Journal of ELSEVIER

Journal of Hydrology 168 (1995) 73-89

Hydrology

[4]

Study of hydrographs of karstic aquifers by means of correlation and cross-spectral analysis Alberto Padilla, Antonio Pulido-Bosch* Departamento de Geodindmica, Facultad de Ciencias, Avda. Fuentenueva s/n, 18071 Granada, Spain

Received 1 January 1994; revisionaccepted 10 October 1994

Abstract

Correlation and cross-spectral analysis can be applied to the study of karstic aquifers to characterize the transformations in these systems between the input function (precipitation) and the output function (discharge). The parameters that can be deduced are the response time, the distinction between quickflow, intermediate flow and baseflow, and the mean delay. This method offers quantifiable and objective criteria for differentiation and comparisons of karstic aquifers.

1. Introduction

In karstic aquifers there are usually very few wells which permit hydrogeological observation, and, when observation is possible, its validity is limited by the great complexity and discontinuity of the medium. In great part, therefore, studies on the functioning and hydrodynamics of karstic aquifers have been based on analysing the hydrograph (depletion and/or recession; Maillet, 1905; Schoeller, 1965; Drogue, 1972; Mangin, 1977; Bonacci, 1987, 1993) or on the complete hydrograph corresponding to an identifiable rainfall event (Galabov, 1972). Most conceptual models for karstic aquifers take into account the highly transmissive conduits and discontinuities together with blocks of low transmissivity, which might account for a much greater volume of the total aquifer (Kiraly, 1975; Drogue, 1980, 1992). The karstification processes acting on the most transmissive discontinuities end in a 'hierarchization' of the flow (Mangin, 1975; Bakalowicz, 1986) in such a way that in the recharge areas many relatively small conduits would increase in size (although also decreasing in numbers) in the direction of the groundwater flow. A well * Corresponding author. 0022-1694/95/$09.50 © 1995 - ElsevierScienceB.V. All rights reserved SSDI 0022-1694(94)02648-3

74

A. Padilla, A. Pulido-Bosch / Journal of Hydrology 168 (1995) 73-89

in the discharge area has a higher probability of providing a greater yield than one in the recharge area (Pulido-Bosch and Castillo, 1984). From these conceptual models of karstic aquifers, we can deduce that after intense precipitation, the flow primarily follows the most transmissive conduits ('quickftow'), whereas in depletion the relative importance of the flow through these great conduits is less, or even negligible ('baseflow'). The quantification of the relative importance of these two types of flow provides valuable information about the karstic organization of the massif. An abrupt and pronounced response to precipitation, translated into a strong discharge over a short period of time, would imply an important, wellorganized karstic network which quickly drains the aquifer. Deconvolution techniques can be used in these calculations (Pulido-Bosch et al., 1987), although correlation and spectral analyses applied to time series of precipitation and discharge have been more frequently applied. In general, this involves daily records over four or more years, for which the hydrographs of discharge reflect numerous rainy periods and as many low-water periods. Mangin (1982a,b) has established a classification of the karstic aquifers from these techniques. Time series analyses, as developed principally by Jenkins and Watts (1968), Hannan (1970), Brillinger (1975) and Box and Jenkins (1976) have been applied in hydrology by Delleur (1971), Yevjevich (1972), Spolia and Chander (1973), Spolia et al. (1980), Ledolter (1978), Lettenmaier (1980) and others. These works have been oriented essentially towards forecasting, completion of data and estimation of parameters of stochastic models. Methods for the description and the functioning of karstic aquifers appear in Mangin (1981), Mangin and Pulido-Bosch (1983), and Mangin (1984). These works analyze primarily the univariate spectral and the crosscorrelation functions, without analyzing or going deeply into the other functions of the cross-spectral analysis. A conventional study of a time series by correlation and spectral methods of analysis uses both univariate and cross analysis. Univariate analysis characterizes the individual structure of the time series, the functions applied being autocorrelation (in the time domain) and spectral density (in the frequency domain). Cross analysis characterizes the transformation of an input function into an output function. The cross-correlation function is analysed in the time domain, and the crossamplitude, phase, coherence and gain functions are analysed in the frequency domain. The interpretation of the correlation and spectral-analysis functions is not difficult when the aquifer systems studied have simple internal structures, such as homogeneous systems of continuous porous media. In these cases, it is sufficient to follow the models and ranges of standardized validity found in the literature. However, the interpretation is complicated when dealing with heterogeneous and discontinuous aquifers, and above all where cross-spectral functions in the frequency domain must be used, as in karstic systems where the quickflow is superimposed on the baseflow. In this study, the intent is to go deeply into the interpretation of the results obtained with correlation and cross-spectral analysis applied to the time series of precipitation and flow in four karstic springs. For a better interpretation of the functions, we begin

