WALSH TRANSFORMS FOR DEPTH ...

WALSH TRANSFORMS FOR DEPTH DETERMINATION OF A FINITE VERTICAL CYLINDER FROM ITS RESIDUAL GRAVITY ANOMALY Mansour A. Al-Garni, King Abdulaziz University, Jeddah, Saudi Arabia

Abstract Walsh functions, which are defined as a set of complete and orthogonal functions of nonsinusoidal waveform, have been used for estimating the top and the bottom depths of isolated vertical cylinder gravity data. Calculating the Walsh transforms for a vertical cylinder, Walsh power spectrum and its analysis proved to be affective in the depths determination. A fully interpretation scheme has been devised from Walsh transforms for the depths determination of a vertical cylinder model. Furthermore, the effect of profile length, sampling interval and noise analysis on the interpretive technique has been investigated. Spectral analysis, which is conventionally referred to the Fourier transform, has been widely utilized for geophysical interpretation. However, Walsh functions, which have only discrete amplitude of +1 and -1, can as well be used in geophysics. Thus, Walsh transform can be implemented for geophysical interpretation which is much faster and simpler than the spectral analysis. A scheme of the interpretation has been applied first to a synthetic data and then to the real data of the famous Humble Dome anomaly to evaluate its validity. The interpreted results of the target showed a good agreement with other published techniques of more commonly utilized methods. ____________________________________________________________________________________ Keywords: Walsh transform; Walsh spectra; sequency octave number; finite vertical cylinder

Introduction The residual anomalies are quantitatively utilized to estimate the location, depth, size, and shape of the causative source. Estimating such parameters from a geophysical signal is known as an inverse problem in the field of geophysics. It is known that the solution of such a problem is ambiguous and not complete either in theory or in practice (Shaw and Agarwal, 1990). Because of some difficulties in the regional/residual anomaly separation and the discrete measurements in the field data, uncertainties problem arises and the ambiguity can be attributed to and fallen under these two factors. Geophysical data processing and interpretation using the spectral analysis, which is conventionally attributed to Fourier analysis, have been widely utilized where causative source parameters that are related to each other in the space domain, can be shown precisely in the frequencydomain analysis (Bhattacharyya and Leu, 1977, Shaw and Agarwal, 1990). (Shaw and Agarwal, 1990), said, "generally, any transform whose kernel functions are a complete and orthogonal set can be used to decompose a function satisfying Dirichlet’s conditions into its constituents” Walsh (1923) showed that a set of complete and orthogonal functions with rectangular waveform, constitutes the kernel of Walsh transform (WT). These functions, similar to sinusoidal functions but rectangular in waveform, have amplitudes either +1 and -1 and known as Walsh functions. Since they are discontinuous in their domain, it is rather difficult to obtain the WT of a given signal in a closed form (Shaw and Agarwal, 1990). Thus, the solution can be evaluated numerically by additive manipulation of the discrete data sequences.

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Generally, the approximation of causative sources by some simple geometrical shapes or their combination are often attempted to represent the geological features. Fourier transform (FT) has been used widely in the interpretation of potential field (Dean, 1958; Odegard and Berg, 1965; Bhattacharyya and Leu, 1977; Bezvoda et al., 1992; Gusp´ıand Introcaso, 2000, Xu et al., 2003). However, some other studies have been implemented in the potential field applications by utilizing the WT (Gubbins et al., 1971; Lanning and Johnson, 1983). Furthermore, a feasibility study using WT has been carried out on some simple geometrically shaped bodies (Shaw and Agarwal, 1990), which shows that the WT can be utilized in the potential field interpretation. Walsh spectra have been used in the gravity interpretations for some simple sources (Shaw, Agarwal and Nandi, 1998). Thus, the application of Walsh transforms to interpret gravity anomalies due to sphere, horizontal and prism-shaped sources (Shaw and Agarwal, 1990) as well as the density mapping from a gravity data have been used (Keating, 1992) which encouraged me to utilize the WT to the problem of finite vertical cylinder (Figure 1). We know that the most popular geometric models that have been investigated in literatures are horizontal cylinders (Gay, 1965, Odegard and Berg, 1965; Abdelrahman, 1990, Abdelrahman et al, 1989). However, a finite vertical cylinder can be appropriately used to resemble a salt dome or a volcanic plug (Nettleton, 1942). The gravity field due to a finite vertical cylinder model is subjected to WT and then to computing Walsh power spectra. Walsh spectrum of such a model is subjected consequently to detailed and careful analyses for estimating the depths (top and bottom) of the causative target. Sequency octave analysis (SOA), which is named after Shaw and Agarwal (1990), is utilized where the whole spectrum is not involved for interpretation process. Shaw and Agarwal (1990) showed that the distribution of the Walsh power spectrum depends on the depth of causative sources. p

