Source Edge Detection of Potential Field Data Using Wavelet Decomposition

Abstract

Edge detection of the sources of potential field anomaly is an important step in the interpretation of subsurface source geometries. The conventional methods based on calculation of horizontal or vertical derivatives identify the edges or center of sources by minima, maxima, or zero values in the transformed data. We present a wavelet source edge detector method (WSED) using wavelet multiresolution analysis to identify potential field sources boundaries. The two-dimensional wavelet decomposition is an effective method to understand the frequency components of the signal in different directions. We use a 2D-discrete wavelet transform using Haar wavelets in resolving lateral edges for source edge detection. We test the method on synthetic magnetic anomalies due to sources of complex geometries generated using prismatic sources. We investigated the robustness of the method on the magnetic data of the Bishop model and found the results useful in resolving the edges. We applied the method to gravity data of the north Delhi fold belt, India, to identify boundaries of different geological formations. Our results indicate distinct properties of the source edges in the wavelet domain, which is for the first time reported for the interpretation of the potential field anomalies.

Appendix 1: Multiresolution Analysis Using DWT

The DWT uses multiresolution analysis to represent a signal in terms of its spatial frequency components (Mallat 1989). It represents a signal using approximation $({\mathbf{V}}_{j} )$ and detail $({\mathbf{W}}_{j} )$ spaces with different levels of resolution.

A sequence $\{ V_{j} \}_{{j \in {\mathbf{z}}}}$ of the closed subspaces of Hilbert space ${\mathbf{L}}^{2} ({\mathbf{R}})$ is a multiresolution approximation if the following properties exist (Mallat 1989):

• 1. $\forall (j,n)\, \in {\mathbf{Z}}^{2} \;\,f(t) \in \,V_{j} \; \leftrightarrow \;f(t - 2^{j} n)\, \in \;V_{j}$

• 2. $\forall j\, \in \,{\mathbf{Z}}\;\;f(t)\; \in \,V_{j} \; \leftrightarrow \,f(t/2)\; \in \,V_{j + 1}$

• 3. $\forall j\, \in \,{\mathbf{Z}}\;\;V_{j + 1} \, \in \,V_{j}$

• 4. $\lim_{j \to + \infty } \;V_{j} = \cap_{j \in z} V_{j} = \{ 0\}$

• 5. $\lim_{j \to - \infty } \;V_{j} = \cup_{j \in z} V_{j} = {\mathbf{L}}^{2} ({\mathbf{R}})$

• 6. There exist $\phi (t)\, \in \,V_{0}$ such that $\phi (t - n)_{n \in z}$ is an orthonormal basis of $\,V_{0}$

For each integer j, the functions are an orthonormal basis for each $V_{j}$. The wavelet spaces $W_{j}$ are introduced from the orthogonal complements of $V_{j}$ in $V_{j - 1} \;(V_{j - 1} = V_{j} \oplus \,W_{j} ,\,\forall j \in \;{\mathbf{z}}).$ One can construct wavelet $\psi$ such that the dilated and translated family is an orthogonal basis of ${\mathbf{L}}^{2} ({\mathbf{R}})$.

In 2D wavelet transform, the scaling and wavelet function are two variable functions, denoted as $\phi (x,y)$ and $\psi (x,y)$.

The scaled and wavelet basis functions expressions are given as (Mallat 1989):

$$\phi_{{j_{0} ,p,q}} \left( {x,y} \right) = 2^{j/2} \phi \left( {2^{j} x - p,{\kern 1pt} {\kern 1pt} {\kern 1pt} 2^{j} y - q} \right),$$

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$$\psi_{j,p,q}^{i} \left( {x,y} \right) = 2^{j/2} \psi^{i} \left( {2^{j} x - p,{\kern 1pt} {\kern 1pt} {\kern 1pt} 2^{j} y - q} \right)\;\;\;\;\;i = \{ H,V,D\} ,$$

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where $\left( {x,y} \right) \in \,{\mathbf{R}}^{2}$ and $\left( {p,q} \right) \in \,{\mathbf{Z}}^{2}$. There is one scaling function and three wavelet functions for each level. If the wavelet function is separable, then these functions can be rewritten as:

$$\phi (x,y) = \phi (x)\,\,\phi (y),$$

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$$\psi^{H} (x,y) = \psi (x)\,\,\phi (y),$$

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$$\psi^{V} (x,y) = \phi (x)\,\,\psi (y),$$

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$$\psi^{D} (x,y) = \psi (x)\,\,\psi (y).$$

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The wavelet family $\{ \psi_{j,p,q}^{H} ,\,\,\psi_{j,p,q}^{V} ,\,\,\psi_{j,p,q}^{D} \}_{{p,q\, \in \,{\mathbf{Z}}^{2} }} \,$ is an orthonormal basis of ${\mathbf{W}}_{j}^{2}$ and $\{ \psi_{j,p,q}^{H} ,\,\,\psi_{j,p,q}^{V} ,\,\,\psi_{j,p,q}^{D} \}_{{(j,p,q)\, \in \,{\mathbf{Z}}^{3} }} \,$ is an orthonormal basis of ${\mathbf{L}}^{2} ({\mathbf{R}}^{2} )$.

The wavelet approximation coefficients for the function $f(x,y)$ of size $M \times L$ can be given as (Mallat 1989):

$$W_{\phi } \left( {j_{0} ,p,q} \right) = \frac{1}{{\sqrt {ML} }}\sum\limits_{x = 0}^{M - 1} {\sum\limits_{y = 0}^{L - 1} {f\left( {x,y} \right)} } {\kern 1pt} {\kern 1pt} \phi_{{j_{0} ,p,q}} \left( {x,y} \right),$$

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where $j_{0}$ is the scale of the coefficients.

The horizontal ($W_{\psi }^{H}$), vertical ($W_{\psi }^{V}$) and diagonal ($W_{\psi }^{D}$) wavelet detail coefficients can be written as (Mallat 1989):

$$W_{\psi }^{H} \left( {j,p,q} \right) = \frac{1}{{\sqrt {ML} }}\sum\limits_{x = 0}^{M - 1} {\sum\limits_{y = 0}^{L - 1} {f\left( {x,y} \right)} } {\kern 1pt} {\kern 1pt} \psi_{j,p,q}^{H} \left( {x,y} \right)\quad j \ge j_{0} ,$$

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$$W_{\psi }^{V} \left( {j,p,q} \right) = \frac{1}{{\sqrt {ML} }}\sum\limits_{x = 0}^{M - 1} {\sum\limits_{y = 0}^{L - 1} {f\left( {x,y} \right)} } {\kern 1pt} {\kern 1pt} \psi_{j,p,q}^{V} \left( {x,y} \right)\quad j \ge j_{0} ,$$

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$$W_{\psi }^{D} \left( {j,p,q} \right) = \frac{1}{{\sqrt {ML} }}\sum\limits_{x = 0}^{M - 1} {\sum\limits_{y = 0}^{L - 1} {f\left( {x,y} \right)} } {\kern 1pt} {\kern 1pt} \psi_{j,p,q}^{D} \left( {x,y} \right)\quad j \ge j_{0} .$$

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This is the general form of the 2D discrete wavelet transform.