A. Padilla, A. Pulido-Bosch / Journal of Hydrology 168 (1995) 73-89

75

by applying correlation and spectral analysis to theoretical time series obtained by deconvolution; the kernel functions (unit hydrographs) used are intended to characterize extreme types of responses in springs of karstic systems that can be found in nature. A second objective includes establishing the duration of the response flow, the mean delay and the differentiation of the quickflow, intermediate flow and baseflow in the four karstic systems studied.

2. M e t h o d s

The expressions used to obtain the coefficients for correlation and cross-spectral analysis functions follow Jenkins and Watts (1968).

2.1. Cross-correlation function Let there be two discretized chronological series: the first, xt (xl, x 2 , . . . , Xn), is the cause of the second, Y, (Yl, Yz,... ,Yn), where n is the total number of data pairs available. The cross-correlation function obtained with the two series is not symmetrical; that is, r+k ~ r-k. The expressions with a truncation point, m, for k = 0, 1, 2 , . . . , m, are: Cyy(k)

(1)

r+k = rxy(k ) = ~C2x(0)C2(0) ¥

--

Cfx(k)

(2)

r k = ryx(k) = ~ / C 2 ( 0 ) C 2 ( 0 )

where 1 n-k

Cxy(k) = n t~=l(xt - x)(y,+k --.V)

(3)

1 n-k Cyx(k) = n t~=l(y' - Y)(Xt+k -- 2)

(4)

cx(o)

=

(x, -

(5)

t=l n

Cy(O)

_ 35)2

(6)

2 and 37 are the means of the series xt and Yt, respectively. The cross-correlation function r+k (Eq. (1)) represents the impulse response of the aquifer when, as in most cases in south Europe, the rainfall series can be considered to be a white noise (Mangin and Pulido-Bosch, 1983).

76

A. Padilla, A. Pulido-Bosch / Journal of Hydrology 168 (1995) 73-89

2.2. Cross spectrum The asymmetry of the cross-correlation function makes it necessary to express the spectral-density function with a complex number:

r x y ( f ) = laxy(f)l exp [-irbxy(f)]

(7)

in which i represents x/Z-f, axy(f) and ~b~y(f) are the values of the cross amplitude and the phase functions for the frequency f , the expressions of which are:

axy(f) = V/~2y(f) + A2xy(f)

(8)

• Axy(f) ~xy(f) = a r c t a n - tI'x, ( f )

(9)

Where the co-spectrum, ~xy(f), and quadrature spectrum, Axy(f), are:

gJxy(f ) =

2

{ rxy(O) + E[rxy(k) m + ryx(k)]Dk

cos (27rfk)

}

(10)

k=l

Axy(f) =2{~__l[rXy(k)-ryx(k)]Dksin(27rfk)}

(11)

in which D k is a weighting function necessary to overcome bias in the coefficients of ~xy(f) and Axy(f). Of the many weighting functions, the one which best adapts to the analysis of the hydrological series (Mangin, 1984) is that proposed by Tukey (Jenkins and Watts, 1968), which is expressed: D k = ½[1 + cos (~rk/m)]

(12)

From the standpoint of its application to hydrological series, the cross-amplitude function, axy(f), can be associated with the duration of the impulse response function, and indicates the filtering of the periodic components of the rainfall data. This characterizes the modulating effect of the aquifer in the short, medium and long term. The phase function, ~bxy(f), shows the delay, for different frequencies, between the precipitation and the flow. Its variation range is 27r, generally between -~r and +Tr. It should be taken into account that the values of 4~xy(f) = ~r + c will give values of (~xy(f) = -Tr + c in the function; this ambiguity should be interpreted according to the tendency of the function. An excessive attenuation of the input data for the system appears in the cross-amplitude and gain functions; the mean delay, d, can be obtained by the slope (equal to 27rd) of the line of best fit to: ~ x y ( f ) = 27rdf

