Example: Gaussian prior model with unknown Gaussian model parameters

One of the most intriguing benefits (in addition to the computational efficiency) of using the FFTMA type a priori model, is that it allows separation of the random component and the covariance model parameters.

This means that one can define an inverse problem, where the a priori model is Gaussian, but where the properties of the Gaussian model (such as the mean, range, anisotropy) can be treated as unknown model parameters (see Hansen et al (2013a,b)).

Each property of the Gaussian prior model that should be treated as an unknown model parameter, must be defined as a separate 1D type type prior model. Further a specific name (identifying the covariance model property it describes) must be defined (see example below), and it must point to the prior model type number for which it describes a covariance model property.

The example below (available here: sippi_AM13_metropolis_gaussian_covariance_inference.m ) describes a 2D FFTMA type a priori model (prior with id 1) with an unknown range (prior with id 2) with an a priori distribution described by a close to uniform distribution between 1.5m and 10.5m:

 im=1;
 prior{im}.type='FFTMA';
 prior{im}.name='Velocity (m/ns)';
 prior{im}.m0=0.145;
 prior{im}.Va='.0003 Sph(6)';
 dx=0.25;
 prior{im}.x=[-1:dx:6];
 prior{im}.y=[0:dx:13];
 prior{im}.cax=[.1 .18];
 i_master=im;

 % range - horizontal
 im=im+1;
 prior{im}.type='gaussian';
 prior{im}.name='range_1'; % the name covariance model property to define
 prior{im}.m0=6;
 prior{im}.min=1.5;
 prior{im}.max=10.5;
 prior{im}.norm=50;
 prior{im}.prior_master=i_master; % point to the id of the prior it describes

Any combination of the following parameters can be set:

 prior{im}.name='range_1; % Range, along direction of angle_1
 prior{im}.name='range_2; % Range, along direction of angle_2
 prior{im}.name='range_3; % Range, along direction of angle_3
 prior{im}.name='ang_1; % Angle 1, degrees from North
 prior{im}.name='ang_2; % Angle 2
 prior{im}.name='ang_3; % Angle 3
 prior{im}.name='sill; % sill,
 prior{im}.name='nu; % the 'nu' parameter, only applies when using the Matern covariance model type.
 prior{im}.name='m0; % A priori mean

As an example consider case where the two ranges, and the angle of

anisotropy fro a 2D Gaussian(FFTMA) a priori type model is treated as

model parameters:

 im=0;

 % velocity field

 im=im+1;

 prior{im}.type='FFTMA';

 prior{im}.name='Velocity (m/ns)';

 prior{im}.m0=0.145;

 prior{im}.Va='.0003 Sph(6)';

 dx=0.25;

 prior{im}.x=[-1:dx:6];

 prior{im}.y=[0:dx:13];

 prior{im}.cax=[.1 .18];

 i_master=im;



 % range - horizontal

 im=im+1;

 prior{im}.type='gaussian';

 prior{im}.name='range_1';

 prior{im}.min=1.5;

 prior{im}.max=10.5;

 prior{im}.norm=50;

 prior{im}.prior_master=i_master;



 % range - horizontal

 im=im+1;

 prior{im}.type='gaussian';

 prior{im}.name='range_2';

 prior{im}.min=1.5;

 prior{im}.max=5.5;

 prior{im}.norm=50;

 prior{im}.prior_master=i_master;





 % rotation

 im=im+1;

 prior{im}.type='gaussian';

 prior{im}.name='ang_1';

 prior{im}.m0=90;

 prior{im}.std=20;

 prior{im}.norm=2;

 prior{im}.prior_master=i_master;

A sample from the corresponding a priori model (FFTMA type) is shown below:

Samples of the a priori distributions for range_1, range_2, and ang_1 are shown here:

As for the examples above, the a posteriori distribution can be samples 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);