It can be seen that in this approach, the linear matrix is built using both the and matrices, and this is a valuable feature. No constraints of the form L_P, L_E, L_Q, and L_ΔS or no structural constraint such as in the filter of Eq. (9) are included in this solution, so there is a flexibility in the shape of the filter. Note that this OI filter does not change with the input data, except if the a priori information error statistics in matrix or the observations errors in change from one input datum to another.

In the following experiments, three tests will be conducted. The OI1 and OI2 solutions refer to the OI estimate with and without the closure term in Eq. (14) (i.e., last term in the equation with G). The solution OI3 will be the OI1 solution plus the postprocessing filtering (PF) step that will be introduced in the following to impose closure in a postprocessing step.

f. A nonlinear neural network weighting technique

In this approach, the estimate of the water components is provided by a nonlinear NN model g:

This is an extension of the SW, CL, or OI solutions that use a linear filtering instead of the nonlinear model g. This nonlinear character of g means that the weighting of the information in _ε is state dependent; this is generally a useful feature.

The details of the NN methodology will not be discussed here. The interested reader is referred to, for example, Bishop (1996) and Aires et al. (2004b). The NN model g is a parametric model, and a training procedure needs to be used to estimate these parameters. The training of g first requires an optimization procedure. The usual back-propagation algorithm (Rumelhart et al. 1986) will be used here. A training dataset is required too: it gathers a set of input–output couples (_ε, ). In this study, these couples will be built with an ensemble of water cycle components that follow a closure constraint from Pan et al. (2012) (see section 3a). Synthetic observations _ε are generated using a specified noise model for ε. The goal of the NN model g is therefore to retrieve the target water cycle components from the perturbed observations _ε. The noise model for the perturbations in _ε is not provided to the NN model; only their implicit characteristics in dataset are available to the NN.

The goal of this study is to retrieve, from the observations _ε, an estimation _a such that its components close the water cycle budget. How can this closure constraint be imposed on the NN outputs? The first solution is easy in practice; it consists of using the learning dataset outputs that respect this closure. In general, a constraint in the learning dataset is reproduced by the trained NN, but this is not a strong explicit constraint on the model g. This is the solution tested in this paper since the dataset described in section 3a respects this budget closure. Another solution would be to introduce the closure constraint (previously defined as _closure) in the loss function that is minimized during the NN training. Such complex constraints have been used, for example, in Aires et al. (1999), but their implementation is complex. In particular, the back-propagation algorithm would need to be redefined in this case. This approach will not be tested here.

In this NN technique, the weights for each water cycle component in _ε can vary with the state of the system. However, there is no constraint on the summation of the weights, as in the L_P, L_E, L_Q, and L_ΔS terms in section 2d. This can potentially be dangerous because compensation phenomena could occur (Aires et al. 2004a): an error in the weight of one observation in _ε is compensated by an opposite error in the weight of another observation. This can lead to very high positive and negative weight values in the NN filter. This is a classic symptom of ill-conditioned problems. In the case of such problems, regularization solutions tend to reduce the potential for such high-value weights, for example, the traditional NN weight decay (Bishop 1996).

As previously mentioned, the learning of the NN requires a dataset that includes the four water cycle component targets that respect the closure constraint. As a consequence, the NN learning inference strategy cannot easily be used on real conditions. However, in the experiments of section 4, this technique will be used to test the state dependency of the obtained filters.

g. The postprocessing filtering

This interesting approach has been presented, for example, in Pan and Wood (2006). It is not an integration method, but rather a postprocessing method to impose the closure constraint on a previously obtained solution _b. Thus, an estimate of the state variables _b = · _ε is first obtained using the observations _ε and some filter . In Pan and Wood (2006), the FG uses also some constraints from the variable infiltration capacity (VIC) macroscale hydrologic model, but other FG solutions (from the SW, CL, OI, or NN estimates) can be used instead. This will be tested in section 4.

A PF process is then applied on the FG solution _b in order to introduce the closure constraint:

where . This can be rewritten as

where _PF = [Id − G^T(G^T)⁻¹G], where Id is the identity matrix. This expression shows that this postprocessing is a filtering of _b based on G with the goal of obtaining the closure. Note again that this filtering is constant and identical for any situation unless there is enough information to have an a priori error matrix dependent on the situation.