x

o

r1

z R

r2

h

Figure 1: Vertical cylinder extending to finite depth.

Theory The first ten Walsh functions and their Fourier corresponding sinusoid functions are shown in Figure 2 for comparison. The concept of frequency (f) as the reciprocal of time period (T) for a periodic function cannot be generalized to Walsh functions since they may not be periodic (Shaw and Agarwal, 1990). Sequency “generalized frequency” in case of Walsh functions is defined as one half of the average number of zero crossings per unit time interval while the sequency and frequency are the same for periodic functions. A difference equation employing lower order sequency functions to develop higher order ones can be used to generate a set of Walsh functions (Beauchamp, 1975, p.21). However; sequency order is preferable in the areas of signal processing and communications (Beauchamp, 1975; Shaw and Agarwal, 1990).

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Now, it is appropriate to introduce Walsh functions which can be derived through difference equation as (Beauchamp, 1975; Shaw and Agarwal, 1990): WAL(2 s q, t ) ( 1) [ s / 2 ]q [ WAL(s,2t ) (-1) sq WAL(s ,2(t - 1/2))]

for 0 t 1 0, 0 t 1,

(1)

with WAL(0,t) = 1, 0 t 1 , where q = 0 or 1 and s = 0, 1, 2, 3, ……represents the sequency order of the Walsh functions where [s/2] means the largest integer smaller or equal to s/2 (Beauchamp, 1975; Shaw and Agarwal, 1990). Walsh functions of even numbered are symmetric and those of odd numbered are asymmetric with respect to the midpoint of the interval (0, 1), which is shown in Figure 2. This is analogous to the relationship of cosine and sine functions which was introduced by Harmuth (1972) as:

Figure 2: The first ten sequency Walsh functions with their equivalent Fourier harmonics (After Ahmed and Rao, 1975).

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WAL(2s , t) CAL( s , t) WAL(2s 1, t) SAL( s, t ) s 0, 1, ..., N/2

(2)

A function f(t) that is integrable in the Lebsque sense, may be presented by an infinite series of Walsh functions over the interval [0, 1] (Beauchamp, 1975, p.40) and the WT pair can be defined as: 

F (s )WAL (s, t)

f (t ) 

(3a)

s 0

 1

F ( s )  f ( t) Wal ( s, t) dt, 0

for s 0,1, 2, ...

(3b)

The integration introduced in equation (3b) may be replaced by summation utilizing the trapezium rule n on N (N evenly spaced=2 , n being a rositure integer) sampling points {x(i)} (Beauchamp, 1975). Therefore, the finite discrete Walsh transform (DWT) is written as: 1 X ( s)  N

N 1

x(i) WAL (s, i),

s 0, 1, 2, ..., N - 1

(4a)

s 0, 1, 2, ..., N - 1

(4b)

i 0

N 1

X (s) WAL (s , i),

x(i ) 

s 0

The Power spectrum of WT P(s) for a signal of finite data point {x(i)}can be easily obtained as (Beauchamp, 1975, p.100, Shaw and Agarwal, 1990) P( 0) X c2 ( 0), P( s ) X c2 ( s) X s2 ( s ),

s 1, 2, ..., N/2 - 1.

