Computing Gaussian derivative waveforms of any order

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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.