Estimation of relative recharge sequence to groundwater with minimum entropy deconvolution

Estimation of relative recharge sequence to groundwater

with minimum entropy deconvolution

Taehee Kima, Kang-Keun Leeb,*

aKorea Institute of Geosciences and Mineral Resources, South KoreabSchool of Earth & Environmental Sciences, Seoul National University, Kwanak Gu San 56-1, Seoul 151 742, South Korea

Received 30 December 2003; revised 25 November 2004; accepted 14 December 2004

Abstract

This study applies the minimum entropy deconvolution (MED) for the estimation of the sequential groundwater recharge

rate. Groundwater recharge rates have conventionally been estimated as an average value over some period of time, e.g. annual

or seasonal recharge rates. Such estimates however, are not suitable for the analysis of the dynamic sequential behavior of the

groundwater head fluctuation. If the sequential groundwater recharge rate is obtained, numerical groundwater models can be

effectively applied to evaluate the effect of any change in the groundwater system following the change in natural or artificial

components. This study successfully applies MED to estimate the sequential recharge rates. As recharge rates are obtained by

relative values, a series of timed observations are necessary. The validity of the estimated sequences of relative recharge rates

can be checked by cross-referencing. Cross-correlations between the applied recharge sequence and the estimated results are

above 0.985 in all study cases. Through the numerical test, the suitability of MED in the estimation of the recharge sequence to

groundwater is investigated.

q 2005 Elsevier B.V. All rights reserved.

Keywords: Groundwater; Recharge estimation; Sequential recharge; Minimum entropy deconvolution; Numerical test

1. Introduction

The estimation of groundwater recharge is a

classical topic in hydrogeology. A large number of

methods have been introduced for the estimation of

groundwater recharge. Scanlon et al. (2002) reviewed

various kinds of techniques for quantifying

0022-1694/$ - see front matter q 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.jhydrol.2004.12.005

* Corresponding author. Tel.: C82 2 880 8161; fax: C82 2 871

3269.

E-mail addresses: [email protected] (T. Kim), kklee@

snu.ac.kr (K.-K. Lee).

groundwater recharge. They subdivided techniques

for estimation recharge into four types, water budget,

techniques based on surface-water studies, techniques

based on unsaturated studies and techniques based on

saturated studies.

The former two categories are related to indirect

approaches to groundwater systems. And the latter

two categories are directly related to physical

groundwater systems. However, from the methodo-

logical viewpoint, it is hard to distinguish the latter

two cases from each other. Therefore, we tried to re-

categorize these methods to estimate groundwater

Journal of Hydrology 311 (2005) 8–19

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T. Kim, K.-K. Lee / Journal of Hydrology 311 (2005) 8–19 9

recharge into four types with the viewpoint of

monitoring data types and resulting types of recharge

estimate focused on whether the resulting estimate is

given by dynamic time-series or a representative

recharge value for some period of time. That is, the

re-categorization criteria are closely related to the

ranges of time period the resulting recharge estimates

can represent. The four types are: (1) the water

balance method, (2) baseflow/springflow recession

analyses, (3) analyses of chemical or physical tracers,

and (4) dynamic recharge estimation from time

series data.

The first type is related to the water balance (WB)

technique. WB is the most conventional method to

estimate the recharge. Bradbury and Rushton (1998)

estimated a runoff-recharge model using the water

balance method, Finch (1998) investigated the

sensitivity of direct groundwater recharge to land

surface parameters with a water balance model, and

Bekesi and McConchie (1999) applied the Monte

Carlo technique to the water balance model to

estimate the groundwater recharge.

The estimation of the groundwater recharge from

baseflow/springflow recession is another type of

method which has been widely employed. Many

studies related to this method have been carried out

(Rutledge and Daniel, 1994; Avery et al., 1999;

Ketchum et al., 2000). But methods related to the

baseflow/springflow recession are directly correlated

with the discharge, rather than the recharge.

The third type is the method of groundwater

recharge estimation by measuring the groundwater

age with environmental tracers or temperature

profiles (Solomon et al., 1993; Leduc et al., 1997;

Bromley et al., 1997; Taniguchi, 1993; Taniguchi

et al., 1999a,b). The applications included in the

above-mentioned three groups are mostly concerned

with the estimation of groundwater recharge as forms

of seasonal average or annual recharge rates. If there

are some studies related to the estimation of a daily

or dynamic recharge rate, those studies are somewhat

or highly related to the following fourth type.