(13)

2.3. Coherence and gain functi'ons With the cross-amplitude function (CAF) and the simple-density spectrum, new functions can be defined, such as the coherence function, nxy(f), and the gain

A. Pudilla. A. Pulido-Bosch / Journal of Hydrology 168 (1995) 73 89

77

function, G x y ( f ) , expressed as:

CtxY(f) ~xy(f) = x/Fx(f)py(f

(14)

)

axy(f) Gxy(f ) - v/~f )

(15)

where F x ( f ) and F y ( f ) are the spectral-density functions of the series x, and y,, respectively. These can be expressed as:

Px(f) = ~1 [ 1 + ~m Okrx(k)cos (27rfk)1

(16)

x=l

where r~(k) is the function of autocorrelation of the xt series, expressed as:

rx(k) -

Cx(k)

1 n-k Cx(k) = n ~ = ( x , - ~)(xt+k - X)

(17)

(18)

The coherence function shows whether variations in the output series respond to the same type of variations in the input series, and thereby indicates the correlation between the periodic variables. The gain function expresses the amplification or attenuation of the input data, attributable to the intervening system.

3. Simulatedtheoreticalexamples With many theoretical simulations possible, we have chosen to include only three in this work, two of which are representative of extreme functioning (extremely and slightly karstified aquifers), the third being intermediate. For the input series we have used the daily rainfall recorded at the meteorological station of El Torcal from 1 September 1974 to 30 October 1981. The output series has been simulated by the following convolution function: n

Qt = Et + Z Pt_jAj j=0

(19)

where E is a random function of the same variance as P; Pt-j is the rainfall on the day t - j ; Aj are the kernel-function coefficients, for which the number and values vary in the different examples studied; in all cases the variation range is between 0 and 1. This is an error function of random characteristics. In all functions applied, the truncation point (m) has been 100.

3.1. Example 1 The results obtained in this first example are presented in Fig. 1. In this case, we

A. Padilla, A. Pulido-Bosch / Journal of Hydrology 168 (1995) 73-89

78 1 0.9

0.8

KERNEL COEFFICIENTS

0.8

Z

o

0.7

0.6

0.6 0.5

o

0.4

L) 0.2

0.3

,.2

,,,

o o

0.2 0.1 0

,-- .,- --,-- -,----, ---,----, ..... .--...---,.- , 45 40 35 30 25 20 15 10 5 0

0 -0.2

-10

0

20

30

40

50

LAGS (days)

J

PERIOD (days) 100 20 10 20 I I I

10

5 I

4 I

PERIOD (days) 2.5 I

100 20 10 ] I 1

5 I

4 I

2.5 L

0.9 m

0.8

15

0.7 >(0 z w rr uJ "70 0

5 lO ,,=c 09 o') o no

5

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.1

0.5

FREQUENCY

PERIOD (days) 100 20 10 3 1 I

5 I

4 1

0.3

0.2

0.4

0.5

FREQUENCY

PERIOD (days) 2.5 I

100 20 10 1 i i I

5 i

4 I

2.5 I

2

0.4

0.5

~0.5

_Z C9

u.I

vvv~

AMPUFICATION

1

0.1

0.2

0.3

FREQUENCY

-0.5

0.4

0.5

0

0.1

0.2

0.3

FREQUENCY

Fig. 1. Plot of the functions obtained in simulated Example 1.

