Ensemble Kalman filtering (EnKF) is an efficient approach to addressing
uncertainties in subsurface groundwater models. The EnKF sequentially
integrates field data into simulation models to obtain a better
characterization of the model's state and parameters. These are generally
estimated following joint and dual filtering strategies, in which, at each
assimilation cycle, a forecast step by the model is followed by an update
step with incoming observations. The joint EnKF directly updates the
augmented state-parameter vector, whereas the dual EnKF empirically employs
two separate filters, first estimating the parameters and then estimating the
state based on the updated parameters. To develop a Bayesian consistent dual
approach and improve the state-parameter estimates and their consistency, we
propose in this paper a one-step-ahead (OSA) smoothing formulation of the
state-parameter Bayesian filtering problem from which we derive a new
dual-type EnKF, the dual EnKF

In modern hydrology research, uncertainty quantification studies have focused
on field applications, including surface and subsurface water flow,
contaminant transport, and reservoir engineering. The motivations behind
these studies were driven by the uncertain and stochastic nature of
hydrological systems. For instance, surface rainfall–runoff models that
account for soil moisture and streamflows are subject to many uncertain
parameters such as free- and tension water storage content, water depletion
rates, and melting threshold temperatures

The EnKF is a filtering technique that is relatively simple to implement,
even with complex nonlinear models, requiring only an observation operator
that maps the state variables from the model space into the observation
space. Compared with traditional inverse and direct optimization techniques,
which are generally based on least-squares-like formulations, the EnKF has
the advantage of being able to account for model errors that are
present not only in the uncertain parameters but also in the external forcings

The EnKF is widely used in surface and subsurface hydrological studies to
tackle state-parameter estimation problems

Several studies argued that the joint EnKF may suffer from important
inconsistencies between the estimated state and parameters that could degrade
the filter performance, especially with large-dimensional and strongly
nonlinear systems

The dual filter has been basically introduced as a heuristic scheme and is
not consistent with the Bayesian filtering framework

Motivated by the promising results of

The rest of the paper is organized as follows. Section 2 reviews the standard
joint and dual EnKF strategies. The dual EnKF

Consider a discrete-time state-parameter dynamical system:

The term

We focus on the state-parameter filtering problem, i.e., the estimation
at each time,

The key idea behind the joint EnKF is to transform the state-parameter system
(Eq.

Starting at time

Forecast step: the parameters vector members,

Analysis step: once a new observation is available, all members,

In contrast with the joint EnKF, the dual EnKF is empirically designed
following a conditional estimation strategy, operating as a succession of two
EnKF-like filters. First, a (parameter) filter is applied to compute

Forecast step: the parameters ensemble,

Analysis step: as in Eq. (

Forecast step: each member,

Analysis step: as in the parameter filter,

To better understand how the dual EnKF differs from the joint EnKF, we focus
on how the analysis members at time

Following a probabilistic formulation, the augmented state system
(Eq.

One can then easily verify that the joint EnKF can be derived from a direct
application of two classical results of random sampling (Properties 1 and 2
in Appendix A) on the following classical generic formulas:

Regarding the dual EnKF, the forecast ensemble of the state and observations
in the parameter filter can be obtained following the same process as in the
joint EnKF. This is followed by the computation of the analysis ensemble of
the parameters using Property 2 and

The classical (time-update, measurement-update) path
(Eqs.

The analysis pdf,

Smoothing step: the one-step-ahead smoothing pdf,

The smoothing step (Eq.

Forecast step: the smoothing pdf at

From Eq. (

Since it is not possible to derive the analytical solution of
Eqs. (

Starting at time

The analysis ensemble,

Starting from an analysis ensemble,

Smoothing step: the state forecast ensemble,

Forecast step: the analysis ensemble of the state

The proposed dual EnKF

The joint EnKF

Despite the smoothing formulation of the dual EnKF

The computational complexity of the different state-parameter EnKF schemes
can be split between the forecast (time-update) step and the analysis
(measurement-update) step. The joint EnKF requires

Approximate computational complexities of the joint EnKF, the
dual EnKF, and the dual EnKF

We adopt in this study the subsurface flow problem of

Plan view of the conceptual model for the 2-D transient groundwater
flow problem. East and west boundaries (

Parameters of the random functions for modeling the spatial
distributions of the reference and perturbed recharge fields. The ranges in

Left panel: daily transient reference pumping rates from wells PW1,
PW2, and PW3. Negative values indicate pumping or groundwater that is being
removed from the aquifer. Right panel: reference heterogenous spatial
recharge values obtained using the sequential Gaussian simulation code

