# Computing a theoretical dispersion curve

This tutorial details all the steps needed to compute a dispersion with gpdc. Another more interactive way to compute dispersion curves is Gplivemodel.

## Creating a model file

Crete a text file (e.g. "test.model"):

```# My first model: two layers over a half-space
# First line: number of layers
3
# One line per layer:
# Thickness(m), Vp (m/s), Vs (m/s) and density (kg/m3)
7.5  500  200 1700
25  1350  210 1900
# Last line is the half-space, its thickness is ignored but the first column is still mandatory
0   2000 1000 2500
```

All lines beginning with '#' are considered as comments and they are ignored. Several models can be chained in one single file, but no comment line is allowed between layers. Column separators can be either TAB or SPACE. The number of consecutive separators does not matter.

## Computing the dispersion curves

To get the dispersion curve of Rayleigh fundamental mode:

```gpdc Test.model
```

You get as output:

```# My first model: two layers over a half-space
# First line: number of layers
# One line per layer:
# Thickness(m), Vp (m/s), Vs (m/s) and density (kg/m3)
# Last line is the half-space, its thickness is ignored but the first column is still mandatory
# 1 Rayleigh dispersion mode(s)
# CPU Time = 1 ms
# Mode 0
0.2 0.00107875907546066
0.209523150557933 0.00107907141898638
[...]
19.0909691332367 0.00525624231593241
20 0.00526288933387501
```

To compute Rayleigh higher modes (4 in this case) as well as the fundamental mode:

```gpdc Test.model -R 5
```

To compute Love higher modes (4 in this case) as well as the fundamental mode:

```gpdc Test.model -R 0 -L 5
```

To compute Love modes, you must set explicitly the number of Rayleigh to 0, it is one by default.

## Saving the dispersion curves

```gpdc Test.model -R 2 > Test_Rayeigh_2modes.disp
```

## Plotting the dispersion curves

We use figue to plot couples of values (X, Y), additionally we provide a make-up file to get the correct type and labeling of axes. Before running the next command you can download Dc.mkup.

```gpdc Test.model -R 2 | figue -c -m Dc.mkup
```

## Going further

### Help about command line options

To get help about all options:

```gpdc -h all
```

### Programming interface

The computation of dispersion is implemented in libQGpCoreWave.so (or equivalent for Windows and Mac). You can access it through various languages: C++, C or Fortran. Complete examples are provided for C and Fortran in the source distribution archive. Download gpdc to have the simplest archive.

#### A C++ example

```#include <QGpCoreWave.h>
[...]
// 3-layer model initialization
LayeredModel model(3);
model.h()=7.5;
model.h()=25.0;
model.slowP()=1.0/500.0;
model.slowP()=1.0/1350.0;
model.slowP()=1.0/2000.0;
model.slowS()=1.0/200.0;
model.slowS()=1.0/210.0;
model.slowS()=1.0/1000.0;
model.rho()=1800;
model.rho()=1900;
model.rho()=2500;
// Initialize frequency sampling, in angular frequency (omega)!!
QVector<double> x;
x.resize(50);
x=...;
[...]
x=...;
// Or use Curve to generate linear or log series
Curve<Point1D> c;
c.line( minFreq, 0.0, maxFreq, 0.0 );
c.resample( nSamples, minFreq, maxFreq, Log | Function );
c.multiply( 2*M_PI ); // convert to angular frequency
QVector<double> x = c.xVector();
// Dispersion computation, fundamental mode only
Rayleigh rayleigh( &model );
//Love love( &model );
Dispersion dispersion( 1, &x);
dispersion.calculate( modelRayleigh, 0 ); // 0 is to avoid ellipticity computation
// Access computed values
Curve<Point2D> * dc = dispersion.curve(0);
// Or directly, fundamental mode, ith sample value.
dispersion.mode(0)[i]
```

To compile:

```g++ [...] -lQGpCoreTools -lQGpCoreWave
```

#### Fortran interface

```call dispersion_curve_init(verbose)
call dispersion_curve_rayleigh(nLayers, h, vp, vs, rho, nSamples, omega, nModes, slowness, group);
call dispersion_curve_love(nLayers, h, vp, vs, rho, nSamples, omega, nModes, slowness, group);
```
dispersion_curve_rayleigh arguments
Argument Type Description
verbose Integer 4 bytes 0 minimal ouput, 1 verbose output
nLayers Integer 4 bytes Number of layers
h Vector of floats 8 bytes (nLayers elements) Thicknesses of layers (m)
vp Vector of floats 8 bytes (nLayers elements) Vp in each layer (m/s)
vs Vector of floats 8 bytes (nLayers elements) Vs in each layer (m/s)
rho Vector of floats 8 bytes (nLayers elements) Density in each layer (kg/m3)
nSamples Integer 4 bytes Number of frequency samples
omega Vector of floats 8 bytes (nSamples elements) Angular frequencies (rad/s)
nModes, Integer 4 bytes Number of modes including fundamental
slowness Vector of floats 8 bytes (nSamples*nModes, elements) Output of slowness values
group Integer 0 for phase, 1 for group

To compile:

```gfortran [...] -lQGpCoreTools -lQGpCoreWave
```

### Theoretical considerations

The equations and methods implemented in libQGpCoreWave.so are fully described from a theoretical point of view in Marc Wathelet's PhD thesis (chapter 3).