To estimate more dynamic fluctuations of ground-

water recharge, a method for sequential estimation of

recharge is necessary. Unfortunately, it is not easy to

find studies on sequential approaches to estimate the

recharge rate to groundwater. We categorized the

fourth type as methods related to the sequential

estimation of groundwater recharge rate. Sometimes,

the methods related to the fourth group are called

‘saturated-volume fluctuation (SVF) analysis

methods’. Ketchum et al. (2000) proposed the storage

accumulation method calibrated with the spring

discharge records. Healy and Cook (2002) tried to

extend the discussion of SVF to the estimation of

specific yield in fractured media. But in these studies,

the local discharge, described with the recession of the

hydraulic head, was not considered. The local

discharge for the studied area is merely considered

as the area average by dividing the spring flow

discharge by the catchment area.

On the other hand, there are many studies on the

estimation of the source sequence with time series

data in geophysics (Broadhead, 1993; Sacchi et al.,

1994; Kaaresen and Taxt, 1998; Sibul et al., 2002).

But there have been only rare applications of

geophysical methods to the estimation of the ground-

water recharge. Even though predictive deconvolution

techniques are one of the most prevalent methods for

the estimation of source wavelets in signal processing,

they have rarely been applied to the estimation of

recharge to groundwater.

In general, the classical predictive deconvolution

technique is based on the minimum phase condition.

However, the minimum phase condition can hardly be

satisfied in nature. On the contrary, the input signal in

a natural system can be considered as a random signal.

To apply the classical predictive deconvolution, some

appropriate sequence satisfying the minimum phase

condition must be picked up. To avoid the strong

restriction of the minimum phase condition, Wiggins

(1978) proposed another deconvolution technique,

called the minimum entropy deconvolution (MED).

Wiggins (1978) insisted that the MED process seeks

the smallest number of large spikes that is consistent

with the data. In other words, MED finds the simplest

signals consistent with the observed data. Wiggins

(1978) represented the simplicity with the varimax

norm. However, the solution process with the varimax

norm is non-linear. To acquire the desired digital

filters for the estimation of source wavelets, some

iteration is needed in the MED proposed by Wiggins.

For the linearization of the MED problem, Carbrelli

(1984) suggested another criterion, the D norm,

instead of the varimax norm. Carbrelli (1984) showed

that the behavior of the D norm is consistent with

T. Kim, K.-K. Lee / Journal of Hydrology 311 (2005) 8–1910

the varimax norm. Using the D norm as a criterion, the

solution process in the MED is linear, and iteration is

not needed. Thus MED was applied and developed in

various geophysical studies (Broadhead, 1993; Sacchi

et al., 1994).

In this study, the MED with the D norm is applied

to the estimation of the sequence of relative recharge

rates. The applicability of MED to evaluation of the

recharge sequence is to be investigated with

synthesized data obtained from numerical modeling.

2. Mathematical framework

In linear systems, the total response to the external

stresses can be described with a convolution equation

hi Z r � ii Cni (1)

where hi is the total response, r is the system response,

ii represents the external stresses, ni is the noise

component, and * denotes the convolution. hi is based

on N observed data. Eq. (1) can be described in matrix

form

H Z RI C n̂ (2)

where H, R, I are the matrices of h, r, and i,

respectively. Now, another matrix, f, is assumed to

operate on both sides of Eq. (2). Eq. (2) is transformed

into

f � hi Z f � ðr � iiÞC f � ni Z yi (3)

The operation of f in Eq. (2) is the convolution.

When f operates on the above equation, f*(r*ii) must

approximate to the original stress, ii and f*n should be

eliminated. The MED is based on the assumption that

the Y obtained from applying the filter F on H has a

simple structure (Carbrelli, 1984). The problem of

composing the matrix with the greatest simplicity

from an infinite set is that of the maximization or the

minimization of a norm that can measure the degree of

simplicity. Wiggins (1978) proposed the varimax

norm as a criterion for the measurement of the degree

of simplicity. The definition of the varimax norm is

VðAÞ ZX

ViðAiÞ (4)

ViðAiÞ Z

Pj a4

ijPj a2

ij

� �2

In statistical sense, Vi(Ai) represents the variance

of the squares of the normalized rows of matrix A.

