Computing Gaussian derivative waveforms of any order

Abstract

The first and second derivatives of a Gaussian, also known as Ricker wavelets, are often used as source waveforms in forward modeling of seismic and electromagnetic wavefields. In applications such as borehole acoustics, the source waveform has more cycles than traditional Ricker wavelets. For such sources, the higher-order derivatives of a Gaussian are a more accurate representation. Such a source waveform can be computed as the product of a Hermite polynomial and the Gaussian; taking the required derivative is not necessary. Explicit expressions for these polynomials are not required because they satisfy a recurrence relation and thus can be computed from the two lowest-order polynomials. The Gaussian is formulated in terms of the dominant frequency of the derivative order of the desired source waveform. Strictly causal waveforms are not possible with a Gaussian because of its infinite length. However, the computed waveform can be made (pseudo-) causal by shifting it along the positive time axis until its amplitude at time zero is less than a user-defined threshold. Numerical tests reveal that amplitudes at zero time are on the order of [Formula: see text] if this shift is equal to the dominant frequency times the square root of the derivative order.

Estimating Geostrophic Shear from Seismic Images of Oceanic Structure*

“…Positive (red) and negative (blue) are shown. Seismogram calculated by convolving I9 with a Ricker wavelet of central frequency 20 Hz, designed to match the seismic bandwidth of the original seismic experiment (Heigl 2007). Note the good correspondence between peaks on r9 and the seismogram (black dots).…”

Fractional integration of seismic wavelets in anelastic media to recover multiscale properties of impedance discontinuities

“…For each ϕ τ ðtÞ seismic wavelet plotted in solid red lines in Figure 1, we determine the associated GDF approximation ϕ 0 τ ðtÞ according to this method: The results, plotted in dashed black lines in Figure 1, show a very good agreement between ϕ τ ðtÞ and ϕ 0 τ ðtÞ. Because the derivative order α is a key parameter to model the shape of seismic source signatures (Heigl, 2007;Wang, 2015b) and to analyze complex discontinuities (Le Gonidec et al, 2002), it is of critical importance to evaluate the accuracy of the GDF approximation as a function of α. We study how different GDF wavelet sources ψðtÞ with a similar peak frequency f p ¼ 180 Hz, but different integer derivative orders α ¼ ½1; 2; 3; 4; 5 (Figure 2a1-2a5, solid black curves) are modified after a propagation of τ ¼ 100 ms, corresponding to a depth of 150 m for a sound velocity of 1500 m∕s.…”

“…This second wavelet is a symmetric wavelet widely used to model seismic source signals in seismic imaging, attenuation estimation, and Q inverse filtering (Wang, 2008(Wang, , 2015a. Higher derivative orders α are used in acoustic logging to model sources with a high number of cycles (Heigl, 2007); we note that fractional derivative orders, which correspond to asymmetric GDF, have been recently used to better represent seismic signals, such as vertical seismic profile (VSP) data waveforms (Wang, 2015b) or to process large frequency bandwidth seismic data (Ker et al, 2013).…”

“…The parameter Ω is the amplitude, the derivative order α can either be an integer or be fractional, and the angular frequency ω 0 is the reciprocal of the reference dilation defined by a 0 ¼ 2∕ω 0 . Note that a GDF can be written as the product of a Hermite polynomial and a Gaussian potential (Heigl, 2007;Ker et al, 2012); i.e., the amplitude spectrumψ of the GDF ψ is always derived from the Gaussian function and is expressed in the frequency domain by (Wang, 2015b) ψðωÞ…”

Finite-difference time-domain methods for anisotropic media with total-field/scattered-field formulation

“…For a basic Gaussian source, the value of the nth derivative can be easily computed from the following product between the nth order Hermite polynomial (H n (t)) and the Gaussian [130], [131]:…”