A. Padilla, A. Pulido-Bosch / Journal of Hydrology 168 (1995) 73-89

79

have simulated the series with a kernel function of three coefficients having decreasing values of between 1 and 0.1 without a delay. Here the intent is to simulate an extremely karstified system which hardly modifies the input function. The rainfall does not remain in the aquifer more than 3 days and the main portion of the water exits by way of the spring during the first day. The cross-correlation function (CCF) is very similar to the impulse response with which the output series was simulated. In the CAF, we find that at virtually no frequency does the system alter the input signal, which remains almost constant; only a small increase in the signal is detected at very low frequencies. In the coherence function (COF), there is a small loss of correlation for frequencies above 0.33 (equivalent to periods of less than 3 days) which coincide with the duration of the kernel function introduced. For these same frequencies the gain function (GAF) indicates a greater attenuation of the signal than for the lesser frequencies. Despite not having introduced any delay, the phase function (PHF) shows a good alignment in the frequencies of less than 0.33. For greater frequencies, considerable distortion occurs, owing to the loss of signal in these frequencies, as can be deduced from the interpretation of the GAF. Although apparently no delay has been introduced, it occurs in reality. It is the mean delay (time from the origin to the centre of the kernel function). The mean delay, calculated by means of the expression (13) proves to be 0.25 days. As can be seen, it is possible to calculate delays smaller than the discretization interval of the series. 3.2. Example 2 In this second example, we have simulated a drainage record on the basis of a kernel function which introduces a control into the long-term rainfall. The number of coefficients is 50, decreasing from A0 = 0.4. This case would correspond to that of a very slightly karstified system, similar to a porous medium, in which there is no quickflow. The five functions obtained applying the correlation and spectral analyses are shown in Fig. 2. The CCF has a gentle slope, characteristic of a system where quickflow is absent; the mean delay of the response is hardly appreciable. The CAF and the GAF show how the system notably filters and attenuates the input signal at the high frequencies and increases at the low frequencies. The values of these functions decrease slowly between the frequencies 0.04 and 0.2, and reach practically null values for frequencies of over 0.2. This is equally observable in the COF. Variations in the precipitation between these two frequencies (corresponding to periods of 25 and 5 days) continue to have a response in the output signal, although it is very filtered and attenuated. The PHF shows an alignment only at frequencies between 0 and 0.2, above which the input signal is very attenuated, distorted and incoherent. From the analysis of the slope of the function in this interval, we deduce that there is a mean delay of 14 days. 3.3. Example 3 Lastly, we have simulated an output series with the kernel function shown in Fig. 3.

A. Padilla, A. Pulido-Bosch / Journal of Hydrology 168 (1995) 73-89

8O 1 0.9

KERNEL COEFFICIENTS

0.8

0.8

z O

0.7

Journal of Hydrology 168 (1995) 73-89

Hydrology

[4]

Study of hydrographs of karstic aquifers by means of correlation and cross-spectral analysis Alberto Padilla, Antonio Pulido-Bosch* Departamento de Geodindmica, Facultad de Ciencias, Avda. Fuentenueva s/n, 18071 Granada, Spain

Received 1 January 1994; revisionaccepted 10 October 1994

Abstract

Correlation and cross-spectral analysis can be applied to the study of karstic aquifers to characterize the transformations in these systems between the input function (precipitation) and the output function (discharge). The parameters that can be deduced are the response time, the distinction between quickflow, intermediate flow and baseflow, and the mean delay. This method offers quantifiable and objective criteria for differentiation and comparisons of karstic aquifers.

1. Introduction

In karstic aquifers there are usually very few wells which permit hydrogeological observation, and, when observation is possible, its validity is limited by the great complexity and discontinuity of the medium. In great part, therefore, studies on the functioning and hydrodynamics of karstic aquifers have been based on analysing the hydrograph (depletion and/or recession; Maillet, 1905; Schoeller, 1965; Drogue, 1972; Mangin, 1977; Bonacci, 1987, 1993) or on the complete hydrograph corresponding to an identifiable rainfall event (Galabov, 1972). Most conceptual models for karstic aquifers take into account the highly transmissive conduits and discontinuities together with blocks of low transmissivity, which might account for a much greater volume of the total aquifer (Kiraly, 1975; Drogue, 1980, 1992). The karstification processes acting on the most transmissive discontinuities end in a 'hierarchization' of the flow (Mangin, 1975; Bakalowicz, 1986) in such a way that in the recharge areas many relatively small conduits would increase in size (although also decreasing in numbers) in the direction of the groundwater flow. A well * Corresponding author. 0022-1694/95/$09.50 © 1995 - ElsevierScienceB.V. All rights reserved SSDI 0022-1694(94)02648-3