The method proposed by Wiggins (1978) composes

the applied filter with a finite length to maximize the

varimax norm in Eq. (4). However, this process

achieves a solution that maximizes the varimax norm

through a non-linear process. To eliminate nonlinear-

ity in arriving at a solution, the D norm was proposed

by Carbrelli (1984). The simplicity proposed by

Carbrelli (1984) was a function of the distance from

y to the points e1,.,e2m where

ei Z ð0;.; 1i;.; 0Þ; i Z 1;.;m

emCi Z ð0;.;K1i;.; 0Þ; i Z 1;.;m(5)

Using 2m unit vectors, a new criterion for the

simplicity of the output y was defined (Carbrelli,

1984)

D1ðyÞ Z min0%i%2m

kðy=kykÞKeik2 (6)

The term, k(y/kyk)Keik2 is rewritten as follows:

kðy=kykÞKeijj2 Z2K2ðyi=kykÞ; i Z1;.;m

kðy=kykÞKeijj2 Z2C2ðyi=kykÞ; i ZmC1;.;2m

(7)

Hence,

min0%i%2m

kðy=kykÞKeijj2 Z 2 1 K max

0%i%mðjyij=kykÞ

� �(8)

Therefore, to minimize the D1(y), the following

term must be maximized (Carbrelli, 1984)

DðyÞ Z max0%i%m

ðjyij=kykÞ (9)

Consequently, D1(y) and D(y) are equivalent to

each other. The newly defined D norm is consistent

with the varimax norm (Fig. 1).

Now, to perform the predictive deconvolution with

MED, a digital filter with finite length l maximizing

D(y) should be calculated. From the mathematical

viewpoint, the following two processes are equivalent,

as both of the two processes are linear (Carbrelli, 1984):

Fig. 1. Plot of norms D(y) and V(y) for output vector with (cos a,

sin a) filter ([a2[0,2p]) and xZ(1,1) (a), (K1,K2) and (100,1)

(Carbrelli, 1984).


(i)
to explore the D norm from vector y and to find the
appropriate filter to maximize the D norm;

(ii)
to find the appropriate filters for each component of
yi/y, compare the yi/y to each other and define the

filter of the maximum yi/y.

The equivalence between the above two statements

can be described as:

supf2R

DðyÞ Z supf2R

½maxiðjyij=jjyjjÞ�

Z maxi

supf2R

ðjyij=jjyjjÞ

" #(10)

It remains to determine the appropriate filter in order

for each of the normalized components to be

maximized. This filter can be found by differentiating

with respect to the coefficients fk of the filter and

equating to zero

v

vfk

ðjyij=kykÞ Z 0; k Z 1;.; l (11)

The differentiated components are defined as

kyk ZX

j

y2j

!1=2

(12)

yi ZX

s

fshiKsC1 (13)

and the derivative of each component is

vyi

vfkZ hiKkC1 (14)

vkyk

vfk

Z kyjjK1X

j

yj

vyj

vfk

Z kyjjK1X

j

yjhjKkC1 (15)

Therefore, the left-hand side of Eq. (11) can be

rewritten as:

v

vfk

ðyi=jjyjjÞ

Z jjyjjK2 hiKkC1jjyjjKjjyjjK1X

j

yjhjKkC1

!yi

( )

Z jjyjjK1 hiKkC1 KyijjyjjK2

!

Xj

Xs

fshjKsC1hjKkC1

!)

(16)

Equating to zero gives

yi=kyjj2X

j

fsX

s

hjKsC1hjKkC1

!Z hiKkC1 (17)

Let

rsKk ZX

s

hjKsC1hjKkC1 (18)

Fig. 2. Conceptual model used in numerical modeling to synthesize

the hydraulic data.