(5)

P( N / 2) X ( N / 2 ) 2 s

Then, it is obvious that the WT of N point signal contains (N/2) + 1 spectral points. Comparison of Walsh and Fourier transform properties The properties of Walsh and Fourier transform have been summarized by Shaw and Agarwal (1990) and Beauchamp (1975, p. 72). Fourier power spectra are invariant under cyclic (linear) shifts, whereas Walsh power spectra are invariant under dyatic, and not under cyclic, shifts of data sequences (Shaw and Agarwal, 1990). However, a small translational shift in the data with respect to a fixed origin creates practically insignificant effects on the Walsh power spectra, based on equation (5) Beauchamp (1975, p. 45, 89). Convolution, correlation, etc., cannot be implemented, as in the Fourier transform in the frequency domain, without any linear time-shift properties of Walsh transform in the sequency domain. However, convolution and correlation would have the same mathematical form in sequency domain as in the frequency domain if one assumes dyatic time delay instead of a linear one (Shaw and Agarwal, 1990). The computation of WT of N(= 2n, n is a positive integer) for the real data utilizing fast WT subroutine is obtained from N n by simple additions without involving any multiplication operation,

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whereas through the fast FT subroutine, the FT of the same signal requires n N complex multiplications and evaluation of N/2 complex exponentials. Bath (1974, p. 116) perceived that for digital filtering, the " Walsh transform is superior in having no Gibbs phenomenon and not requiring any special windowing in time-domain" comparing to FT. Beauchamp (1975, p. 173) and Shaw and Agarwal (1990) showed that " The results of WT are obtained in one-eighth of the computational time of the FT calculations and give almost the same information" for a matched filtering example.

Formulation of The Problem Nettleton (1942) showed that the gravity field at an observation point P caused by a finite vertical cylinder, given in Figure 1, is expressed as the following relation:  1  1 g ( x) A 2  2 2 0. 5 2 0. 5  ( x h )  (x z )

(6)

where A R 2 G, z is the depth to the top of the body, h is the depth to the body bottom, R is the radius of the cylinder, ρis the density contrast, and G is the gravity constant. The main objective of this paper is utilizing the Walsh transform to estimate the depths (top and bottom) of a finite vertical cylinder. First, the Walsh power spectra of gravity fields are computed over a finite vertical cylinder with known body parameters and then are developed suitable interpretation schemes using some pattern(s) of the spectra, which constitutes the forward problem. Second, the schemes have been applied to the real data, which were taken from the published literatures. Sequency Octave Analysis Walsh power spectrum of a vertical gravity anomaly over a finite vertical cylinder is shown in Figure 3. We observed that the maximum Walsh power spectra of gravity field over a finite vertical cylinder exists at the same sequency number. The parameters of causative body cannot be interpreted by considering the whole power spectrum as a result of the randomness behavior of the spectrum (Shaw and Agarwal, 1990). 0

-2

log10P

-4

-6

-8

-10 0

20

40

60 k

80

100

120

Figure 3: Walsh power spectrum of isolated anomaly of vertical gravity filed due to a finite vertical cylinder of radius =1 unit, z =5 units, h=10 units and density contrast = 1 unit.

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It can be observed that the spectral peaks are located at a certain sequences Q j(=2j-1, j = 0, 1, 2, ….., n-1), which are termed after Shaw and Agarwal (1990) as "sequency octave numbers", when peak spectral power from each group is considered. Figure 4 shows the power distribution of the spectral points Q j and their relation to the depth of the causative sources. The present analysis employs spectral j-1 power separated repeatedly by one octave as 2 points on the sequency axis. It shows also the sequency octave power spectra of gravity fields over a finite vertical cylinder. The sequency octave power spectra exhibits more flatness of the octave spectra as the depth decreased, which may be attributed to the performance of SOA as a simple averaging with different window lengths of integral powers of 2. Thus, SOA depends on the horizontal gradient of the signal. 1

0.1

0.2

0.01

0.3 0.4 0.5 0.9

0.7

0.001 0

1

2

3

4

5

6

7

Sequency Octave Number

Figure 4: Sequency octave power spectra of gravity anomaly due to a finite vertical cylinder for indicated depths ratio (z/h).

Interpretation Scheme The observed signal is Walsh transformed and analyzed using SOA where the pattern of using SOA is correlated with the body parameter of the source. The sequency octave power spectrum represents a dependence on the data length. Thus, normalizing the sequency with profile length should be applied for depth estimation, which is fixed where the domain of normalized octave sequency ( Q0 Q j / N ) lies between 0 and 1/2.

