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# trunk/src/green/disan/phadi.f
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(XXX macro: "PageOutline")
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Navigation: [[Mainpage], [wiki:trunk], [wiki:branches], [wiki:tags], [wiki:vendor], [wiki:export], [wiki:admin], [wiki:docs|WikiStart]]
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**Calculate dispersion relation from phase differences**
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## Phase-slowness analysis
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The theory behind this program is described by
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[[Forbriger (2003. Inversion of shallow-seismic wavefields: I. Wavefield transformation. Geophys. J. Int., 153, 719-734, appendix A)|http://dx.doi.org/10.1046/j.1365-246X.2003.01929.x]]:
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Consider *u(ω,r)*
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being a seismogram at offset *r* and
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```latex
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\begin{equation*}
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\tilde{u}(\omega,r)=\int\limits_{-\infty}^{+\infty}u(t,r)\,e^{i\omega t}\,\text{d}t
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\end{equation*}
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```
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being its Fourier transform.
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Phase velocities may be derived from the phase *φ(ω,r)*
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of the Fourier coefficients
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```latex
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\begin{equation*}
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\tilde{u}(\omega,r)=A(\omega,r)\,\exp\left[i\varphi(\omega,r)\right]
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\end{equation*}
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```
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if the corresponding waveforms are single-mode plane waves (then *A(ω,r)*
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will vary only slowly with *r*).
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If they are not, the outcome of phase-difference techniques is
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unpredictable.
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The phase *φ(ω,r)=p(ω)ωr*
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of a single plane mode is a linear function of the offset.
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The derivation of its absolute value from the Fourier coefficients is
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ambiguous by an additive constant of an integer multiple of 2*π*.
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This is due to the non-uniqueness of the involved arctan- or
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complex ln-function.
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However, for a wide frequency-range the phase increment from *r,,l,,* to
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*r,,l+1,,'' is certainly less than 2''π*.
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For this reason we use the advantage of a dense geophone spread and
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derive the phase traveltime
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```latex
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\begin{equation*}
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T(\omega,r_l)=\sum\limits_{k=2}^{l}\frac{-i}{\omega}
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\ln\left[\tilde{u}(\omega,r_k)/\tilde{u}(\omega,r_{k-1})\right]
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\end{equation*}
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```
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relative to offset *r,,1,,*.
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Then we fit a straight line
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```latex
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\begin{equation*}
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T_{\text{fit}}(\omega,r)=p(\omega)\,r+c
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\end{equation*}
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```
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searching for *p(ω)'' and ''c* such that
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```latex
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\begin{equation*}
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\sum\limits\limits_{l=1}^{N}\left|T_{\text{fit}}(\omega,r_l)-T(\omega,r_l)\right|^2
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\end{equation*}
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```
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is minimized (linear regression).
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Some offsets at both ends of the spread may be discarded if this improves
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the fit.
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The gradient *p(ω)* is the sought phase-slowness at angular frequency
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ω.
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## Practical considerations
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1. If the wave does not consist of a single mode, *T(ω,r,,l,,)'' as a function of ''r* will not appear like a straight line. The outcome of the analysis then is undefined.
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1. If receiver distances *r,,l,,-r,,l-1,,'' are too large, signals are aliased. The actual phase difference of the Fourier signals between ''r,,l,,'' and ''r,,l-1,,'' then is larger than 2''π''. The algorithm will then not be able to obtain the correct phase slowness. Presenting phase slowness ''p(ω)'' as a function of ''ω* it will have discontinuities. |