HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-20-555-2016Joint inference of groundwater-recharge and hydraulic-conductivity fields from head data using the ensemble Kalman filterErdalD.daniel.erdal@uni-tuebingen.deCirpkaO. A.https://orcid.org/0000-0003-3509-4118Center for Applied Geoscience, University of Tübingen, 72074 Tübingen, GermanyD. Erdal (daniel.erdal@uni-tuebingen.de)1February20162015555697May201512June201515December201528December2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://hess.copernicus.org/articles/20/555/2016/hess-20-555-2016.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/20/555/2016/hess-20-555-2016.pdf
Regional groundwater flow strongly depends on groundwater recharge and
hydraulic conductivity. Both are spatially variable fields, and their
estimation is an ongoing topic in groundwater research and practice. In this
study, we use the ensemble Kalman filter as an inversion method to jointly
estimate spatially variable recharge and conductivity fields from head
observations. The success of the approach strongly depends on the assumed
prior knowledge. If the structural assumptions underlying the initial
ensemble of the parameter fields are correct, both estimated fields resemble
the true ones. However, erroneous prior knowledge may not be corrected by the
head data. In the worst case, the estimated recharge field resembles the true
conductivity field, resulting in a model that meets the observations but has
very poor predictive power. The study exemplifies the importance of prior
knowledge in the joint estimation of parameters from ambiguous measurements.
Introduction
Regional groundwater flow depends on spatially variable properties of the
subsurface, notably the hydraulic conductivity field, and boundary conditions
such as groundwater recharge. In practical groundwater-modeling applications,
parameters of both aquifer properties and boundary conditions are estimated
from measurements of hydraulic heads at a limited number of observation
locations e.g.,. While many theoretical studies on
parameter estimation in aquifers have concentrated on the assessment of the
spatially variable hydraulic-conductivity field, also groundwater recharge is
known to be highly variable in both time and space
e.g.,. Among the different techniques of estimating
recharge reviewed by , we consider here numerical
approaches in which measured time series of hydraulic head are used to
estimate groundwater recharge. The key question to be addressed in the
present study is under which conditions it is possible to infer both the
recharge field (a space–time function) and the spatial distribution of
hydraulic conductivity from the same data set of hydraulic-head measurements.
In engineering practice, the model domain is typically subdivided into a
small number of zones with given geometry, and uniform values of the material
properties are assigned to each zone. Likewise, the land surface is
subdivided into zones with uniform recharge values, reflecting land use, soil
types, and local climate variability. As an alternative, parameter values may
be estimated at a limited number of points and interpolated in between
e.g.,. By construction, these approaches can only
determine spatial structures of the parameter fields meeting the prescribed
shapes. A particular difficulty of this approach is that the variability
within the given zones may be bigger than between the zones, while the
internal variability is completely neglected in the parameter estimation.
The estimation of hydraulic conductivity as a continuous field has been
intensively investigated in the past (see the reviews of
; and recently ). In these
approaches discretization of the domain leads to a formal number of
parameters to be estimated that is identical to the number of cells or grid
points. Typical 2-D applications result in O (104) parameters,
whereas 3-D numerical domains may easily be made of O (106) cells.
As the number of measurement points is by orders of magnitude smaller, this
inverse problem is inherently ill-posed without additional constraints. Some
authors therefore rely on flexible sets of shapes, such as polynomial trends
or Voronoi polygons e.g., rather than
estimating O (104–106) parameter values. In standard geophysical
inversion, Tikhonov regularization is the common approach to estimate
distributed parameter fields from a limited set of measurements. Here, the
parameters are assumed to be continuous spatial functions, but large
gradients, curvatures, or deviations from prior values are penalized
applications to subsurface hydrology are given by, amongst others,. In subsurface hydrology,
however, the geostatistical framework is more common.
and independently showed that the two approaches are
mathematically equivalent to each other.
In geostatistical inversion, the parameter field to be estimated is assumed
to be an autocorrelated random space function. This prior knowledge is used
in Bayesian inference, where the statistical distribution of the parameters
is conditioned on the measurements of dependent quantities, such as hydraulic
heads. A variety of schemes target a single smooth spatial distribution
approximating the conditional mean of the parameter field using Gauss–Newton-
or conjugate-gradient-type estimation schemes
e.g.,.
These methods can be extended to the generation of multiple conditional
realizations by the method of smallest modification
e.g.,. However, the computational
costs to obtain a single conditional realization is identical to those of the
smooth best estimate. Also, the Gauss–Newton method requires the evaluation
of the sensitivity of each measurement with respect to all parameter values,
involving the solution to as many adjoint problems as there are measurements,
which may become unbearable in the case of many measurements, such as those
obtained from transient processes. In the context of the present study it may
be noteworthy that many geostatistical approaches have focused on the
exclusive estimation of hydraulic conductivity; some include storativity
e.g.,, but most assume
that the boundary conditions are deterministic. An exception is the study of
, who used the geostatistical approach of
sequential self-calibration to jointly estimate the fields of hydraulic
conductivity and groundwater recharge from head measurements. The authors
considered the problem of a well-capture zone, in which they estimated
hydraulic conductivity as a continuously varying spatial field, whereas
recharge was parameterized by zones with uniform values.
