An approximation of the inverse Ricker wavelet as an initial guess for bidirectional deconvolution

Finite approximation

It is known that a Ricker wavelet is the second-order derivative of a Gaussian function. For computation, we use a finite and discrete approximation to a Ricker wavelet as a replacement to the infinite and continuous real second-order derivative of a Gaussian function. We use a second-order finite-difference operator to approximate a second-order derivative and binomial coefficients to approximate a Gaussian function.

In the Z domain,

$$= -(1-Z)^2(1+Z)^{2N} .$$ (6)

The parameter $N$ is half the order of the binomial we used. Here we use $2N$ in equation 6 instead of $N$ simply to keep the order of the binomial even to facilitate the later separation. In practice, we would choose the value of $N$ parameter according to the wavelength (or principle frequency component) of the wavelet in our data. The larger the value, the wider the wavelet.

Figures 1(a) and 1(b) show this fourth-order ($N$ =4) finite approximation of the Ricker wavelet in the time and frequency domains. Here we use the fourth-order as an example, but we can use a different-order implementation as long as the approximate Ricker wavelet has the same wavelength (or principle frequency component) as the wavelet of our data.

An approximation of the inverse Ricker wavelet as an initial guess for bidirectional deconvolution