USING THE POISSON-HARDY WAVELET TO DETERMINE THE SOURCE PROPERTIES FOR THE MAGNETIC ANOMALIES OF MEKONG DELTA

¹DƯƠNG HIẾU ĐẬU, ¹TRƯƠNG THỊ BẠCH YẾN, ²ĐẶNG VĂN LIỆT

¹Cần Thơ University, Cần Thơ City,
²University of Sciences, Việt Nam National University, Hồ Chí Minh City

Abstract: Determination of source properties for the gravity and magnetic data is an important role in the inverse potential field problems. In recent years, several researchers have been successfully using the method based on the continuous wavelet transform with wavelet functions made of the first horizontal derivatives of Poisson kernel to estimate the horizontal location and the depth of the sources. In this paper, the authors construct the Poisson-Hardy wavelet to determine the position of sources and then use the relations of multi-scale continuous wavelet transform to calculate the structural index of simple homogeneous anomaly sources. The results of this estimation on the magnetic data of the Mekong Delta are compatible with the results of the other methods.

I. INTRODUCTION

In the 1980’s decade, Thompson et al. [13] provided the solution for the inverse problem of magnetic prospecting using the method called as “Euler convolution”. They also introduced a concept about the structural index N or “attenuation rate” of the simple, homogeneous magnetic sources. In the next decade, Moreau et al. [6, 7], Sailhac et al. [9, 10, 11], and Fedi et al. [4] described an interpretation technique based on continuous wavelet transform (CWT) and the Poisson wavelet families to estimate the homogeneous structure of the simple magnetic anomalies. In this item, we have proposed a suitable wavelet function named as Poisson-Hardy wavelet [1] to determine the position and the depth of the magnetic sources. In this paper, the authors also apply the real part of this wavelet function to the Moureau’s theory [6] to determine the structural index N of magnetic sources.

II. CONTINUOUS POISSON-HARDY WAVELET FUNCTION

The continuous wavelet transform of 1-D signal f(x)Î L²(R) can be given by:

(1)

Where, s, b Î R⁺ are scale and translation (shift) parameters; L²(R) is the Hilbert space of 1-D wave functions having finite energy; is the complex conjugate function of y(x), an analyzing function inside the integral (1). In particularly, CWT can operate with various complex wavelet functions, if the wavelet function looks like the same form of the original signal.

Using the method based on Multi-Edge Detection (MED) to determine the horizontal positions and depth of magnetic sources, we designed a complex wavelet function from the Poisson kernel of the upward continuation filtering (Blakely) [2]. The upward continuation filtering is given by the well-known Dirichlet integral:

(2)

Where, H(x, -h) is the field measured on a plane a distance h above the datum plane (z = 0); F(x,0) is the field measured on the datum plane, and:

(3)

is the Poisson kernel, that plays the role of smoothing function.

For the MED method, the position and the depth of the gravity/magnetic source were determined by local maxima points corresponding to the wavelet function transform y⁽¹⁾ (the first derivative of the smoothing function) or inflection points (called ‘zero crossing’ points) corresponding to the wavelet function transform y⁽²⁾ (the second derivative of the smoothing function). The y⁽¹⁾ and y⁽²⁾ are given:

(4)

(5)

Grosman et al. [3] proposed a process to compute the “zero crossing” points using the phase of a complex wavelet function called as “Hardy wavelet” form:

y⁽⁴⁾ = y⁽²⁾ + i y⁽³⁾ (6)

Where, y⁽³⁾(x) is the Hilbert transform of y⁽²⁾(x). In this paper y⁽³⁾ is given as:

(7)

Here, we put h = 1 in the formula (4), (5) and (6) to satisfy the wavelet function conditions. From equations (5) and (7), we construct a new complex wavelet function named as Poisson-Hardy wavelet function as type (6) [1].

III. DETERMINATION OF STRUCTURAL INDEX

According to the Thompson’s theory [13], the homogeneous field source f(x, y, z) can be expressed as following equation for any constant l non zero:

(8)

Where, a is a coefficient of homogeneity, that is related to the structural index of the magnetic source by the relation:

(9)

The Table 1 shows the structural index of various simple magnetic sources written by Reid [8].