74

A. Padilla, A. Pulido-Bosch / Journal of Hydrology 168 (1995) 73-89

in the discharge area has a higher probability of providing a greater yield than one in the recharge area (Pulido-Bosch and Castillo, 1984). From these conceptual models of karstic aquifers, we can deduce that after intense precipitation, the flow primarily follows the most transmissive conduits ('quickftow'), whereas in depletion the relative importance of the flow through these great conduits is less, or even negligible ('baseflow'). The quantification of the relative importance of these two types of flow provides valuable information about the karstic organization of the massif. An abrupt and pronounced response to precipitation, translated into a strong discharge over a short period of time, would imply an important, wellorganized karstic network which quickly drains the aquifer. Deconvolution techniques can be used in these calculations (Pulido-Bosch et al., 1987), although correlation and spectral analyses applied to time series of precipitation and discharge have been more frequently applied. In general, this involves daily records over four or more years, for which the hydrographs of discharge reflect numerous rainy periods and as many low-water periods. Mangin (1982a,b) has established a classification of the karstic aquifers from these techniques. Time series analyses, as developed principally by Jenkins and Watts (1968), Hannan (1970), Brillinger (1975) and Box and Jenkins (1976) have been applied in hydrology by Delleur (1971), Yevjevich (1972), Spolia and Chander (1973), Spolia et al. (1980), Ledolter (1978), Lettenmaier (1980) and others. These works have been oriented essentially towards forecasting, completion of data and estimation of parameters of stochastic models. Methods for the description and the functioning of karstic aquifers appear in Mangin (1981), Mangin and Pulido-Bosch (1983), and Mangin (1984). These works analyze primarily the univariate spectral and the crosscorrelation functions, without analyzing or going deeply into the other functions of the cross-spectral analysis. A conventional study of a time series by correlation and spectral methods of analysis uses both univariate and cross analysis. Univariate analysis characterizes the individual structure of the time series, the functions applied being autocorrelation (in the time domain) and spectral density (in the frequency domain). Cross analysis characterizes the transformation of an input function into an output function. The cross-correlation function is analysed in the time domain, and the crossamplitude, phase, coherence and gain functions are analysed in the frequency domain. The interpretation of the correlation and spectral-analysis functions is not difficult when the aquifer systems studied have simple internal structures, such as homogeneous systems of continuous porous media. In these cases, it is sufficient to follow the models and ranges of standardized validity found in the literature. However, the interpretation is complicated when dealing with heterogeneous and discontinuous aquifers, and above all where cross-spectral functions in the frequency domain must be used, as in karstic systems where the quickflow is superimposed on the baseflow. In this study, the intent is to go deeply into the interpretation of the results obtained with correlation and cross-spectral analysis applied to the time series of precipitation and flow in four karstic springs. For a better interpretation of the functions, we begin

A. Padilla, A. Pulido-Bosch / Journal of Hydrology 168 (1995) 73-89

75

by applying correlation and spectral analysis to theoretical time series obtained by deconvolution; the kernel functions (unit hydrographs) used are intended to characterize extreme types of responses in springs of karstic systems that can be found in nature. A second objective includes establishing the duration of the response flow, the mean delay and the differentiation of the quickflow, intermediate flow and baseflow in the four karstic systems studied.

2. M e t h o d s

The expressions used to obtain the coefficients for correlation and cross-spectral analysis functions follow Jenkins and Watts (1968).

2.1. Cross-correlation function Let there be two discretized chronological series: the first, xt (xl, x 2 , . . . , Xn), is the cause of the second, Y, (Yl, Yz,... ,Yn), where n is the total number of data pairs available. The cross-correlation function obtained with the two series is not symmetrical; that is, r+k ~ r-k. The expressions with a truncation point, m, for k = 0, 1, 2 , . . . , m, are: Cyy(k)

(1)

r+k = rxy(k ) = ~C2x(0)C2(0) ¥

--

Cfx(k)

(2)

r k = ryx(k) = ~ / C 2 ( 0 ) C 2 ( 0 )

where 1 n-k

Cxy(k) = n t~=l(xt - x)(y,+k --.V)

(3)

1 n-k Cyx(k) = n t~=l(y' - Y)(Xt+k -- 2)

(4)

cx(o)

=

(x, -

(5)

t=l n

Cy(O)

_ 35)2

(6)

2 and 37 are the means of the series xt and Yt, respectively. The cross-correlation function r+k (Eq. (1)) represents the impulse response of the aquifer when, as in most cases in south Europe, the rainfall series can be considered to be a white noise (Mangin and Pulido-Bosch, 1983).

