# Programm src/green/disan/phadi.f

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## Phase-slowness analysis

Consider *u(ω,r)*
being a seismogram at offset *r* and

```
\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

```
\begin{equation*}
\tilde{u}(\omega,r)=A(\omega,r)\,\exp\left[i\varphi(\omega,r)\right]
\end{equation*}
```

if the corresponding waveforms are single-mode plane waves (then *A(ω,r)*
will vary only slowly with *r*).
If they are not, the outcome of phase-difference 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 non-uniqueness of the involved arctan- or
complex ln-function.
However, for a wide frequency-range 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

```
\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_{k-1})\right]
\end{equation*}
```

relative to offset *r,,1,,*.
Then we fit a straight line

```
\begin{equation*}
T_{\text{fit}}(\omega,r)=p(\omega)\,r+c
\end{equation*}
```

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

```
\begin{equation*}
\sum\limits\limits_{l=1}^{N}\left|T_{\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 phase-slowness at angular frequency
ω.

## Practical considerations

- 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.
- 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 2π. 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.