In groundwater hydrology, sequential data assimilation and Kalman filter
methods have long been used e.g.,. Particularly, and
increasingly, popular is the ensemble Kalman filter (EnKF)
or versions thereof. Although the EnKF was primarily
constructed to update model-state variables, in subsurface hydrology it is
commonly used to estimate hydraulic conductivity. For this purpose
, , ,
, and , among others, showed that the
use of head observations in an EnKF framework can help improve the
conductivity estimates, while and , among
others, considered tracer tests for the same purpose. Most parameter
estimations used 2-D models, as these are conceptually simpler, faster, and
easier to constrain and display. However, EnKF has also successfully been
applied to infer 3-D hydraulic-conductivity fields e.g.,.
An important step in setting up an EnKF to estimate parameters is the choice
of initial ensemble. This choice is the most straightforward way of allowing
prior information – such as ideas about correlation lengths, mean values, or
spatial pattern – to influence the filter process. From a technical point of
view, the issue of initial sampling is how to represent the prior knowledge
in an ensemble that is as small as possible, by, for example, adding ensemble
subspace restriction and requirements on the sampling
e.g.,. From a practical point of view,
especially in subsurface modeling, the issue is that our prior knowledge of
the parameters, their mean values, deterministic trends, and spatial
correlation structure is often limited. This may be seen as a more severe
problem than choosing a sufficiently large ensemble size to actually capture
the assumed variability by the ensemble. To overcome the limited knowledge
about true parameters values, the use of synthetic test cases for methods
testing and evaluation is very common in subsurface hydrology
e.g.,. Here, the prior knowledge is only
limited to what the modeler considers a reasonable assumption, and it is not
uncommon in the groundwater-EnKF context that the synthetic true parameter
field is a single realization generated the same way as the initial ensemble
e.g.,. Hence, perfect knowledge about the statistics of the estimated
parameters is implicitly assumed, which is a highly unrealistic assumption.
The impact of the prior assumptions in groundwater modeling was considered,
for example, by , who concluded that it was possible to
estimate reasonable log-conductivity fields using the EnKF despite wrong
priors, although the result was worse than when using correct information,
and by , who showed that it is possible to
use the EnKF to correct a biased prior mean and partly correct a wrong prior
variance, but not erroneous prior correlation lengths.
In this work we study the impact of the prior knowledge when jointly
estimating conductivity and recharge from hydraulic-head data only. We use an EnKF setup in which the
initial ensemble is drawn using different assumptions of the spatial pattern
of the parameters. Section discusses why the conductivity and
the recharge are so difficult to estimate jointly if only pressure-head data
are available. Section explains the ensemble Kalman filter
and the synthetic example used throughout this paper, while results and
discussions are found in Sect. . We end with conclusions
in Sect. .
Theory
In regional-scale groundwater-flow problems, we typically rely on the
validity of the Dupuit assumption, stating that variations in hydraulic head
and groundwater velocity are restricted to the horizontal directions. Under
this condition, the depth-averaged, two-dimensional groundwater-flow equation
for a phreatic aquifer reads as
Sy∂h∂t-∇⋅Kh-z0∇h=R,
subject to initial and lateral boundary conditions. Sy(x) [–]
is the specific-yield field, which is the drainage-effective porosity of the
formation, K(x) [L T-1] denotes the depth-averaged
hydraulic-conductivity field, R(x, t) [L T-1] is the field of
groundwater recharge, z0(x) [L] denotes the geodetic height of
the aquifer bottom, h(x, t) [L] is the hydraulic-head field to be
simulated, t [T] is time, and x [L] is the vector of horizontal
spatial coordinates.
The term K(h-z0) may be interpreted as a transmissivity field
T(x,t) [L2 T-1], varying in space and time. We now consider
a confined surrogate aquifer with an assumed transmissivity field
Tass(x) [L2 T-1] that differs from the true one
(e.g., an incorrectly estimated transmissivity field). The logarithm of the scaling
factor between the two transmissivities is denoted f(x, t) [–]
f=lnK×h-z0Tass.
Substituting Eq. () into Eq. () yields
Sy∂h∂t-∇⋅Tassexp(f)∇h=R.
Applying the chain rule of differentiation to the divergence in
Eq. (), applying the product rule of differentiation to
∇exp(f), and dividing by exp(f) results in
exp(-f)Sy︸:=Sapp∂h∂t-∇⋅Tass∇h=exp(-f)R+∇f⋅∇hTass︸:=Rapp⇒Sapp∂h∂t-∇⋅Tass∇h=Rapp,
subject to the same initial and lateral boundary conditions as above. In
Eq. (), Sapp(x, t) [–] and
Rapp(x, t) [L T-1] are apparent specific-yield and
groundwater-recharge fields. Equation () results in exactly
the same hydraulic-head distribution as the original groundwater-flow
Eq. (), even though the transmissivity field is
different. Note that exp(-f) is positive, so the apparent specific
yield Sapp remains positive, whereas no sign restrictions apply to
∇f⋅∇h, resulting in both positive and negative
Rapp values. In the case of a phreatic aquifer, the true transmissivity
varies with hydraulic head, so the apparent parameters change with time.
If the water-filled thickness of the true aquifer does not change with time,
which is the case for confined aquifers, the apparent fields are time-invariant.
