

# trunk/src/green/disan/phadi.f



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**Calculate dispersion relation from phase differences**






## Phaseslowness analysis



The theory behind this program is described by



[[Forbriger (2003. Inversion of shallowseismic wavefields: I. Wavefield transformation. Geophys. J. Int., 153, 719734, appendix A)http://dx.doi.org/10.1046/j.1365246X.2003.01929.x]]:






Consider *u(ω,r)*



being a seismogram at offset *r* and



```latex



\begin{equation*}



\tilde{u}(\omega,r)=\int\limits_{\infty}^{+\infty}u(t,r)\,e^{i\omega t}\,\text{d}t



\end{equation*}



```



being its Fourier transform.



Phase velocities may be derived from the phase *φ(ω,r)*



of the Fourier coefficients



```latex



\begin{equation*}



\tilde{u}(\omega,r)=A(\omega,r)\,\exp\left[i\varphi(\omega,r)\right]



\end{equation*}



```



if the corresponding waveforms are singlemode plane waves (then *A(ω,r)*



will vary only slowly with *r*).



If they are not, the outcome of phasedifference techniques is



unpredictable.






The phase *φ(ω,r)=p(ω)ωr*



of a single plane mode is a linear function of the offset.



The derivation of its absolute value from the Fourier coefficients is



ambiguous by an additive constant of an integer multiple of 2*π*.



This is due to the nonuniqueness of the involved arctan or



complex lnfunction.



However, for a wide frequencyrange the phase increment from *r,,l,,* to



*r,,l+1,,'' is certainly less than 2''π*.



For this reason we use the advantage of a dense geophone spread and



derive the phase traveltime



```latex



\begin{equation*}



T(\omega,r_l)=\sum\limits_{k=2}^{l}\frac{i}{\omega}



\ln\left[\tilde{u}(\omega,r_k)/\tilde{u}(\omega,r_{k1})\right]



\end{equation*}



```



relative to offset *r,,1,,*.



Then we fit a straight line



```latex



\begin{equation*}



T_{\text{fit}}(\omega,r)=p(\omega)\,r+c



\end{equation*}



```



searching for *p(ω)'' and ''c* such that



```latex



\begin{equation*}



\sum\limits\limits_{l=1}^{N}\leftT_{\text{fit}}(\omega,r_l)T(\omega,r_l)\right^2



\end{equation*}



```



is minimized (linear regression).



Some offsets at both ends of the spread may be discarded if this improves



the fit.



The gradient *p(ω)* is the sought phaseslowness at angular frequency



ω.






## Practical considerations



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.



1. If receiver distances *r,,l,,r,,l1,,'' are too large, signals are aliased. The actual phase difference of the Fourier signals between ''r,,l,,'' and ''r,,l1,,'' 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. 