In this paper, this approach is used, for example, with , that is, the SW solution of section 2c. Note that this postprocessing step does not use the observations _ε (used for the FG _b only), so the observations are not used twice to estimate , which would have been a problem (Rodgers 2000).

a. Hydrological dataset

The numerical experiments of this paper use the terrestrial water cycle dataset produced in Pan et al. (2012). This dataset includes the P, E, Q, and ΔS components at the monthly time scale for 23 years (1984–2006) and over 32 major basins. Therefore, the dataset is a matrix of dimension 276 × 4 for each basin. In this dataset, the terrestrial water budget closure is respected since this was the goal of its construction. The method used to impose this closure was based on the PF method, with an FG provided by a combination of observations and modeling constraints. Figure 1 represents the four water components in for the Mississippi basin. The budget is not represented in this figure, but, by construction, it is equal to zero for all time steps. It can be noted that large interannual variations are present, but no easy long-term trend can be identified here.

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The four water cycle components (P, E, Q, and ΔS) over the Mississippi basin for 1984–2006. Dataset is from Pan et al. (2012).

Citation: Journal of Hydrometeorology 15, 4; 10.1175/JHM-D-13-0148.1

The tests conducted in this paper are performed over two basins only: Mississippi and Niger. These basins are major hydrological regions. The mean and standard deviation (STD) of the four water components for these two basins are presented in Table 1. The average values are close for both basins but the variability can differ: evapotranspiration has a higher STD for Mississippi, but precipitation is the more variable component for Niger. In both cases, the runoff Q has the lowest variability.

Table 1.

Statistics on the Mississippi and Niger water components and a priori information on observation uncertainty characteristics for observation Y_ε.

From this dataset , synthetic observations are built for testing the ability of the integration methods of section 2 to retrieve . The perturbed observations will be built based on the following structure:

Each water component will be observed three times, except for runoff Q, which is observed twice only. In the following Monte Carlo experiments, each one of these observations will be chosen equal to the true component in plus a random perturbing noise using Eq. (3).

The a priori information on observation uncertainty is given in Table 1 (right columns). These uncertainties are synthetic and have been chosen based on an educated physical guess. There is no given information on potential biases in the observations (bias will be present in the following Monte Carlo experiments). It will be seen in the following section that the actual observation errors used in the Monte Carlo experiments will be different from this a priori information in order to get closer to real conditions. For example, the observation error covariance matrix is well determined for the simulation of the experimental datasets, but our knowledge of is only partial when testing the integration techniques. Tests have been conducted with a perfect knowledge of these observation errors; obviously, the retrieval results are better, but this is not a realistic situation.

b. Perturbation model

To test the integration methodologies presented in section 2, a Monte Carlo approach is applied to the dataset of the previous section. A strategy could be to first perturb the data once to obtain one synthetic observation _ε and then test which method retrieves better from _ε. However, the results obtained using this strategy would be highly dependent on the chosen perturbation ε. The Monte Carlo approach consists of drawing a large number of perturbations ε in order to obtain reliable statistics on the quality of the integration methods.

The first very important component of a Monte Carlo experiment is therefore the statistical model for the perturbations ε. The perturbation model represents both errors in observation _ε and uncertainties of our knowledge of these observation errors. In this study, the goal is to obtain a model that represents a bias and variance error on each one of the observations in _ε. The model also needs to represent the uncertainties on these observation errors, since most of the time, the user of the integration models has only a limited knowledge of these observation errors.

Let S(t) be the time series of one water cycle component. The time series are perturbed using the following model:

with a(t) following a normal distribution a(t) ~ (0, σ_a), where σ_a ~ (0, +20), that is, a uniform distribution between 0 and +20, and b ~ (−10, +10), also a uniform distribution. When the random errors a(t) + b are such that the term S(t) + a(t) + b is negative, the sampling is redone in order to avoid negative P, E, or Q components. The random bias b is independent from the time index t, so it represents a random bias applied for the whole time series for each N Monte Carlo members. The term a(t) is a random Gaussian noise with no bias and a variance that is drawn uniformly from −10 and +10 for each time series. Different parameters for the two uniform distributions (0, +20) and (−10, +10) could have been chosen for each observation _ε (from P₁ to ΔS₃). For simplicity of presentation, they are chosen here to be equal for each observation type. These noise characteristics constitute the synthetic errors of observations that will be used in the Monte Carlo experiment, but our a priori knowledge of these errors is imperfect and is provided in Table 1.

Other distributions of errors could have been chosen, for instance, a lognormal distribution for precipitation. Also, the error model can be chosen to be state dependent, for example, the random term is not the same for a low and a high runoff. Again, for simplicity, we have chosen to use a Gaussian distribution of the random errors because it is a good assumption when no other information on the uncertainties is available, and because often, errors become Gaussian when a first guess is used. A uniform distribution for parameters σ_a and b was chosen because it is the “no information” distribution in Bayesian statistics and because it represents well the type of a priori knowledge that is generally available about uncertainties.

c. Monte Carlo algorithm

The Monte Carlo scheme for the estimation of the integration error statistics is described below to facilitate the understanding of the approach. It uses N = 1000 samples; the stability of the results to this number has been validated. The inputs are the datasets _Mississippi and _Niger (i.e., two 276 × 4 matrices for Mississippi and Niger) gathering the time series of the state variables for both basins. The Monte Carlo algorithm draws random samples _ε using the perturbation model. The outputs of the Monte Carlo are the retrieval statistics of ( − _a) for the various methodologies. We will test only six methodologies: SW, OI, NN, and for each one of them, their PF-filtered versions. No off-diagonal error in for the input datasets is considered in these experiments.