The derivation given above exemplifies that the same hydraulic-head field can
be obtained with different hydraulic-conductivity fields by modifying
recharge and, in the case of transient flow, the specific yield. It is noteworthy
that the apparent recharge depends on the gradient of the original
transmissivity field. Hence, a large – positive or negative – apparent
recharge is expected at locations where the transmissivity changes
drastically. Though we have shown that modifications of recharge and specific
yield can always replace the conductivity, the opposite case is not
guaranteed, because the conductivity has clear physical limitations: notably
it cannot be negative.
The fact that conductivity variation can be exchanged by recharge and
specific-yield variations renders the joint estimation of hydraulic
conductivity, recharge (and specific yield) an inherently ill-posed problem
even when the hydraulic-head field is known at every point in the domain (and
every time point).
We may illustrate the problem by the example of an unconfined aquifer at
steady state, shown in Fig. . The original simulation (left
column in Fig. ) exhibits a square-shaped inclusion of low
permeability in an otherwise uniform high-permeability field (first row; 2
orders of magnitude difference in K) and a constant low recharge rate
(second row). As boundaries, we employ a significant head drop from west (50
m) to east (8 m) and no flow boundaries on the north and south sides. The
resulting head field is shown in the third row of Fig. , and
the corresponding field of Darcy velocities in the fourth row of Fig. .
If the inclusion is removed, and the recharge remains the same, the system
shows a perfectly homogeneous behavior (middle column of Fig. ).
The third column in Fig. , on the other
hand, shows exactly the same hydraulic-head field as the original simulation,
but the permeability field is uniform, whereas the recharge field shows
strong fluctuation. From Fig. we can note that, in
accordance with Eq. (), the strong positive and negative
recharge rates are introduced at the interface of the removed inclusion.
Also, while the head fields of the original and surrogate models are
identical, the velocity fields are quite different, because the
conductivities are different. The latter implies that transport would be
strongly different between the two cases. It becomes also clear that, without
additional constraints, a unique joint estimation of both recharge and
conductivity fields is strictly impossible.
Illustrative example of replacing a heterogeneous conductivity field
(left column panels) with a homogeneous conductivity and an effective
recharge (right column panels). Please note the different scale on the third
recharge plot.
In classical model calibration, the ambiguity between transmissivity and
groundwater recharge may cause problems of ill posedness, but assuming
known zones with block-wise uniform parameter values restricts the
solution of the inverse problem. As example, the strong positive and negative
recharge values of the surrogate model in Fig. would most
likely not be obtained in standard model calibration because the recharge
zones would hardly be chosen as embedded rectangular frames. In shape-free
inversion, using either Tikhonov regularization or geostatistical methods, by
contrast, the solution space is much less restricted, and chances that
unresolved transmissivity variations are traded for recharge fluctuations are
in principle fairly high. The question thus arises under which conditions the
estimated fields are reasonable despite the ambiguity of aquifer properties
and boundary conditions.
MethodsEnsemble Kalman filter
In the following we briefly repeat the basic assumptions of deriving the
EnKF within a Bayesian framework. While it is
possible to have a much more pragmatic view on EnKF as an extended
least-square estimator, we believe that the transparency of the Bayesian
framework with respect to the underlying assumptions is beneficial. In
particular, the Bayesian framework explains the choice of the initial
ensemble as prior knowledge and the conceptual importance of the prior
knowledge in the estimation procedure, while a frequentist's point of view is
in contrast to making use of prior knowledge altogether. For further
transparency, we first explain the extended Kalman filter (see similar
derivations by ).
We denote the vector of all parameters (recharge values and log-hydraulic
conductivities of all cells) Φ. Prior to considering
measurements, they are assumed to be random functions following a
multi-Gaussian distribution, which is fully characterized by the prior mean
μΦ′ and covariance matrix
PΦΦ′. Throughout this paper, the prior values are
denoted by a single prime, and the posterior by a double prime. If we assume that the covariance
function PΦΦ′(v) is stationary with the distance vector v
and known structural parameters (variance, correlation lengths, rotation
angles), the element (i, j) of the covariance matrix
PΦΦ′ is PΦΦ′(|xi-xj|). The full
matrix is constructed by all grid points.
The vector of simulated hydraulic heads ht at time level t
depends on the heads ht-1 at the previous time level and on the
parameters Φ. Because the old heads ht-1
depend on Φ, they are random variables, too. In the
combination of data assimilation and parameter estimation applied here, the
vector of all simulates states (the heads ht in all cells) and
the vector of all parameters Φ are concatenated to a single
vector xt of states and parameters, assumed to be random
multi-Gaussian functions with unconditional mean μx′ and
covariance matrix Pxx′, in which the prior statistics of
ht are obtained by linearized uncertainty propagation of the
statistics of ht-1 and Φ.
For convenience, we denote running the model and simulating the observations
(which is here just picking the heads at the observation locations) as
ft(ht-1, xt). It should be noted that
f here, hence, denotes both the forward model and the observation
operator. This model outcome is contrasted to the measurements of heads at
time level t, here denoted yt. The true (unknown) heads at the
measurement locations are considered to be a vector of random variables with
a multi-Gaussian distribution, characterized by the measurement vector
yt as the mean and the covariance matrix R, reflecting
measurement error.