76

A. Padilla, A. Pulido-Bosch / Journal of Hydrology 168 (1995) 73-89

2.2. Cross spectrum The asymmetry of the cross-correlation function makes it necessary to express the spectral-density function with a complex number:

r x y ( f ) = laxy(f)l exp [-irbxy(f)]

(7)

in which i represents x/Z-f, axy(f) and ~b~y(f) are the values of the cross amplitude and the phase functions for the frequency f , the expressions of which are:

axy(f) = V/~2y(f) + A2xy(f)

(8)

• Axy(f) ~xy(f) = a r c t a n - tI'x, ( f )

(9)

Where the co-spectrum, ~xy(f), and quadrature spectrum, Axy(f), are:

gJxy(f ) =

2

{ rxy(O) + E[rxy(k) m + ryx(k)]Dk

cos (27rfk)

}

(10)

k=l

Axy(f) =2{~__l[rXy(k)-ryx(k)]Dksin(27rfk)}

(11)

in which D k is a weighting function necessary to overcome bias in the coefficients of ~xy(f) and Axy(f). Of the many weighting functions, the one which best adapts to the analysis of the hydrological series (Mangin, 1984) is that proposed by Tukey (Jenkins and Watts, 1968), which is expressed: D k = ½[1 + cos (~rk/m)]

(12)

From the standpoint of its application to hydrological series, the cross-amplitude function, axy(f), can be associated with the duration of the impulse response function, and indicates the filtering of the periodic components of the rainfall data. This characterizes the modulating effect of the aquifer in the short, medium and long term. The phase function, ~bxy(f), shows the delay, for different frequencies, between the precipitation and the flow. Its variation range is 27r, generally between -~r and +Tr. It should be taken into account that the values of 4~xy(f) = ~r + c will give values of (~xy(f) = -Tr + c in the function; this ambiguity should be interpreted according to the tendency of the function. An excessive attenuation of the input data for the system appears in the cross-amplitude and gain functions; the mean delay, d, can be obtained by the slope (equal to 27rd) of the line of best fit to: ~ x y ( f ) = 27rdf

(13)

2.3. Coherence and gain functi'ons With the cross-amplitude function (CAF) and the simple-density spectrum, new functions can be defined, such as the coherence function, nxy(f), and the gain

A. Pudilla. A. Pulido-Bosch / Journal of Hydrology 168 (1995) 73 89

77

function, G x y ( f ) , expressed as:

CtxY(f) ~xy(f) = x/Fx(f)py(f

(14)

)

axy(f) Gxy(f ) - v/~f )

(15)

where F x ( f ) and F y ( f ) are the spectral-density functions of the series x, and y,, respectively. These can be expressed as:

Px(f) = ~1 [ 1 + ~m Okrx(k)cos (27rfk)1

(16)

x=l

where r~(k) is the function of autocorrelation of the xt series, expressed as:

rx(k) -

Cx(k)

1 n-k Cx(k) = n ~ = ( x , - ~)(xt+k - X)

(17)

(18)

The coherence function shows whether variations in the output series respond to the same type of variations in the input series, and thereby indicates the correlation between the periodic variables. The gain function expresses the amplification or attenuation of the input data, attributable to the intervening system.

3. Simulatedtheoreticalexamples With many theoretical simulations possible, we have chosen to include only three in this work, two of which are representative of extreme functioning (extremely and slightly karstified aquifers), the third being intermediate. For the input series we have used the daily rainfall recorded at the meteorological station of El Torcal from 1 September 1974 to 30 October 1981. The output series has been simulated by the following convolution function: n

Qt = Et + Z Pt_jAj j=0

(19)

where E is a random function of the same variance as P; Pt-j is the rainfall on the day t - j ; Aj are the kernel-function coefficients, for which the number and values vary in the different examples studied; in all cases the variation range is between 0 and 1. This is an error function of random characteristics. In all functions applied, the truncation point (m) has been 100.