• Data: Mississippi and Niger data: _Mississippi and _Niger.
• Result: Retrieval accuracies of the integration methodologies.
• Define and (i.e., a priori information on observation and first guess errors);
• For i = 1 to N (i.e., number of Monte Carlo samples, 1000) do
  • For j = 1 to J (i.e., number of observations, 276) do
    • Draw bias b from (−10, +10);
    • Draw σ_a from (0, +20);
    • For t = 1 to T (number of time steps) do
      • Draw a(t) from (0, σ_a);
    • End
    • Obtain observation noises ε_j(t) = a(t) + b for each time step t.
  • End
  • Obtain _ε = · + ε(t) for each time step.
  • Estimate using Eq. (7);
  • Estimate using Eq. (17);
  • Estimate using Eq. (19);
  • Apply the PF filtering to each estimate and obtain: , , ;
• End
• Perform the (_a − ) root-mean-square (RMS) statistics on the pooled (N samples) dataset for the six methods.

a. Filters

The first filter considered here is the PF _PF of Eq. (21). Figure 2 represents this 4 × 4 nonsymmetric matrix. It can be seen that _PF is close to the identity matrix, but the two more uncertain variables, ΔS and even more P, have a lower diagonal amplitude, which means that these variables can be modified more than the two others to reach the closure. The closure is possible thanks to the off-diagonal terms. This closure filter can be applied a posteriori to the SW, CL, OI, and NN filters.

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The values indicate how the weighting is done in the PF filter between observational water components and resulting components with closure.

Citation: Journal of Hydrometeorology 15, 4; 10.1175/JHM-D-13-0148.1

In Fig. 3, the 4 × 11 matrix filters of the SW and OI methods are represented without and with the PF. These filters are identical for Mississippi and Niger since they are not dependent on the state variables or observations _ε. They would change if our a priori information on observation uncertainty in Table 1 was different for the two basins. It can be seen that the SW filter has no off-diagonal terms: the three observations for precipitation are used only for the precipitation, and similarly for evapotranspiration, runoff, and terrestrial water storage change. This is a structural constraint. Again, the observations with less uncertainties have a larger weight. The application of the PF filter on the SW solution keeps a structure that is relatively close to the SW filter, but off-diagonal terms are introduced to obtain the water budget closure. The application of the PF filter on the OI solution has a relatively small impact because a closure constraint was already introduced in the OI filter [Eq. (17)]. However, the weights on the precipitation data are particularly low, meaning that the determination of the precipitation in is almost as much determined by the precipitation observations in _ε as by the other water cycle components and the closure constraint.

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Filters for SW, SW + PF, OI2, and OI3 = OI1 + PF. The values in these matrices indicate how the weighting is performed by each filter on each input dataset, in order to obtain the results integrated dataset of the four water cycle components.

Citation: Journal of Hydrometeorology 15, 4; 10.1175/JHM-D-13-0148.1

Figure 4 represents the Jacobians of the neural network integration model. They are the analog of the linear filters SW or OI since these Jacobians represent a linearization of the nonlinear NN function g. As such, they can be directly compared to Fig. 3. The Jacobians are dependent on the actual observations _ε, so they are represented here for both Mississippi and Niger, and for January and July means (i.e., monthly averages over the 23 years of data). It can be noted that January and July Jacobians are not identical. Mississippi and Niger Jacobians differ too. This means that the NN is able to optimize its weighting of the observations in _ε to the particular location and time. However, the overall structure of the filters is similar and close to the filters of Fig. 3, which is sign of robustness of the integration methodologies. The closure constraint introduced with PF has almost no impact on the NN filters (not shown) because the closure is already good with the NN integration solution.

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NN Jacobians (analog of the linear filters of Fig. 3) for (from top to bottom) Mississippi in January and July and Niger in January and July.