Since we assume multi-Gaussian distributions, finding the best conditional
estimate μx′′ of the entire head
field at the new time level and the parameters by application of Bayes'
theorem results in minimizing the following objective function
W(xt):
Wxt=xt-μxt′TP′xtxt-1xt-μxt′+ftht-1,xt-ytTR-1ftht-1,xt-yt,
which is done by setting the derivative of W(x) to zero
e.g.,. In the linearized version,
ft(ht-1, xt) is linearized about the prior
mean μxt′, and the linearized conditional covariance matrix
Pxtxt′′ of xt is obtained by inverting the
Hessian of W(xt), using the same linearization. Kalman filtering
is based on these approximations. Here, the data are successively accounted
for, considering one time level after the other. Then, the posterior mean
μxt′′ and covariance matrix Pxtxt′′ of
time level t are propagated to the next time level t+ 1 to obtain
the corresponding prior mean and covariance matrix.
By applying rules of matrix identities, it can be shown that linearization
about the prior mean μxt′ leads to the
following expression for the conditional mean and covariance matrix:
μxt′′=μxt′+Pxtyt′Pytyt′+R-1yt-ftμht-1,μxt′,Pxtxt′′=Pxtxt′-Pxtyt′Pytyt′+R-1Pytxt′,
in which Pytxt′=JPxtxt′
is the cross-covariance matrix between yt and xt,
Pxtyt′=P′ytxtT, and
Pytyt′=JPxtxt′JT
is the propagated covariance matrix of yt, expressing the
uncertainty of yt caused by the uncertainty of xt.
J denotes the sensitivity matrix of ft with respect to
xt, derived about the prior mean.
The scheme described so far is known as extended Kalman filter. It relies on
linearization about the prior mean and has the disadvantages that the full
sensitivity matrix J must be evaluated, which can be
computationally very costly. Also, even slight nonlinearities in
ft(ht-1, xt) imply that the propagated
covariance matrices are not correct.
A popular alternative to the original Kalman filter is the
EnKF , in which the linearization is performed
about an entire ensemble of state and parameter values, and no sensitivity
matrices are computed. The prior statistics are given by
μxt′=1n∑i=1nx′t(i),μyt′=1n∑i=1nfth′′t-1(i),x′t(i),Pxtxt′=1n-1∑i=1nx′t(i)-μxt′⊗x′t(i)-μxt′,Pxtyt′=1n-1∑i=1nx′t(i)-μxt′⊗fth′′t-1(i),x′′t(i)-μyt′,Pytyt′=1n-1∑i=1nfth′′t-1(i),x′t(i)-μyt′⊗fth′′t-1(i),x′t(i)-μyt′,
in which n is the number of ensemble members, the superscript (i) denotes
the ith member, and a⊗b is the tensor product
of vectors a and b. Upon initialization, the original
ensemble members x0(i) are drawn from the unconditioned
multi-Gaussian distribution of x, whereas the updating of the
individual ensemble members follows the procedure outlined above:
x′′t(i)=x′t(i)+bPxtyt′Pytyt′+R-1yt+ε(i)-fth′′t-1(i),x′t(i),
in which ε(i) is a vector of random observation
noise drawn from a multi-Gaussian distribution with zero mean and covariance
matrix R. The factor b is the so-called damping parameter
e.g.,, which serves to slow down the update
of states and parameters. It is an ad hoc tuning parameter that is
primarily required for small ensemble sizes; few guidelines exist on how to
select it. In this work, the damping is set to 0.6 for the updates of the
head values and 0.05 for the parameter update, though, since the ensemble size
is large and there are many temporal observations (see below), the choice is
not crucial in any sense. For a more in-depth description of the filter
algorithm, the interested reader can consult or
for general filter details or and
for in-depth details on the actual implementation used
in this study.
It should be noted that the ensemble Kalman filter still relies on the same
assumptions as the original Kalman filter. Notably, the combined vector of
states, parameters, and observations is assumed to be a multi-Gaussian random
variable, which means that xt is multi-Gaussian, the model
ft depends linearly on xt, and the measurement error is
multi-Gaussian, too. These conditions are not strictly met, so the EnKF
solution is only a linearized estimate. However, in contrast to the extended
Kalman filter, in EnKF the linearization is performed by considering an
entire ensemble rather than by taking derivatives at a single point
e.g.,. The large ensemble sizes used in this work as
well as the repeated application over many time steps alleviates the effects
of nonlinearity to some extent, by allowing a generous use of the dampening
factor. Hence the filter is slowed down and the possible erroneous updates
resulting from the linearization have a less strong effect on the update.
Further, the model considered is only weakly nonlinear, so in total the
effects of the linearlizations are likely small compared to other sources of
errors (e.g., prior uncertainties, as discussed later). For a detailed
discussion of the linearization operated by the ensemble Kalman filter
applied to groundwater models, see .
* Pumps are numbered as in Fig. ; z0
and poro are the homogeneous bedrock elevation and porosity, respectively.