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Figure 5 shows the plots of spectral value P ( P / Pmax ) Q0 versus Q0 for the causative target of different depths. It exhibits peaks at certain normalized octave sequencies Qmax . The value of Qmax is govern by the depth to the finite vertical cylinder. Therefore, the empirical equations to estimate the top (z) and the bottom (h) of vertical cylinder, respectively, as follow: Q max z 0.2 x

(7a)

z Q max  0.02 x  Q max z 0.02 h x h

(7b)

where x is the distance of sampling interval. Figure 6 shows the interpretation scheme based on the empirical equations for analyzing field data. To evaluate the possible application of any transformed domain such as this, I have examined carefully the effects of profile length and sampling interval by numerical computation, which shows the effects on the Walsh power spectrum by numerical computation.

0.04

0.03

0.2

0.02 0.3 0.4 0.5

0.01

0.7 0.9

0 0

0.1

0.2

0.3

0.4

0.5

Figure 5: Plot of P Q 0 versus normalized octave sequency Q0 for a finite vertical cylinder body of gravity field showing the dependence of the peak position Q max on depth ratio (z/h) as indicated on the curves.

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Start Digitize the field anomaly data where the maximum is located at the center of the profile

Apply Walsh transform and generate the power spectra.

Perform SOA and calculate P Q 0

Find the normalized octave sequency,Q max . This can be found using an interpolation scheme.

Using equation (7) to calculate the top and bottom depths of the causative body.

End Figure 6: Flow chart of a generalized approach for depth estimation.

Effect of Profile Length and Sampling Interval Table 1 shows the computed values of the product of normalized Walsh power spectra P and normalized octave sequency Q0 with octave number j for profile length varying from 512 to 64 data points considering finite vertical cylinder causative source at R= 1 unit, z = 5 units and h=10 units. The maximum amplitude Amax of P Q0 and its corresponding normalized octave sequency Qmax are also shown in Table 1. It is obvious from the table that a profile length of 10 to 15 times the depth value is adequate to approximate the depth by the use of SOA of the Walsh power spectra. Table 2 represents the computation for the same model for a fixed profile length of 512 units with sampling interval equals to 1, 2 and 4 units. It is very clear that for different sampling intervals, the value of Qmax changes and the correct values of approximated depths are obtained from equations (7a and 7b) in all three cases.

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Table 1: Effect of profile length on sequency octave analysis for a vertical cylinder at z=5 units and h=10 units. Sequency P Q 0 with unit sampling interval and profile length of Octave Number 512 256 128 64 0 0.000000 0.000000 0.000000 0.000000 1 0.001973 0.003894 0.007387 0.012218 2 0.005825 0.010978 0.017686 0.018798 3 0.012767 0.020419 0.021122 0.011776 4 0.021800 0.022376 0.012143 0.005419 5 0.023037 0.012400 0.005388 0.002700 6 0.012563 0.005432 0.002650 7 0.005454 0.002600 8 0.002600 A max 0.0240 0.0235 0.022 0.019 0.041 0.041 0.040 0.039 Qmax Computed 4.878 4.878 5.000 5.128 top depth (z) Actual top depth 5.000 5.000 5.000 5.000 (z) Computed 9.999 9.999 10.000 9.999 bottom depth (h) Actual 10.000 10.000 10.000 10.000 bottom depth (h)

Table 2: Effect of sample interval on sequency octave analysis for a vertical cylinder at z=5 units and h=10 units. Sequency Octave P Q0 for a fixed profile length of 512 units with sampling Number interval of

0 1 2 3 4 5 6 7 8 Amax Q max Computed top depth (z)

1 0.60653 1.24298 2.36767 4.30447 7.39905 10.9445 11.7090 7.91216 0.024 0.041

2 0.390106 0.948763 1.874387 3.366806 5.54117 7.39602 5.912035 0.047 0.081

4 0.322175 0.782793 1.493266 2.507065 3.62237 3.561439 0.091 0.161

4.878

4.938

4.969

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Actual top depth (z) Computed bottom depth (h) Actual bottom depth (h)

5.000

5.000

5.000

9.999

9.999

10.000

10.000

10.000

10.000

Noise Analysis Random noises effect has been investigated on the interpretive technique where white Gaussian random noises have been added to the gravity anomaly (Fig. 7). It shows also the added white Gaussian noise of 50 dB, 30 dB and 20 dB to the gravity anomaly where the spectral quantity P Q0 versus Q0 is computed for each one. It is noticed that there is not much difference in depths determination from the noise free anomaly. Thus, the effect of white Gaussian random noise up to 20 dB is almost negligible on the presented technique (Table 3).