We consider a dynamically complex experimental setting involving various
time-dependent external forcings. The recharge is assumed spatially
heterogenous and sampled using the GCOSIM3D toolbox

Prior to assimilation, a reference run is first conducted for each experimental setup using the prescribed parameters above, and is considered as the truth. We simulate the groundwater flow system over a year-and-a-half period using the classical fourth-order Runge–Kutta method with a time step of 12 h. The initial hydraulic head configuration is obtained after a 2-years model spin-up starting from a uniform 15 m head. Reference heterogenous recharge rates are used in the setup as explained before. The water head changes (in m) after 18 months are displayed with contour lines in the left panel of Fig. 3. One can notice larger variations in the water head in the lower left corner of the aquifer domain, consistent with the high conductivity values in that region. The effects of transient pumping in addition to the heterogenous recharge rates are also well observed in the vicinity of the pumping wells.

Groundwater flow contour maps obtained using the reference run (left panel) and the perturbed forecast model (right panel) after 18 months of simulation. The well locations from which head data are extracted are shown by black asterisks. In the left panel, we show the first network consisting of nine wells. In the right panel, the other network with 25 wells is displayed.

To imitate a realistic setting, we impose various perturbations on the reference model and set our goal to estimate the water head and the hydraulic conductivity fields using an imperfect forecast model and perturbed data extracted from the reference (true) run. This experimental framework is known as “twin-experiments”. In the forecast model, we perturb both transient pumping and spatial recharge rates. The perturbed recharge field, as compared to the reference recharge in Fig. 2, is sampled with different variogram parameters as shown in Table 2. Pumping rates from PW1, PW2, and PW3 are perturbed by adding a Gaussian noise with mean zero and standard deviation equal to 20 % of the reference transient rates. The flow field simulated by the forecast (perturbed) model after 18 months is shown in the right panel of Fig. 3. Compared to the reference field, there are clear spatial differences in the hydraulic head, especially around the first and second pumping wells.

To demonstrate the effectiveness of the proposed dual EnKF

To initialize the filters, we follow

The filter estimates resulting from the different filters are evaluated based
on their average absolute forecast errors (AAE) and their average ensemble
spread (AESP):

Mean average ensemble spread (AESP) of the water head and the
hydraulic conductivity for three different ensemble sizes. The reported
values are given for the joint EnKF, dual EnKF, joint EnKF

Filter inbreeding indicator: Ratio of the mean average absolute error (AAE)
and mean average ensemble spread (AESP) of the water head and the hydraulic
conductivity for three different ensemble sizes. The reported values are given
for the joint EnKF, dual EnKF, and the proposed dual EnKF

We first study the sensitivity of the three algorithms to the ensemble size,

Furthermore, we examined the estimated uncertainties about the forecast
estimates by computing the average spread of both the hydraulic head and
conductivity ensembles. To do this, we evaluated the time-averaged AESP of
both variables and tabulated the results for the three ensemble sizes in
Table 3. For all schemes, increasing the ensemble would increase the spread
of the hydraulic head ensemble due to the natural variability of the
considered subsurface system. In contrast, the AESP conductivity decreases as

AAE time series of the hydraulic head and conductivity using the
joint EnKF, dual EnKF, joint EnKF

One could also exploit the computed AAE and AESP to assess whether the
filters suffer from the inbreeding problem. Filter inbreeding occurs when the
variance of the state and parameters ensemble is increasingly reduced over
time. This may not only deteriorate the quality of the estimated filter error
covariance matrices, but also wrongly suggests more confidence in the
forecast and strongly limits the filter update by the incoming observation.
One standard test for examining inbreeding is to compute the ratio of the AAE
to the AESP

In terms of computational cost, we note that our assimilation results were
obtained using a 2.30 GHz workstation and four cores for parallel looping
while integrating the ensemble members. The joint EnKF is the least intensive
requiring 70.61 s to perform a year-and-a-half assimilation run using
50 members. The dual EnKF and dual EnKF

In the second set of experiments, we test the filters' behavior with different temporal frequency of observations; i.e., the times at which head observations are assimilated. We implement the three filters with 100 members and use data from nine observation wells perturbed with 0.10 m noise.