An important constraint is that the scheme, like any other Bayesian method,
depends on the choice of the unconditional mean and covariance structure of
the parameters Φ. It is important to keep in mind that our
application (estimating spatial patterns of both hydraulic conductivity and
recharge from hydraulic-head data) is based on, at least partially, ambiguous
data, as outlined in Sect. . Bayesian parameter-estimation
schemes are well posed even in the presence of non-informative or ambiguous
data due to the prior information: in the case of non-informative data, the
likelihood of the data shows no dependence on the parameters, and the
posterior falls back to the prior. Thus, while the updating procedure leads
to modifications of the parameters, the original prior knowledge carries
over. Spatial patterns that are in contradiction to the prior knowledge
cannot be recovered by the scheme. This would of course be different if the
observations were in strong contradiction to the prior. If so, we could see a
departure from the prior, both in terms of absolute values and in
terms of structure. This point will be discussed in more detail in
Sect. . In our application, however, Φ contains
parameters describing both aquifer properties and boundary conditions, and, as
we have shown above, the effects of these two types of parameters on the
measured heads can be similar. Hence, the data can be non-unique with respect
to the parameters, and the prior knowledge may determine which patterns of
conductivity and which patterns of recharge can be jointly inferred by the
scheme. If the prior knowledge is erroneous, the estimated fields may also be erroneous.
Setup of the synthetic test case used for the parameter field
estimations.
Setup of a synthetic experiment
For testing the possibilities and limitations in jointly estimating
conductivity and recharge, we have set up a synthetic 2-D example of
transient flow in an unconfined aquifer. The model setup is shown in
Fig. and consists of spatially variable recharge with a
temporal seasonal trend, spatially variable conductivity, a temporally
variable southern boundary corresponding to a river, and five pumping
wells. The actual recharge is calculated by multiplying the trend parameter
with the shown recharge field. More technical details about the setup are
found in Table . Observations of groundwater heads are
taken daily at 45 observation wells spread throughout the domain during a
1-year simulation and assuming an observation error of 1 cm. The recharge
and log-conductivity fields are both sampled as random fields with
anisotropic, exponential covariance functions and strong rotation of the
principal directions of anisotropy (Table ). It should be
noted that in the current example the reference conductivity and the
reference recharge fields are generated as fields that are uncorrelated to
each other. This could, for example, represent a scenario in which the
recharge is primarily controlled by variable land use and vegetation while
the conductivity is a material property that varies spatially but is constant
over time.
Parameters and properties used for the generation of the synthetic
conductivity and recharge fields*.
*μ is the mean; σ is the variance; α is the
rotation angle; and lx and ly are the correlation lengths in x and
y direction, respectively.
For the estimation of the recharge and conductivity fields, we apply the
ensemble Kalman filter using an ensemble of 2000 members. As this work aims
at exploring which prior knowledge is required for the estimation process,
three different cases of prior knowledge are considered. In the first, the
initial ensemble members are drawn from the same (hence correct) distribution
as the reference (true) field. The second case is identical to the first
apart from the rotation angle of the anisotropy being randomly chosen for
each ensemble member. In the third case, the rotation angle is fixed but
wrong. Here, the recharge is sampled using the rotation angle and correlation
lengths of the true conductivity field and vise versa, creating a rather
problematic initial ensemble. A plot of the three correlation structures can
be found in the bottom of Fig. in Sect.
where the three initial ensembles are called the “good”, “random”, and
“wrong” ones. Please note that the correlation plot for the random initial
is only meant as an illustration of the fact that each ensemble has a unique
rotation angle and does not show the actual angles considered.
Normalized root mean square error for the prediction
period*.
Good Random Wrong OLETE-1TE-2OLETE-1TE-2OLETE-1TE-2R1.31.20.0011.61.30.0011.91.80.002K4.70.90.0087.80.90.00813.63.10.029R and K4.60.80.0097.61.10.01112.22.40.019
* According to Eq. () for three setups of
prior knowledge (good, random, wrong) to estimate recharge alone (R),
to estimate conductivity alone (K), and to jointly estimate conductivity and recharge
(R and K). OLE is observation location error, and TE is total error.
Estimation of stand-alone recharge. Upper panels show the final ensemble
mean after all assimilation steps, and lower plots the covariance function used to generate the initial
ensemble. Please note that the random covariance functions imply drawing the
rotation angle from a uniform distribution between 0 and 2π, whereas only
a few illustrative examples are shown.
The goodness of the resulting fields are judged in two ways. First, the
ensemble mean of the fields are visually compared to the reference fields and
subjectively judged to be similar or not. Second, the normalized root mean
square error of the simulated heads in the 45 observation wells is computed by:
NRMSE=1ntnnodes∑t=t1t2∑i=1nnodeshtrue(i,t)-h(i,t)‾2σh2(i,t)
where nt is the number of temporal observations between t1
and t2, nnodes the number of nodes considered,
h(i,t)‾ is the ensemble mean head observation at position i and
time t, htrue is the corresponding true value, and
σh is a standard deviation used to normalize the error. As
this is a virtual experiment, we can calculate the NRMSE both using the
observation locations as nodes and using all nodes in the model. The former
corresponds to what can be done with real experimental data and is denoted
OLE (observation location error), while the latter only works for a virtual
experiment and is denoted TE (total error). For OLE, the normalizing standard
deviation σh(i,t) is the measurement uncertainty of
hydraulic-head observations, hence here a fixed value decided prior to the
EnKF simulations, while for TE this corresponds to the conditional ensemble
standard deviation, variable in both space and time. Due to the choice of the
normalizing standard deviation, the total error does not give a direct
indication of the goodness of the ensemble mean solution but rather indicates
how well the predicted heads fit the ensemble standard deviation. Therefore,
a second version of the total error is also considered, where the normalizing
standard deviation is replaced by normalizing the error with the mean of the
two head values (i.e., σh(i,t)= 0.5 ⋅ (htrue(i, t) +h(i,t)‾)). The
two versions of the total error are abbreviated as TE-1 and TE-2.
