WALSH FUNCTIONS

The Walsh functions form an orthogonal and complete set of functions representing a discretized function (Rao, 1983). As in the FFT, the discrete function must be padded before decomposition, so that the total number of samples reaches the nearest power of two. Figure a shows the first sixteen Walsh functions, which form a complete set describing sequences with sixteen samples. Two important characteristics of the Walsh functions are their compactness (representing the lower order functions requires fewer samples), and the simplicity and quickness of their construction.

 

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Figure 1 (a) The first sixteen Walsh functions. The frequency (order number) increases from bottom to top: W⁰, W¹, W^2,1, W^2,2, W^3,1 to W^3,4, and W^4,1 to W^4,8. (b) Superposition of the four Walsh functions of order 3 to form the wavelet transform component of the same order.

The following rules are used to generate the subset of Walsh functions corresponding to order N:

• The number of samples of the functions in this subset is $\; \; 2^N$.
• Except for order 0 (zero frequency), the number of Walsh functions is given by

  k_N = 2^N-1

  .
• Each function of order N-1 generates two functions of order N, one by contraction and repetition, the other by contraction and repetition with a change of sign. Thus

  $$W^{N,2j-1}_i = W^{N,2j-1}_{i+k_N} = W^{N-1,j}_i, \mbox{\hspace{1cm}} i=1,k_N$$

  $$W^{N,2j}_i = -W^{N,2j}_{i+k_N} = W^{N-1,j}_i, \mbox{\hspace{1cm}} i=1,k_N,$$

  where i is the sample index and j is the index of one of the functions of order N.

Beginning with the functions of order 0 (W⁰=1) and 1 (W¹=1,-1), it is possible to generate a complete set by using this iterative process.

There is a direct relationship between Walsh functions and the particular wavelet transform described by Ottolini (1990), which had a two-sample signal function as the basis wavelet. In contrast to the Walsh set, the wavelet set has only one component of order N, but the number of parameters associated with this component is equal to the number of Walsh functions of the same order. Figure b shows that adding all Walsh components of a given order results in the wavelet component of the same order. In general, any wavelet component can be obtained as a superposition of all the Walsh components of same order.

The Walsh functions have some properties that make them suitable for certain geophysical applications. Lanning and Johnson (1983) used the Walsh decomposition to develop an automated algorithm to identify rock boundaries in well-log data. They claim that the discrete transitions in signal level of these functions make them ideal for describing layered media. Figure a shows a sonic-log from the Gulf of Mexico and its Walsh spectrum, while Figure b compares the sonic smoothed with a triangle filter, with a band-limited (order 0 to 6 for a maximum order of 12) Walsh composition of the same sonic-log. Both curves show the same basic features, but the Walsh composition manifest a ``blocky" structure that can be better correlated with the geologic layers.

 

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Figure 2 (a) A sonic log (left), and its Walsh spectrum (right). The zero frequency component was removed. (b) A Walsh band-limited version (continuous line), and smoothed version (dotted line) of the sonic log.

In traveltime inversion the advantage of using Walsh instead of Fourier decomposition lies in the simplicity of the forward modeling computation, since the samples of the basis functions can only assume values 1 or -1, and only a few samples are required to describe the lower frequency components. Replacing square functions by Walsh functions in the nonlinear scheme described in the preceding section leads to a solution similar to the set of equations (7).