Eq. (18) then substitutes into Eq. (17)

yi=kyjj2X

j

fsrsKk

!Z hiKkC1 (19)

Eq. (19) can be converted into matrix form

yi=kyjj2R$f Z hi (20)

where

R Z

r0 r1 / rlK1

r1 r0 / «

« « r1

rlK1 / r1 r0

26664

37775

hi Z½hi hiK1 / hiKlC1 �T; if iKk!0; hiKlC1 Z0

R here is the autocorrelation matrix.

Using equations from (11) to (20) (Carbrelli, 1984),

some appropriate filters and the D norm for each filter

are calculated, and the maximum for each of the D

norms is identified. Through this process, the desired

filter can be obtained.

3. Numerical test for the applicability of MED

For the applicability of MED in the estimation of the

recharge sequence to groundwater, a numerical test

was performed in a two-dimensional domain. For an

unconfined aquifer, the Boussinesq equation was

adapted as the governing equation, as follows

v

vxhKx

vh

vx

� �C

v

vyhKy

vh

vy

� �Cq Z Sy

vh

vt(21)

where h denotes hydraulic head (saturation thickness),

K denotes hydraulic conductivity, Sy is specific yield, q

is recharge, x, y denote coordinates, and t is time.

Eq. (21) for an unconfined aquifer is nonlinear, and this

equation cannot be solved directly with a numerical

model. A linearized Boussinesq equation was therefore

applied for the numerical test. In the equation for an

unconfined aquifer, vZh2 is substituted

v

vxKx

vv

vx

� �C

v

vyKy

vv

vy

� �C2q Z

Syffiffiffi�v

pvv

vt(22)

3.1. Numerical tests with simple aquifer systems

Basically, a deconvolution technique can be applied to

the linear system However, the Boussinesq equation is

nonlinear, even though the equation would be linearized.

The tested domain is 200 m!200 m. The boundary

conditions and data sampling points are identified in

Fig. 2. The boundary conditions are set with specified

hydraulic heads and saturation thickness. Head variation

along with recharge sequence was obtained at three

points: 218, 221, 224 nodal points (Fig. 2). To confirm the

applicability of MED to the unconfined aquifer, two

numerical tests were performed with two simple systems:

(a) homogeneous aquifers, (b) simple heterogeneous

aquifers with 40, 50, 60, 70, 80, 90, 100 m saturation

thickness (Fig. 3). The applied sequence of recharge in

these cases is presented in Fig. 4(a) and hydraulic

parameters are assigned as below:

KZ1.0 m/day, SyZ1.0!10K1;

KZ9.0 m/day, SyZ1.0!10K1 in the dark part.

3.2. Numerical test with random field of hydraulic

conductivity

A conceptual model based on the geostatistical

approach, approximating the real aquifer, was applied.

Fig. 3. Distribution of hydraulic parameters in the vertically layered

systems: (a) case 1, (b) case 2.


The spatial distribution of hydraulic conductivity was

constructed through random generation with the turning

band method (Tompson et al., 1989). The statistical

model for random generation is:

†
assumed distribution: lognormal distribution;
†
expectation of hydraulic conductivity: 3.0 m/day;
†
assigned variance (log): 0.4;
†
correlation length (l): 10 m;
†
applied covariance structure: spherical model
CðrÞ Z 1 K3r

2lK

r3

2l3; 0%r% l

0; rR l

8<: (23)

The assigned expectation of hydraulic conductivity is

within the conventional ranges in weathered, fractured

rocks. The generated hydraulic conductivity field is

shown in Fig. 5. In this case, the value of the specific

yield is also set as 1.0!10K1. Also the initial saturation

thickness is 30 m. The sequences of hydraulic head

through the numerical simulation are in Fig. 6. For the

application of the boundary condition, the recharge

sequence was randomly generated. The sequence of the

random recharge is set out in Fig. 4(b).

3.3. Numerical test with induced noise

To make the synthesized sequences more similar to

the real monitoring data set, three sequences of white

noise were imposed on the synthesized results through

the random field of hydraulic conductivities The

standard deviations (sn) of the applied noise sequences

are 0.025, 0.05, 0.075 m. When we monitor the

hydraulic sequences with commercial pressure transdu-

cers—30 m of maximum pressure head—the maximum

error can be up to 0.03 m from the sensor itself.

Synthesized sequences with white noise are shown in

Fig. 8(a).