The use of NRMSE gives a quantitative metric
of judging the actual performance of the estimated model. We assimilate head
observations from day 50 to day 300, while the remaining 65 days of the
1-year data are used to test the model's predictive capabilities. This
results in an assimilation error for judging how well the assimilation went
and a prediction error for judging the model's predictive powers. It should be
noted that, to properly asses the predictive power of the model in a scenario
different to the one used for the assimilation, one of the four wells shown
in Fig. only starts pumping at day 301.
We have combined the three different prior distributions with three different
estimation problems, namely the estimation of (a) recharge alone,
(b) hydraulic conductivity alone, and (c) recharge and hydraulic conductivity
together, leading to a total of nine different scenarios. In the stand-alone
scenarios, all other parameters and settings are assumed known and are set to
their true values. As can be seen from Fig. , the
recharge shows not only a strong spatial pattern but also a temporal trend.
In the estimations shown below, this temporal trend is assumed known.
Normalized root mean square error for the assimilation period.
Good Random Wrong OLETE-1TE-2OLETE-1TE-2OLETE-1TE-2R0.30.90.0020.41.00.0020.51.80.004K1.20.90.0070.90.80.0073.82.10.019R and K1.90.80.0082.70.90.0093.71.70.014
Estimation of stand-alone conductivity. Upper panels show the final
ensemble mean after all assimilation steps, and lower plots the covariance function used to generate the
initial ensemble. Please note that only a few illustrative examples of the
random orientation angle of anisotropy are shown.
Results and discussionStand-alone estimation of recharge or conductivity
The simplest of the estimation problems presented in this study is the
stand-alone estimation of recharge, since the hydraulic heads depend linearly
on recharge. This is reflected in Fig. , showing the ensemble
mean of the estimated recharge fields. As expected, the best results are
achieved with the best initial estimate (second column). However, the
estimates using the covariance functions with the random and wrong
orientations of anisotropy are also acceptable, with patterns similar to those of
the true recharge field. Table quantitatively confirms
these qualitative findings by low values of the normalized root mean square
error of predicted heads. From the last column in Fig. we see
that, although the filter manages to produce a reasonable ensemble mean of
the recharge field, the similarity with the covariance function used to
create the initial ensemble is still very prominent. This is especially so if
one starts considering individual ensemble members (not shown), and it
demonstrates how sensitive the EnKF method is to the initial guess, even in
this linear problem.
It is important to keep in mind that the ensemble size is large, so the
plots of the ensemble means shown in Fig. are smoothed. It
is not expected that the smooth ensemble estimate exhibits the same extreme
values as those seen in the true parameter distribution, whereas individual
ensemble members should show the same variability as the (unknown) reference field.
Joint estimation of recharge (top row panels) and conductivity
(middle row panels). Shown is the final ensemble mean after all assimilation steps and the covariance functions
used to generate the initial ensembles (bottom row panels).
In comparison to estimating the recharge fields, the estimation of
conductivity fields alone is more complicated. Here, the nonlinearities of
Eq. () affect the estimation. More importantly, the
orientation of the anisotropy of heterogeneity plays a vital role in the
behavior of groundwater flow. This is also seen in the final estimates of the
conductivity fields, shown in Fig. , where the only reasonable
result is achieved if the right pattern is assumed in the prior knowledge
(second column) or if the prior pattern is random (third column). The
reasonable performance of the prior distribution with diffuse knowledge about
the anisotropy orientation may be explained by the large initial ensemble
containing some members with reasonable patterns and decent behavior. In the
case that the orientation of anisotropy is assumed erroneously in the prior
knowledge (fourth column), the filter completely fails to produce any result
similar to the truth. This finding does not depend on the ensemble size. The
prediction errors listed in Table clearly confirm the
visual impression. The result shows similarities with the results of
and , who both managed to correct a
wrong prior mean and variance of conductivity fields (here corresponding to
the good and the random priors), but not the correlation lengths (here
corresponding to having a wrong prior).
The prediction errors, listed in Table , emphasize that
estimating recharge leads to smaller errors in predicting heads than the
estimation of the hydraulic-conductivity field. This could indicate that
improvements of the estimated conductivities are more important for lowering
the prediction error, which would follow the findings of
. As pointed out above, the higher errors when
estimating conductivities are likely related to the head value in a cell
depending not only on the conductivity of that cell but also on the macroscopic
anisotropy of hydraulic conductivity in the entire aquifer.
Joint estimation of recharge and conductivity
As derived in Sect. , joint estimation of recharge and
conductivity fields is impossible without prior knowledge about either of the
two quantities. In Bayesian inversion methods, however, prior knowledge is
assumed anyway. In the EnKF method, the prior information is conveyed by the
initial ensemble drawn from the prior distribution. By this, the jointly
estimated recharge and conductivity fields are unique and reproducible in a
statistical sense. The remaining question is whether these estimates also
resemble the true fields and whether they are good for prediction purposes.
