## Using SIGM in Matrix Headers

Each of the coordinate and coordinate-difference observation group header records (2DC, 2DD, 3DC, and 3DD) takes an optional "sigma ID". A "sigma-ID" refers to a previously defined SIGM record.

Note that if centering errors are used in the sigma-record, the covariance matrix coordinate system must be LG (local geodetic).

For the GENC matrix header, all elements of the covariance matrix are initially set to zero before the following revisions are made. For all other matrix header types, the elements of the covariance matrix are initially given (either directly or indirectly) by the ELEM records.

The "Addition constant (entire matrix)" and the "Addition constant (diagonal)" are first added directly to the covariance matrix. The "PPM (diagonal)" value is used to calculate a standard deviation (using either a geocentric radius for 2DC and 3DC groups, or the vector lengths for 2DD and 3DD groups) and that value is squared and added to the diagonal of the covariance matrix.

Next, for 3DC and 3DD groups, the "Addition constant (height/Z)" and "Factor (height/Z)" values are used to revise the covariance matrix. If the matrix coordinate system is CT then the Z or Z-difference diagonal elements are revised. If the matrix coordinate system is LG then the height or height-difference diagonal elements are revised. The revision is performed as follows (E denotes the current matrix element, A the "Addition constant (entire matrix)" value, and F the "Factor (height/Z)" value:

E = (E + A) * F.

Finally, the "Factor (entire matrix)" and the "Factor (diagonal)" are applied to the covariance matrix.

**The elements of the sigma-record are used in these groups of measurements as follows:****Standard deviation**: This square of this value is added to the "Addition constant (diagonal)" value specified in the matrix header record.**PPM**: This value is added to the "PPM (diagonal)" value specified in the matrix header record**Centering errors**: For 2DC and 3DC groups, the square of the at-centering value is added to the "Factor (diagonal)" value specified in the matrix header record. For 2DD and 3DD groups, the sum of the squares of the from-centering and to-centering values is added to the "Factor (diagonal)" value specified in the matrix header record.**Auxiliary parameter**: Adds the specified auxiliary parameter to the observation group.Note that if centering errors are used in the sigma-record, the covariance matrix coordinate system must be LG (local geodetic).

**The covariance matrix header record can be one of the following record types:**- COV: Specifies that covariance matrix elements follow in a set of ELEM records.
- CORR: Specifies that correlation matrix elements and standard deviation values follow in a set of ELEM records
- GENC: Specifies that the covariance matrix will be built from the fields of the GENC record (see below) and that no matrix elements are specified (no ELEM records follow).
- WGT: Specifies that weight matrix elements follow in a set of ELEM records.

**The "matrix revision" fields of the matrix header record (COV, CORR, GENC, and WGT) are as follows:**- 015-024: Addition constant (entire matrix)
- 026-035: Factor (entire matrix)
- 037-046: Addition constant (diagonal)
- 048-057: Factor (diagonal)
- 059-068: PPM (diagonal)
- 070-079: Addition constant (height/Z)
- 081-090: Factor (height/Z)

For the GENC matrix header, all elements of the covariance matrix are initially set to zero before the following revisions are made. For all other matrix header types, the elements of the covariance matrix are initially given (either directly or indirectly) by the ELEM records.

The "Addition constant (entire matrix)" and the "Addition constant (diagonal)" are first added directly to the covariance matrix. The "PPM (diagonal)" value is used to calculate a standard deviation (using either a geocentric radius for 2DC and 3DC groups, or the vector lengths for 2DD and 3DD groups) and that value is squared and added to the diagonal of the covariance matrix.

Next, for 3DC and 3DD groups, the "Addition constant (height/Z)" and "Factor (height/Z)" values are used to revise the covariance matrix. If the matrix coordinate system is CT then the Z or Z-difference diagonal elements are revised. If the matrix coordinate system is LG then the height or height-difference diagonal elements are revised. The revision is performed as follows (E denotes the current matrix element, A the "Addition constant (entire matrix)" value, and F the "Factor (height/Z)" value:

E = (E + A) * F.

Finally, the "Factor (entire matrix)" and the "Factor (diagonal)" are applied to the covariance matrix.