Figure 5 plots the mean AAE of the hydraulic conductivity estimated using the
three filters for the six different observations sampling frequencies. The
dual and joint EnKFs lead to comparable performances, but the latter
performs slightly better when data are assimilated more frequently, i.e.,
every 5 and 3 days. The performance of the proposed
dual EnKF

Mean average absolute errors (AAE) of log-hydraulic conductivity,

Reference (dashed) and predicted (solid) hydraulic head evolution at
monitoring wells MW1, MW2, and MW3. Results are obtained using the joint EnKF
and the dual EnKF

Time series of AAE of hydraulic head (left panel) and conductivity
(right panel) using the joint EnKF, dual EnKF, and dual EnKF

We have also compared the hydraulic head estimates for different sampling
frequencies of observations. Similar to the parameters, the improvements of
the dual EnKF

One effective way to evaluate the estimates of the state is to examine the
evolution of the reference heads and the forecast ensemble members at various
aquifer locations. For this, we plot in Fig. 6 the true and the estimated
time-series change in hydraulic head at the assigned monitoring wells as they
result from the joint EnKF, dual EnKF, and the dual EnKF

Spatial maps of the reference, initial and recovered ensemble means
of hydraulic conductivity using the joint EnKF, dual EnKF, and
dual EnKF

We further examine the robustness of the proposed dual EnKF

To further assess the performance of the filters we analyze the spatial
patterns of the estimated fields. To do so, we plot and interpret the
ensemble mean of the conductivity as it results from the three filters using
nine observation wells. We compare the estimated fields after 18 months
(Fig. 8) with the reference conductivity. As can be seen, the joint and
the dual EnKFs exhibit some overshooting in the southern (low conductivity)
and central regions of the domain. In contrast, the dual EnKF

Mean AAE of the hydraulic conductivity using the joint EnKF,
dual EnKF, and dual EnKF

In the last set of sensitivity experiments, we fix the number of wells to
nine, the sampling period to 5 days, and test with different standard
deviations of measurement error to perturb the observations. We plot the
results of 10 different observational error scenarios in Fig. 9 and
compare the conductivity estimates obtained using the joint EnKF, dual EnKF,
and the dual EnKF

Finally, we investigated the time evolution of the ensemble variance of the
conductivity estimates as they result from the dual EnKF and the
dual EnKF

Left panel: ensemble variance map of the initial conductivity field.
Right sub-panels: ensemble variance maps of estimated conductivity after
6 and 18 month assimilation periods using the dual EnKF and the proposed
dual EnKF

Reference (dashed) and predicted (solid) hydraulic head evolution at
the control well: CW. Results are obtained using the joint EnKF, dual EnKF,
and the dual EnKF

To further assess the system performance in terms of parameters retrieval, we
have integrated the model in prediction mode (without assimilation) for an
additional period of 18 months starting from the end of the assimilation
period. We plot in Fig. 11, using the final estimates of the conductivity
as they result from the three filters (after 18 months), the time evolution
of the hydraulic head at the CW. The ensemble size is set to 100,
sampling period is 1 day, number of data wells is 25, and measurement
noise is 0.5 m. The reference head trajectory at the CW decreases from
17.5 to 16.9 m in the first 2 years, and then slightly increases to 17.2 m in
the rest of the years. The forecast ensemble members of the joint EnKF at this
CW fail to capture to reference trajectory of the model. This is due to the
large measurement noise imposed on the head data. The dual EnKF performs
slightly better and predicts hydraulic head values that are closer to the
reference solution. The performance of the dual EnKF

Finally, in order to demonstrate that our results are statistically robust,
10 other test cases with different reference conductivity and heterogeneous
recharge maps were investigated. In each of these cases, we sampled the
reference fields by varying the variogram parameters, such as variance,

We presented a one-step-ahead smoothing-based dual ensemble Kalman filter
(dual EnKF

Performance of the joint/dual EnKF and the proposed
dual EnKF

We tested the proposed dual EnKF

The proposed scheme is easy to implement and only requires minimal
modifications to a standard EnKF code. It is further computationally
feasible, requiring only a marginal increase in the computational cost
compared to the dual EnKF. This scheme should therefore be beneficial to the
hydrology community given its consistency, high accuracy, and robustness to
changing modeling conditions. It could serve as an efficient estimation tool
for real-world problems, such as groundwater, contaminant transport, and
reservoir monitoring, in which the available data are often sparse and noisy.
Potential future research includes testing the dual EnKF

Only simulated data were used in this study. Please contact the corresponding or first author for the details about the algorithms and the codes.

The following classical results of random sampling are extensively used in the derivation of the ensemble-based filtering algorithms presented in this paper.

We show here that the samples,

Now, using the samples

Research reported in this publication was supported by King Abdullah University of Science and Technology (KAUST). Edited by: M. Giudici Reviewed by: three anonymous referees