Citation: Journal of Hydrometeorology 15, 4; 10.1175/JHM-D-13-0148.1

b. Integration results

Figure 5 represents the time series of the precipitation, evapotranspiration, runoff, water storage change, and terrestrial water budget for year 1984 for the Niger basin. There are three observations for each water component (red). These observations are deviated from the target cycle (dashed black line). The SW solution (green) is a weighting of the observations. It can be seen in the time series of the budget that this weighting of the observations is already a good improvement, even if no closure constraint is used, since the residuals are much lower. The NN estimate (blue) is an even better solution: the residuals are, in this case, negligible. It confirms that the NN is able to obtain the closure even if this constraint was only implicitly imposed in the learning dataset .

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Time series of the P, E, Q, ΔS, and terrestrial water budget for year 1984 for the Niger basin. Observations are in red, the SW estimates are in green, the NN estimate in blue, and the target is dashed black.

Citation: Journal of Hydrometeorology 15, 4; 10.1175/JHM-D-13-0148.1

The RMS errors of the retrieval of the four water cycle components and the residuals of the terrestrial water budget are represented in Table 2 for the Mississippi and Niger basins, for the whole 1984–2006 record. These estimates result from the Monte Carlo experiments described in section 3. The STD of each variable is represented first for comparison purposes. Typical errors for the observations are also represented: set 1 is for (P₁, E₁, Q₁, ΔS₁) in Eq. (22) and set 2 is for (P₂, E₂, Q₂, ΔS₂). The RMS errors for set 1 are the same for both basins since no location dependency has been introduced in the observation noise. However, for each water component, errors from the two sets are different.

Table 2.

RMSE statistics from the Monte Carlo experiments for the retrieval of P, E, Q, ΔS, and the resulting water cycle budget for the Mississippi and Niger basins. The STD of each variable is represented for comparison purposes. Integration methodologies are SW, SW + PF, OI1, OI2, OI1 + PF, NN, and NN + PF. Statistics are performed for the whole 1984–2006 period.

The results of this table are also represented in Fig. 6 for Mississippi (continuous lines) and for Niger (dashed lines). Only the SW, SW+PF, and NN results are presented since the other integration methods are comparable to one or the other of these three methods. Each integration technique is better than the use of a single set of observations. This is expected, of course, since the combination of multiple observations is better than the use of only one. SW is a very good improvement over the use of a single set of observations; the RMS errors can be divided by a factor of 2. However, the resulting budget is still not acceptable (RMS about 9.25 mm for Mississippi). The introduction of the postprocessing filtering PF to SW solves this problem: the budget reaches closure. In addition, the introduction of the closure constraint has not degraded the retrievals of SW; it has actually improved it significantly (Ko and Bitmead 2007), for example, from 5.38 to 4.36 for precipitation. This means that the closure constraint is good a priori information and that its use in the integration process is very valuable.

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RMSE of the retrieved water component (P, E, Q, and ΔS) using method SW (red), SW + PF (green), and NN (blue). The other integration methods are similar to one of these curves. The resulting water cycle budget is also represented. Results are for the whole 1984–2006 time series. Mississippi basin is in continuous lines and Niger in dashed lines.

Citation: Journal of Hydrometeorology 15, 4; 10.1175/JHM-D-13-0148.1

The RMS errors of OI1 are almost identical to SW. This is surprising since more information is used in the OI integration techniques, that is, the observing operator . However, this information is relatively limited in this experiment because of the simplicity of the structure. Its use in the integration might only be useful for more sophisticated models such as when is the linearization of the radiative transfer in numerical weather prediction centers for traditional weather forecasting. The introduction of the closure constraint in the OI2 estimation has a very strong positive impact in the water budget closure that decreases from 9.25 for the OI1 estimate, to 0.01 with OI2, again for the Mississippi case. This demonstrates the ability of the OI technique to utilize external constraints such as the closure. Again, the introduction of the closure constraint also improves the estimate of the water components. Note that results for the SW + PF and the OI2 solution are almost identical. Furthermore, the OI1 + PF solution, which includes the PF filtering, performs as well as the OI2 estimate. This means that the introduction of the closure can be done simultaneously with the observations or as a postprocessing step.

The NN approach appears to be quite a good solution. It outperforms the other methods in terms of retrieval accuracy. Furthermore, it is possible to confirm that the closure constraint imposed only implicitly in the dataset used to train the NN is respected in the NN outputs since the NN budget has RMS error close to zero. The use of the PF filter does not improve the solution: the retrieval uncertainties are the same and the closure of the budget is only slightly reduced. The fact that the NN follows the closure well using only an implicit constraint could be surprising. However, the PF processing to explicitly impose the closure (section 2g) is just a 4 × 4 matrix close to identity (Fig. 2), so if the NN “learns” during the training that such postprocessing is useful to obtain optimal retrievals, it is easy for the NN to represent it in its model g. Another remark is that these NN experiments have been performed without any regularization imposing constraints on the weight amplitude (section 2f), so the fact that the results appear robust means that the problem is well posed.