0.03 0.025 = 0 .041

Qmax

0.02

P Q0

0.015 Q

0

0.01 0.005 0 0

0.1

0.2

0.3

0.4

0.5

Q0 0.025 = 0.040

0.02

Qmax

0.015

P Q0

0.01 0.005 0 0

0.1

0.2

0.3

0.4

0.5

Q0 0.025 0.02

= Qmax

0.039

0.015

P Q0

0.01 0.005 0 0

0.1

0.2

0.3

0.4

0.5

Q0

Figure 7. Shows white Gaussian noise of 50 dB, 30 dB and 20 dB with their corresponding P Q0 versus Q0 .

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Table 3. Effect of white Gaussian random noise on the interpretive technique. Parameters Top Depth (z) in arbitrary Bottom Depth (h) in arbitrary units units Assumed values 5.000 10.000 Interpreted values free noise 4.878 9.999 Interpreted values with 50 dB 4.878 9.999 Interpreted values with 30 dB 5.000 10.000 Interpreted values with 20 dB 5.128 9.999

Field Example The presented technique has been tested on Humble Dome anomaly, Houston, USA. A Bouguer gravity profile along AA' of the gravity map of Humble Dome (Nettleton, 1976; Figure 8.17, p. 265) given in Figure 8 has been digitized at an interval of 0.178 km using bicubic spline. This digitized data have been subjected to Walsh transformation and SOA. The Walsh power spectrum of this profile is shown in Figure 9. The product of P Q 0 versus Q 0 is shown in Figure (10) where Q max is determined. Hence, the depth estimates to the top and the bottom of the causative body are of about 4.50 km and 9.98 km. The results of the presented study with those of the other workers are summarized in Table 4.

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

0 x (km)

-2

-4 -6

-8 -10 Observed gravity

-12

-14 mGal -16

Figure 8: Residual gravity anomaly profile of Humble Salt Dome, Harris County, Texas, USA (Kara and Kanli, 2005).

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0

-2

log1 0P

-4

-6

-8

-10

0

10

20

30 k

40

50

60

Figure 9: Walsh power spectrum of Humble Dome anomaly.

Figure 10: Sequency octave analysis of Humble Dome anomaly.

Table 4: Comparison between depths estimated from the present technique and from those of the other authors. Worker Method Depth Nettleton (1976) Characteristic Curves 4.97 km Mohan et al. (1986) Mellin Transform 4.96 km Shaw and Agarwal (1990) Walsh Transform 4.98 km Abdelrahman and least-square minimization 4.92 km El-Arabi (1993) Abdelrahman et al. (2001) least-square minimization 4.96 km Kara and Kanli (2005) Nomograms h= 4.58 km z=7.59 km Al-Garni Present technique h= 4.50 km z=9.98 km

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Conclusions I have shown that the Walsh transforms can be used for interpretation of gravity anomalies caused by finite vertical cylinder bodies. A methodology for interpretation to estimate depths, based on the SOA, has been developed which is independent of the data length of the field signal. Study of the effect of data interval shows stable accuracy in depths determination. It shows that the ratio of depth to sample interval assumes a value as low as two for the accuracy of this technique. The depth determinations to the top and the bottom of the finite vertical cylinder are approximated using this technique with simple practical equations. Furthermore, the presented technique is stable up to 20 dB of white Gaussian random noise. The analysis of the field data of Humble Dome anomaly demonstrated the applicability of this technique, which is simple and less computation time consuming.

Acknowledgments The author is thankful to Fadlallah Farouq for revising the manuscript and to the Department of Geophysics, Faculty of Earth Science, King Abdulaziz University, for providing necessary facilities to implement this work.

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