Resolution Of Converted Xave

Converted wave resolution

Converted wave resolution as a function of bandwidth
Arnim B. Haase and Charles P. Ursenbach ABSTRACT A 1D-algorithm for the computation of trace envelopes when attenuation is present has been programmed. With this algorithm it is confirmed that adding low-frequency bandwidth can increase resolution. However, beyond a critical depth, adding low frequencies decreasesresolution in these computations. A second, more realistic algorithm includes geometrical spreading and a noise floor. Assuming that any signal above the noise floor can be restored to a flat spectrum, pulse widths are computed from boxcar windows. Comparing the resulting pulse widths for C-waves and P-waves shows that adding low-frequency bandwidth improves resolution. Also, resolutioncrossover depths are moved deeper. INTRODUCTION The impact of attenuation on converted wave resolution has been investigated by Bale and Stewart (2002) and also by Deffenbaugh et al. (2000). It is observed that, with increasing depth, high frequencies are increasingly attenuated. Because of this high frequency loss, bandwidth is narrowed with increasing depth and resolution deteriorates. The idea ofadding bandwidth at the low frequency end of the spectrum in order to counteract attenuation controlled bandwidth loss at the high end of the frequency spectrum has been investigated by Martin and Stewart (1994) as well as by Bale and Stewart (2002). In this study we seek to provide computational tools that allow one to estimate resolution that could be gained by adding low frequencies. The practicalquestion of simultaneous low frequency noise contamination is not considered at this stage. Martin and Stewart (1994) point out that, with the advent of 24-bit recording systems, there is sufficient dynamic range available to recover signal when highamplitude noise is present. RESOLUTION CROSSOVER DEPTH Deffenbaugh et al. (2000) define a resolution crossover depth for situations where Qp > Qs .They state that C-waves offer better resolution than P-waves at shallow targets before attenuation becomes significant. They also give an equation for the calculation of this crossover depth. Their equation implies a frequency of zero Hz for the lower limit of their seismic band, which is not too useful for investigating the role of lower frequencies. The purpose of this study is to developcomputational tools that can be used to determine pulse width (wavelet resolution) as a function of bandwidth, which in turn is a function of depth. Then, comparing pulse width as a function of depth for C-waves and P-waves, the resolution crossover depth can be found. The prediction by Martin and Stewart (1994) as well as Bale and Stewart (2002) of more compact wavelets and enhanced C-wave resolutionbecause of added low-frequency content can thus be put to the test. COMPUTATIONAL ALGORITHM The original plan had been to develop a rule of thumb for estimating pulse width from bandwidth as a function of depth. When assuming a homogeneous earth for simplicity,
CREWES Research Report — Volume 17 (2005) 1

Haase and Ursenbach

attenuation alone causes the loss of high frequencies withincreasing depth. Attenuation is the result of anelasticity, which also leads to velocity dispersion of seismic waves. An equation for this frequency dependence of seismic velocities is given by Aki and Richards (1980):

⎛ ⎞ ln ⎛ ω / ω ⎜ ⎜ ref ⎟ ⎝ ⎠ − i v (ω ) = v ⎜1 + ref ⎜ πQ 2Q ⎜ ⎝

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

(1)

Because v(ω) is part of the exponent when calculating attenuation, this exponent isdifferent at every frequency point and no simplified equation could be found. The approach taken here is to compute signal strength as a function of depth, one frequency point at a time, followed by an inverse Fourier transform. Offset is taken to be constant in these computations. The roles of depth and offset can also be interchanged. Pulse width is measured at one half of maximum amplitude of trace...