The minimum-frequency property of complex trace analysis

Abstract A waveform has the property of minimum frequency if its instantaneous frequency equals the Hilbert transform of its instantaneous bandwidth plus the spectral low-cut frequency. Minimum frequency is characterized by an amplitude spectrum that is maximally skewed toward the low frequencies. The associated property of maximum frequency is characterized by an amplitude spectrum that is maximally skewed toward the high frequencies. Minimum and maximum frequency are the time-domain counterparts of the spectral properties of minimum and maximum delay. Many waveforms have the property of minimum frequency, such as the Ricker wavelet. Waveforms that have the property of maximum frequency include Klauder wavelets that emphasize high frequencies. Impulsive wavelets can have both minimum frequency and minimum delay. The properties of minimum and maximum frequency suggest time-variant measures to estimate the spectral low-cut and high-cut frequencies of seismic traces and produce an instantaneous frequency attribute that is free of spikes. The property of minimum frequency has long been known but remains unfamiliar to geophysics. It is explained in the context of complex trace analysis illustrated with seismic wavelets and traces.