According to Moreau et al. [6], we denote f(x, z = 0) as measuring data in the ground (z = 0) due to a homogeneous source located at x = 0, z = z₀ with the structural index N. When we carry out the continuous wavelet transformation on measuring data with the wavelet functions that are the horizontal derivative of q_up(x) we obtain an equation related to the wavelet coefficients at two scale levels s and s’:

Table 1. Structural index for various homogeneous magnetic sources

No	Type of sources	N
1	Sphere	3
2	Vertical cylinder	2
3	Horizontal cylinder	2
4	Dike	1
5	Contact or Fault	0

(10)

Where, b = (a - g) (11)

With g being the order of derivatives of analyzing wavelet functions, N can be calculated from (9) and (11):

N = - b - g - 1 (12)

For different positions x and x’, the relation of scale parameters s and s’ is given [6] as follows:

(13)

In this paper, we determine the structural index N of anomaly sources by the continuous wavelet transform with the wavelet function y² (g = 2), Thus we can rewrite the equation (10) as follows:

(14)

Using short notation and taking the logarithm for both sides of (14), we derive a new expression:

(15)

Where, c is the constant related to the const. in the right side of equation (14). The structural index determination will be done by the estimation for the slope of a straight line:

(16)

Where,

IV. EXPERIMENTAL MODEL

The magnetic source is a cylinder hollow iron tank with the length and radius of 120 cm and 80 cm, respectively. It was buried in a shallow ditch so that the level of the tank top is the ground surface (Fig. 1). The measurement profile is perpendicular to the strike of the tank and the length of the profile is about 30 m long, with the step size of 0.5 m. The PM-2 proton magnetometer with the accuracy up to 1.0 nT was used. The sensor was hold at the altitude of 2 m from the ground, so the position of the source is: x = 0, z = 2 m.

The total intensity magnetic anomaly of the tank is showed in Fig. 2. The determination of the horizontal position and the depth of the souce comprise two steps: a) computing the first horizontal gradients, and b) computing the wavelet transform of the first horizontal gradient with y⁽¹⁾ and y⁽⁴⁾, respectively. Fig. 3 shows the modulus of y⁽¹⁾ and phase of Poisson-Hardy wavelet y⁽⁴⁾, this figure is an useful tool to determine the position and the depth of the tank. The result (x = 0, z = 2m) shows that, it is suitable with the position and the depth of the model.

Fig. 4 shows the logarithm curve of wavelet transform y⁽²⁾(x)/s² vs. to logarithm of (s+z). In the Fig. 4, using least square method, we get the equation for a straight line: Y = - 4,7x + 13, then we estimate b » -5, so the structural index is N = 2 (Equ. 12, g = 2). It is suitable with the tank model having the cylinder shape (N = 1, Table 1).

V. INTERPRETATION OF MAGNETIC DATA FROM THE MEKONG DELTA

The total intensity of aeromagnetic map at the 1/500.000 scale (Department of Geology and Minerals of Việt Nam) has been used. In this map we slice a profile 177 km long from Ngọc Hiển (Cà Mau) to Thoại Sơn (Châu Đốc), then the data are interpolated into regular points (step size 1.0 km) by cubic spline. Using the International Geomagnetic Reference Field (IGRF) from Kyoto University, we calculate the total intensity magnetic anomalies of the profile; the results are showed in Fig. 6.

The Fig. 7 shows the modulus y⁽¹⁾ and phase y⁽⁴⁾ of the horizontal gradient of magnetic anomalies of the profile Ngọc Hiển - Thoại Sơn. There are 3 anomaly sources of this profile and the results are presented in Table 2.

Fig. 8 is the logarithm curve of wavelet transform [y⁽²⁾(x)]/s² vs. to logarithm of (s+z) of the anomaly source located at position of 137 km. Using the least square method to determine the equation of linear line Y = - 4.0 + 10, so b » - 4 (Equ. 16) so the structural index is N = 1 (Equ. (12), g -2). Consequently, the source may be a dyke (Tab. 2).

For verifying the result, we present the magnetic anomaly of the anomaly located at position of 137 km from x = 130 km to x = 146 km in Fig. 9a. We also computed the theoretical model of the magnetic anomaly of an inclined dyke 87^o from the horizontal plane with the inclination I = 4^o (the values is the average value of inclination of Mekong delta area) (Fig. 9b). The two Fig. 9a and 9b are similar, so they prove the reasonable result for the estimated structural index (Fig. 8). In addition, the angle of the theoretical inclined dike could be recognized as the inclined source located at position of 137 km.

VI. CONCLUSIONS

We used the Poisson-Hardy wavelet function came from the second derivative of the kernel of the upward continuation filtering to solve the inverse problem of potential field with the determination of the positions, depths and structural index. The results of the experimental model show that the interpretation using Poisson-Hardy gives a good result. We interpreted also the Ngọc Hiển - Thoại Sơn magnetic anomaly profile, the results show that there are two anomaly sources with the dike form and one source having the cylindric form.

Since this method can only be applied for the simple and separate sources, we could not determine the depths and the structural index of some anomaly sources near by. This technique will be improved for further research to interpret closed sources.

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