data: Data and data uncertainties/noise

data is a Matlab structure that defines any number of data and the associated uncertainty/noise model.

data{1} defines the first data set (which must always be defined), and any number of additional data sets can be defined in data{2}, data{3}, ...

This allows to consider for example seismic data in data{1}, and electromagnetic data in data{2}.

For each set of data, a Gaussian noise model (both correlated and uncorrelated) can be specified. The noise model for different data types (e.g. data{1} and data{2} are independent).

Once the noise model has been defined, the log-likelihood related to any model, m, with the corresponding forward response, d, can be computed using

[d,forward,prior,data]=sippi_forward(m,forward,prior,data)
logL=sippi_likelihood(data,d)

where d is the output of sippi_forward.

The specification of the noise model can be divided into a description of the measurement noise (mandatory) and the modeling error (optional).

Gaussian measurement noise

Uncorrelated Gaussian measurement noise

To define a set of observed data, [0,1,2], with an associated uncorrelated uncertainty defined by a Gaussian model with mean 0 and standard deviation 2, use

data{1}.d_obs=[0 1 2]';
data{1}.d_std=[2 2 2]';

which is equivalent to (as the noise model for each data is the same, and independent)

data{1}.d_obs=[0 1 2]';
data{1}.d_std=2;

One can also choose to define the uncertainty using a variance as opposed to the standard deviation

data{1}.d_obs=[0 1 2]';
data{1}.d_var=4;

Correlated Gaussian measurement noise

Correlated Gaussian measurement uncertainty can be specified using the Cd field, as for example

data{1}.Cd=[4 1 0 ; 1 4 1 ; 0 1 4];

Note that data{1}.Cd must be of size [NDxND], where ND is the number of data in data{1}.d_obs.

Gaussian modeling error

The modeling error refers to errors caused by using for example an imperfect forward model, see HCM14.

A Gaussian model of the modeling error can be specified by the mean, dt, and the covariance, Ct.

For example

data{1}.dt=[0 0 0];
data{1}.Ct=[4 4 4; 4 4 4; 4 4 4];

is equivalent to

data{1}.Ct=4

which implies a zero mean modeling error with a covariance model where all model parameters has a covariance of 4.

sippi_compute_modelization_forward_error can be used to estimate the modeling error related to using an approximate forward model. See the tomography example, for an example of accounting for correlated modeling errors, following HCM14.