Figure shows the results of the joint estimation using the
three different initial ensembles and Fig. shows the
corresponding spatial distributions of the estimation variance. If the
initial ensemble is good – that is, the reference fields are drawn from the
same statistical distribution as the initial ensemble – it is possible to
estimate both conductivity and recharge with reasonable precision, given the
number and accuracy of observations (second column). When the initial
ensemble is poor, however, the result is rather poor for the recharge and
more blurry for the conductivity (third column), or we infer fields that look
good but are wrong (last column).
Joint estimation of recharge (top row panels) and conductivity
(bottom row panels). Shown is the final ensemble variance after all
assimilation steps.
As shown theoretically in Sect. , it is always possible to
compensate a missing or wrong conductivity field with a wrong recharge field.
An effect of this compensation is also clearly seen in the last column of
Fig. : even after 250 days of data assimilation, the estimated
recharge shows remarkable similarity with the reference conductivity field.
The long assimilation time is important, since, if there had been no
compensation, the estimated fields would not retain their erroneous
structures for so many filter updates. This shows that the issue of trading
one quantity for the other is not only a theoretical issue but also relevant
in practice. It should be noted here that the cause of the original poor
estimations is not the compensation mechanism described in this paper but
the false prior sampling. However, the compensation mechanism sustains the
poor estimates when the observations are, as in this work, non-unique.
The lacking ability of the random and wrong initial ensemble estimates with
respect to predicting heads under conditions not encountered in the
calibration period is documented as OLE and TE-2 in Table ,
where the prediction errors caused by the poorly estimated fields are often
an order of magnitude larger then those resulting from a really good
estimation. It is interesting to note that the error at the observation
locations obtained throughout the assimilation, shown in
Table , are not a good indicator for the predictive
capabilities of the various models, as quantified by the prediction errors
listed in Table . Although there are differences in the
assimilation error, both within and between the different estimation setups,
it would be difficult to predict any model performance from these errors.
That the joint estimation is performing much better with the good prior
compared to the poorer ones is only obvious if the full table is available.
When comparing the errors in Tables and ,
it is easily detectable that the errors, especially at
the observation locations, are much smaller during the assimilation than
during the prediction. This marks the fact that we have not estimated exactly
the true fields, and as soon as we stop assimilating head data into the
system, the models may start to deviate. Depending on the goodness of the
estimated fields, the models will deviate more or less, but all estimation
setups show an increase in error at the observation location during
prediction. It should also be noted that, during a resimulation using a
different initialization for the randomness in the EnKF, the values of the
observation location errors during prediction were notably different for most
cases in which conductivity fields were estimated. This indicates that the
estimated conductivity fields and, hence, the head predictions are
influenced by the randomness in the EnKF. Therefore, the exact error values
presented here should be used with care. For the comparative conclusions and
discussions in this paper, however, the EnKF simulations are considered
stable enough, as the errors are always on the same order and no differences
are directly visible on the final mean estimated fields.
Interesting to note in Table is that the highest errors,
both at the observation location and for the total errors, are found when
only conductivity fields are estimated using the wrong prior. Hence, using
the wrong priors while jointly estimating recharge and conductivity gives a
better prediction of the heads. As is obvious from Figs.
and , both estimated conductivity fields are similarly poor;
hence, this quantitatively confirms the existence of the aliasing
problem: jointly wrong estimated fields can compensate each other to simulate
more correct head fields.
The values of the total normalized errors (TE-1), also listed in
Table , must be interpreted in light of the standard
deviation of estimation used for normalization. A value of unity would
indicate that the mismatch between predicted and true hydraulic heads follows
exactly the predicted uncertainty of the hydraulic-head estimation. A value
significantly smaller than unity indicates that the conditional ensemble is
too wide, whereas a value significantly larger than unity points to an
erroneous estimate with erroneously small error bounds. As can be seen from
Table , both the true and random priors lead to a
combination of head mismatch and associated uncertainty close to unity,
slightly overestimating the uncertainty, whereas the wrong prior leads not only
to wrong patterns of the hydraulic conductivity fields but also to
erroneous head predictions with too-small prediction uncertainty. As can also
be seen in Fig. , the conductivity field estimation with
the wrong prior shows a much smaller variance than using the other priors.
This likely leads to an underprediction of the variance in the head fields
and, hence, the high value observed for TE-1 during the prediction (Table ).
The issue of low errors in the assimilation period is further illustrated
with an example of two observations wells in Fig. , from
which it is clearly seen that all approaches show a good fit during
assimilation but that the heads using the wrong prior deviate in the prediction.
From a practical point of view this highlights the importance of having
relevant validation data to test the predictive power of a model when the
parameters are inferred using sequential data assimilation.
Two head observations plotted over time for the joint estimation of
recharge and conductivity. Shown is the ensemble mean. Assimilation is
performed from day 50 to day 300, while the remaining days are considered for
prediction.