4. Results of the application of MED to synthesized

data sets

MED with D norm was applied to estimate the

sequence of relative recharge with synthesized data sets.

In actual application of the MED, it is the most sensitive

problem how the length of the applied filter can be

determined. The statistical concepts of MED come from

factor analysis in multivariate statistics. In factor

analysis, the number of factors is determined by

eigenvalue analysis. In general, the number of eigen-

values above 1 is applied as the number of factors in

factor analysis. In this study, to determine the length of

the filter, a householder transform was applied to

evaluate the number of eigenvalues above 1 from the

constructed autocovariance matrix, firstly. In most

cases for the models tested the filter length is above 20.

Fig. 4. Applied recharge sequence to numerical test: (a) for cases 1 and 2, (b) for random field of hydraulic conductivity.


With a very simple sequence of assumed recharge, the

above method using eigenvalues may be used to estimate

the sequence of recharge. However, when synthesized

sequences from numerical modeling were considered,

the results showed some time shift and dissipation. Then

trial and error were used in this study to determine the

length of the filter. In most cases, it was revealed that 2 is

appropriate for the length of the filter to estimate the

recharge sequence in groundwater phenomena in applied

models. All the applied lengths of filters in this study

were 2. The physical meaning of this length will be

discussed later. The results of the estimations are

presented in Figs. 7 and 8, and Tables 1–4.

Fig. 5. Spatial distribution of hydraulic conductivity

To evaluate the applicability of this method, cross-

correlations between the applied recharge sequence and

the estimated results were made, because the results of the

methods proposed in this study are presented as relative,

not absolute values. The similarity in the signal shape is

the measure of the fit in cross-correlation between the

plots of differently timed series of observations. The more

similar any two compared sequences, the closer their

cross-correlation fit is to 1. When the cross-correlation is

1, the sequences are identical in shape. In this study, all

cross-correlations that were made achieved a conformity

higher than 0.985 (simple aquifer systems with various

saturation thicknesses, Tables 1 and 2) and higher

in arithmetic scale (a), and in log scale (b).

Fig. 6. Results from numerical modeling with random hydraulic conductivity field.


than 0.992 (random field of hydraulic conductivity,

Table 3, Fig. 7). In cases with simple aquifer systems,

cross-correlations are slightly increased along with the

increase of the saturation thickness. This means that the

unconfined aquifer could approximate to the linear

system as the saturation thickness gets thicker and

thicker. And, from the results of the tested cases, it can be

concluded that MED can be applied to estimate the

recharge sequences in the unconfined case. Based on

the results from the simple aquifer cases, we applied the

MED to the results synthesized with the random field

of hydraulic conductivity without white noise and with

three cases of white noise sequences. In the case without

white noise, the estimated results show very good

correlations with the assigned sequence of recharge

(Table 3, Fig. 7). Even in the cases with white noise

sequences, the cross-correlations show values higher

than 0.83 (Table 4, Fig. 8).

5. Discussion

In this study, the full procedure using the MED to

estimate the sequence of relative recharge with the

sequence of hydraulic head was discussed with a

numerical test. In numerical tests, the applicability of

MED to the estimation of the relative recharge

sequences was investigated. For the investigation, a

linearized Boussinesq equation was employed. In

addition, the conceptual model based on the geostatis-

tics, approximating the real aquifer, was applied. The

results of these numerical tests demonstrated the

applicability of the MED. All the values of cross-

correlations in the numerical tests are above 0.993

(Table 1), and the resultant wavelets of recharge are

almost the same as the original sequence of recharge.

We also tried to justify the application of MED when

some random noise is introduced into the monitored data

Fig. 7. Estimated sequence of relative recharge with MED in numerical test.


set. Even though the cross-correlations decreased as the

standard deviations of induced white noise increase in

tested cases (Table 4), cross-correlations with daily

fluctuating data sets are higher than 0.98 (snZ0.025 m),

0.92 (snZ0.05 m), and 0.83 (snZ0.075 m). In general,

a cross-correlation higher than 0.8 means that two

sequence are highly similar in the statistical sense.