3.1. Example 1 The results obtained in this first example are presented in Fig. 1. In this case, we

A. Padilla, A. Pulido-Bosch / Journal of Hydrology 168 (1995) 73-89

78 1 0.9

0.8

KERNEL COEFFICIENTS

0.8

Z

o

0.7

0.6

0.6 0.5

o

0.4

L) 0.2

0.3

,.2

,,,

o o

0.2 0.1 0

,-- .,- --,-- -,----, ---,----, ..... .--...---,.- , 45 40 35 30 25 20 15 10 5 0

0 -0.2

-10

0

20

30

40

50

LAGS (days)

J

PERIOD (days) 100 20 10 20 I I I

10

5 I

4 I

PERIOD (days) 2.5 I

100 20 10 ] I 1

5 I

4 I

2.5 L

0.9 m

0.8

15

0.7 >(0 z w rr uJ "70 0

5 lO ,,=c 09 o') o no

5

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.1

0.5

FREQUENCY

PERIOD (days) 100 20 10 3 1 I

5 I

4 1

0.3

0.2

0.4

0.5

FREQUENCY

PERIOD (days) 2.5 I

100 20 10 1 i i I

5 i

4 I

2.5 I

2

0.4

0.5

~0.5

_Z C9

u.I

vvv~

AMPUFICATION

1

0.1

0.2

0.3

FREQUENCY

-0.5

0.4

0.5

0

0.1

0.2

0.3

FREQUENCY

Fig. 1. Plot of the functions obtained in simulated Example 1.

A. Padilla, A. Pulido-Bosch / Journal of Hydrology 168 (1995) 73-89

79

have simulated the series with a kernel function of three coefficients having decreasing values of between 1 and 0.1 without a delay. Here the intent is to simulate an extremely karstified system which hardly modifies the input function. The rainfall does not remain in the aquifer more than 3 days and the main portion of the water exits by way of the spring during the first day. The cross-correlation function (CCF) is very similar to the impulse response with which the output series was simulated. In the CAF, we find that at virtually no frequency does the system alter the input signal, which remains almost constant; only a small increase in the signal is detected at very low frequencies. In the coherence function (COF), there is a small loss of correlation for frequencies above 0.33 (equivalent to periods of less than 3 days) which coincide with the duration of the kernel function introduced. For these same frequencies the gain function (GAF) indicates a greater attenuation of the signal than for the lesser frequencies. Despite not having introduced any delay, the phase function (PHF) shows a good alignment in the frequencies of less than 0.33. For greater frequencies, considerable distortion occurs, owing to the loss of signal in these frequencies, as can be deduced from the interpretation of the GAF. Although apparently no delay has been introduced, it occurs in reality. It is the mean delay (time from the origin to the centre of the kernel function). The mean delay, calculated by means of the expression (13) proves to be 0.25 days. As can be seen, it is possible to calculate delays smaller than the discretization interval of the series. 3.2. Example 2 In this second example, we have simulated a drainage record on the basis of a kernel function which introduces a control into the long-term rainfall. The number of coefficients is 50, decreasing from A0 = 0.4. This case would correspond to that of a very slightly karstified system, similar to a porous medium, in which there is no quickflow. The five functions obtained applying the correlation and spectral analyses are shown in Fig. 2. The CCF has a gentle slope, characteristic of a system where quickflow is absent; the mean delay of the response is hardly appreciable. The CAF and the GAF show how the system notably filters and attenuates the input signal at the high frequencies and increases at the low frequencies. The values of these functions decrease slowly between the frequencies 0.04 and 0.2, and reach practically null values for frequencies of over 0.2. This is equally observable in the COF. Variations in the precipitation between these two frequencies (corresponding to periods of 25 and 5 days) continue to have a response in the output signal, although it is very filtered and attenuated. The PHF shows an alignment only at frequencies between 0 and 0.2, above which the input signal is very attenuated, distorted and incoherent. From the analysis of the slope of the function in this interval, we deduce that there is a mean delay of 14 days. 3.3. Example 3 Lastly, we have simulated an output series with the kernel function shown in Fig. 3.

A. Padilla, A. Pulido-Bosch / Journal of Hydrology 168 (1995) 73-89

8O 1 0.9

KERNEL COEFFICIENTS

0.8

0.8

z O

0.7