Like in the scenarios in which only recharge or only conductivity were
estimated, the mean joint estimate lacks the extreme values of the reference
fields. As discussed above, such behavior is expected for the smooth best
estimate even in cases where the scheme works perfectly fine. Individual
ensemble members show significantly stronger variability, as can be seen also
from the maps of the estimation variance in Fig. . We
consider the results from the good initial ensemble as good, since they
capture the main patterns of the parameter fields well and have, overall,
reasonable absolute parameter values. For purposes of transport
predictions, we would recommend using the entire ensemble rather than the
ensemble mean. In the case of the estimates using the wrong prior knowledge, in
particular where the orientation of anisotropy is chosen randomly, the
fluctuations cannot be aligned well in the right direction, and averaging
over features oriented in all directions leads to particularly smooth
estimates of the mean.
Conclusions
In the present study we have shown that it is possible to jointly estimate
reasonable fields of hydraulic conductivity (or its logarithm) and recharge
as spatially fluctuating fields from pure head observations provided that the
statistics of the true fields are fairly well understood. Starting with wrong
assumptions about conductivity and recharge patterns can lead to aliasing, in
which undetected features of hydraulic conductivity are traded for
erroneous fluctuations in recharge.
In real-case applications, the prerequisite of a good prior can pose a severe
problem because the true spatial patterns may be widely unknown. From a more
technical point of view it may be noteworthy that a rather common way of
setting up a synthetic groundwater-EnKF test is to generate a large ensemble
of realizations and use one of them as the truth and the rest as the initial
ensemble. By this it is guaranteed that the statistics of the initial
ensemble are perfect, and, as shown here, a good result can be expected.
Unfortunately, in real-world applications the geostatistics of
(log-)hydraulic conductivity are typically quite uncertain, so the good
performance of a scheme, involving both the measurement strategy and the
inverse method, in an overly optimistic test case regarding prior knowledge
may not be transferable. We thus highly recommend designing realistic test
cases that include potential bias in prior knowledge.
In the present work, we only used head data for data assimilation and
parameter estimation. As shown in Sect. , however, conductivity
and recharge are not simultaneously identifiable if considered as parameters
that can vary unrestricted in space and time: even if hydraulic heads were
observed everywhere at all times, exactly the same head field could be achieved
with different combinations of conductivity and recharge fields. Therefore,
the joint estimation is impossible without prior information about the
parameter fields, implying that wrong prior information cannot be corrected
by the head data. Other types of observations could, of course, also be
considered. Ideally we would have (plenty of) observations of subsurface
fluxes or of conductivity. In this case, the total data set would become
highly informative and the prior would be significantly less important.
However, this is not realistic for applications in subsurface hydrology.
Fluxes cannot be measured as such, and conductivity measurements are, if
existing and trusted, very local. Further observations could be obtained from
tracer tests, which are time consuming, or age tracer data, which may be costly
and require very long simulation times. Head observations are, in this
respect, common and trustworthy measurement. Hence, our example can be
considered rather realistic for a real-world scenario of estimating aquifer parameters.
In real-world applications, vague guesses of the hydraulic conductivity
distribution may exist from drilling logs, slug tests, and pumping tests
e.g.,. All of these tests are
independent of recharge so that making use of this information may alleviate
the problem of non-uniqueness outlined in this paper to some extent. Vague
guesses of K can find their way into parameter estimation either by means of
an improved prior of K or by explicitly accounting for the additional
measurement types in the EnKF procedure, including the full observation
operator. For recharge, the patterns should in principle reflect land use and
soil types, which are accessible information. Further, spatially variable
recharge may also be constrained by the use of remote-sensing information
e.g.,. These type of data could
be either used as direct observations in the assimilation (if we trust them)
or considered as prior information and used to condition the initial ensemble
. The latter could also be seen as a way of
discarding initial samples that contain unfeasible conductivity–recharge
combinations. This would create a much more appropriate initial ensemble.
Hence, as shown in this work, the filter would have an increased chance of
successfully estimating the parameters when the prior is good. The idea of
improving the initial ensemble can also be related to the popular method of
multiple-point geostatistics. Here, training images which should
represent relevant spatial correlation patterns have been used to condition
conductivity fields (see ). The combination of
assimilating head data and the use of training images to condition the
ensembles has also been tested, with promising results .
The combination of these approaches could prove a possible way to achieve a
more correct prior sample and, hence, to improve the performance of the joint
estimation of conductivity and recharge fields by lowering the risk of
conductivity-to-recharge aliasing due to wrong prior knowledge.
In the presented work, we consider a rather standard formulation of the
ensemble Kalman filter without iterations, smoothing, and many ad hoc
features. For the joint estimation of recharge and conductivity, an iterative
approach such as the dual-state filter , locally
iterative filters , or fully iterative filters
or smoothers could be considered. The advantage
would be a separation between the update of the recharge and the update of
the conductivity. This could, potentially, reduce the risk for
conductivity-to-recharge aliasing. The iterative approaches have been
reported to have improved performance and physical consistency, but they tend to
come with longer simulation times.
Acknowledgements
Financial support from the Deutsche Forschungsgemeinschaft (DFG) under CI 26/13-1
in the framework of the research unit FOR 2131 “Data Assimilation for
Improved Characterization of Fluxes across Compartmental Interfaces” is
gratefully acknowledged.
Edited by: M. Giudici
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