**Table of Contents**

- Examples of A priori models
- Polynomial line fitting
- Cross hole tomography
- Reference data set from Arrenæs
- Travel delay computation: The forward problem
- AM13 Gaussian: Inversion of cross hole GPR data from Arrenaes data with a Gaussian type a priori model
- AM13 Gaussian, accounting for modeling errors
- AM13 Gaussian with bimodal velocity distribution
- AM13 Gaussian with unknown Gaussian model parameters

- Probilistic covariance/semivariogram indeference

SIPPI can be used as a convenient approach for unconditional an conditional simulation.

In order to use SIPPI to solve inverse problems, one must provide the solution to the forward problem. Essentially this amounts to implementing a Matlab function that solves the forward problem in using a specific input/output format. If a solution to the forward problem already exist, this can be quite easily done simply using a Matlab wrapper function.

A few implementations of solutions to forward problems are included as examples as part of SIPPI. These will be demonstrated in the following

A prior model consisting of three independent 1D distributions (a Gaussian, Laplace, and Uniform distribution) can be defined using

ip=1; prior{ip}.type='GAUSSIAN'; prior{ip}.name='Gaussian'; prior{ip}.m0=10; prior{ip}.std=2; ip=2; prior{ip}.type='GAUSSIAN'; prior{ip}.name='Laplace'; prior{ip}.m0=10; prior{ip}.std=2; prior{ip}.norm=1; ip=3; prior{ip}.type='GAUSSIAN'; prior{ip}.name='Uniform'; prior{ip}.m0=10; prior{ip}.std=2; prior{ip}.norm=60; m=sippi_prior(prior); m = [14.3082] [9.4436] [10.8294]

1D histograms of a sample (consisting of 1000 realizations) of the prior models can be visualized using ...

sippi_plot_prior_sample(prior);

The FFT-MA type a priori model allow separation of properties of the covariance model (covariance parameters, such as range, and anisotropy ratio) and the random component of a Gaussian model. This allow one to define a Gaussian a priori model, where the covariance parameters can be treated as unknown variables.

In order to treat the covariance parameters as unknowns, one must define one a priori model of type `FFTMA`

, and then a number of 1D `GAUSSIAN`

type a priori models, one for each covariance parameter. Each gaussian type prior model must have a descriptive name, corresponding to the covariance parameter that is should describe:

prior{im}.type='gaussian'; prior{im}.name='m_0'; % to define a prior for the mean prior{im}.name='sill'; % to define a prior for sill (variance) prior{im}.name='range_1'; % to define a prior for the range parameter 1 prior{im}.name='range_2'; % to define a prior for the range parameter 2 prior{im}.name='range_3'; % to define a prior for the range parameter 3 prior{im}.name='ang_1'; % to define a prior for the first angle of rotation prior{im}.name='ang_2'; % to define a prior for the second angle of rotation prior{im}.name='ang_3'; % to define a prior for the third angle of rotation prior{im}.name='nu'; % to define a prior for the shape parameter, nu % (only applies when the Matern type Covariance model is used)

A very simple example of a prior model defining a 1D Spherical type covariance model with a range between 5 and 15 meters, can be defined using:

im=1; prior{im}.type='FFTMA'; prior{im}.x=[0:.1:10]; % X array prior{im}.m0=10; prior{im}.Va='1 Sph(10)'; prior{im}.fftma_options.constant_C=0; im=2; prior{im}.type='gaussian'; prior{im}.name='range_1'; prior{im}.m0=10; prior{im}.std=5 prior{im}.norm=80; prior{im}.prior_master=1; % -- NOTE, set this to the FFT-MA type prior for which this prior type % should describe the range

Note that the the field `prior_master`

must be set to point the to the FFT-MA type a priori model (through its id/number) for which it should define a covariance parameter (in this case the range).

10 independent realizations of this type of a priori model are shown in the following figure

Such a prior, as all prior models available in SIPPI, works with sequential Gibbs sampling, allowing a random walk in the space of a prior acceptable models, that will sample the prior model. An example of such a random walk can be performed using

prior{1}.seq_gibbs.step=.005; prior{2}.seq_gibbs.step=0.1; clear m_real; for i=1:150; [m,prior]=sippi_prior(prior,m); m_real(:,i)=m{1}; end

An example of such a set of 150 dependent realization of the prior can be seen below

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