AM13 Gaussian with bimodal velocity distribution
------------------------------------------------

A Matlab script for the following example is avalable at
`sippi\_AM13\_metropolis\_bimodal.m <https://github.com/cultpenguin/sippi/blob/master/examples/case_tomography/sippi_AM13_metropolis_bimodal.m>`__.

The `GAUSSIAN <#prior_gaussian>`__ and `FFTMA <#prior_fftma>`__\ a prior
types implicitly assume a normal distribution of the model parameter.

It is however possible to change the Gaussian distribution to any shaped
distribution, using a normal score transform. Note that when this is
done the given semivariogram model for the `FFTMA <#prior_fftma>`__ a
priori model will not be reproduced. If this is a concern, then the
`VISIM <#prior_visim>`__ type a priori model should be used.

The data and forward structures is identical to the one described in the
`previous <#AM13_gaussian>`__ example.

::

     %% Load the travel time data set from ARRENAES
     clear all;close all
     D=load('AM13_data.mat');
     options.txt='AM13';

     %% SETUP DATA
     id=1;
     data{id}.d_obs=D.d_obs;
     data{id}.d_std=D.d_std;
     data{id}.Ct=D.Ct+1; % Covariance describing modeling error

     % SETUP THE FORWARD MODEL USED IN INVERSION
     forward.forward_function='sippi_forward_traveltime';
     forward.sources=D.S;
     forward.receivers=D.R;
     forward.type='fat';forward.linear=1;forward.freq=0.1;

The desired distribution (the 'target' distribution) must be provided as
a sample of the target distribution, in the ``data{id}.d_target``
distribution.

::

     %% SETUP PRIOR
     im=1;
     prior{im}.type='FFTMA';
     prior{im}.name='Velocity (m/ns)';
     prior{im}.m0=0.145;
     prior{im}.Va='.0003 Sph(6)';
     dx=0.15;
     prior{im}.x=[-1:dx:6];
     prior{im}.y=[0:dx:13];
     prior{im}.cax=[.1 .18];

     % SET TARGET
     N=1000;
     prob_chan=0.5;
     dd=.014*2;
     d1=randn(1,ceil(N*(1-prob_chan)))*.01+0.145-dd; %0.1125;
     d2=randn(1,ceil(N*(prob_chan)))*.01+0.145+dd; %0.155;
     d_target=[d1(:);d2(:)];
     prior{im}.d_target=d_target;

5 realizations from the corresponding a priori model looks like Figure
`figure\_title <#gaussian_bimodal_dist>`__ compares the distribution
from one realization of both prior models considered above.

As for the examples above, the a posteriori distribution can be sampled
using e.g.

::

     options.mcmc.nite=500000; % optional, default:nite=30000
     options.mcmc.i_sample=500; % optional, default:i_sample=500;
     options.mcmc.i_plot=1000; % optional, default:i_plot=50;
     options=sippi_metropolis(data,prior,forward,options);

     % plot posterior statistics
     sippi_plot_posterior(options.txt);