Therefore, MED can be applied under the conditions of

white noise due to the accuracy of the sensor and/or the

measuring error. In this study, we did not consider the

artificially stressed conditions such as pumping, artificial

drain, and injection because those kinds of disturbances

can be removed with a simple superposition model.

It is noticeable that the length of filter in every case

is 2. The length 2 has an important mathematical and

physical meaning in that the system response is an

exponential decay. In all results through the MED,

small negative spikes follow the estimated recharge

(a positive spike). This is due to the length of filter, 2.

The length, 2, means that the system response shows

an exponential behavior. It depends on the system

design whether the system response is exponential or

not. The apparent response with a discrete time

domain can be described in terms of an exponential

decay. The assumption of the exponential decay

introduces an error into the results of estimation with

Fig. 8. Results from numerical modeling with induced noise (a), and estimated sequence of relative recharge with MED (b) at node 221.


the MED, the negative spikes following the estimated

impulses. For simple comparison, the other sequence

of hydraulic head was synthesized through the time

domain convolution with a response function that

Table 1

Correlation between applied recharge and estimated results with the simp

Saturation

thickness

40 50 60 7

218 9.85!10K1 9.87!10K1 9.89!10K1 9

221 9.85!10K1 9.87!10K1 9.89!10K1 9

224 9.85!10K1 9.87!10K1 9.89!10K1 9

Table 2

Correlation between applied recharge and estimated results with simple h

Saturation

thickness

40 50 60 7

218 9.90!10K1 9.92!10K1 9.94!10K1 9

221 9.91!10K1 9.93!10K1 9.94!10K1 9

224 9.90!10K1 9.92!10K1 9.94!10K1 9

decays exponentially with time. The same sequence of

recharge in Fig. 4(b) was assumed to work on the

assumed exponential system. Using a synthesized head

sequence, the sequence of relative recharge was

le homogeneous aquifer

0 80 90 100

.90!10K1 9.91!10K1 9.92!10K1 9.93!10K1

.90!10K1 9.91!10K1 9.92!10K1 9.93!10K1

.90!10K1 9.91!10K1 9.92!10K1 9.93!10K1

eterogeneous aquifer

0 80 90 100

.95!10K1 9.92!10K1 9.96!10K1 9.97!10K1

.96!10K1 9.93!10K1 9.97!10K1 9.98!10K1

.95!10K1 9.96!10K1 9.96!10K1 93.97!10K1

Table 3

Correlation between applied recharge and estimated results in the

case with random field of hydraulic conductivity

Node 218 221 224

Correlation 9.92!10K1 9.92!10K1 9.92!10K1

Hydraulic

conductivity

(m/day) around

the node

4.502 1.871 7.782!10K1

2.372 1.336 2.417

3.239!10K1 1.630 1.346

2.836 2.204 3.393

Table 4

Correlation between applied recharge and estimated results with

induced noise

Standard

deviation (m)

218 221 224

2.5!10K2 9.81!10K1 9.82!10K1 9.82!10K1

5.0!10K1 9.23!10K1 9.27!10K1 9.23!10K1

7.5!10K2 8.37!10K1 8.43!10K1 8.43!10K1

Fig. 9. Comparison of results from numerical modeling a


estimated with the MED (Fig. 9(d)). Cross-correlation

conformity is over 0.994. When the negative spikes

are removed from the estimations, cross-correlation

conformity in the study increases greatly, up to 0.999.

These results indicate that two sequences, the applied

sequence of recharge in numerical modeling and the

estimated sequences of relative recharge with the

MED, are almost identical in their shape.

The MED has a significant limitation in that the result

of estimation is provided in the form of a relative

sequence, not absolute values. Therefore, for the exact

estimation of the recharge sequence, the level of

recharge should be recorded at least one time or as the

total amount of recharge over a period of time. It is

difficult to measure the recharge rate for the ground-

water, and long-term monitoring of the recharge

sequence is very expensive. From this viewpoint,

utilization of the MED can be a powerful and convenient

method in the estimation of the recharge sequence for

groundwater.

nd convolution with exponential response function.


Acknowledgement

This study was supported by grants from Sustainable

Water Resources Research Center of the Frontier

Research Program (#3-2-2 and #3-4-2).

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Estimation of relative recharge sequence to groundwater with minimum entropy deconvolution

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