Bernese GPS Software Version 5.0 - Bernese GNSS Software

7.4 Weighting and Correlations
The normalized residuals (option “Type of computed residuals” in panel “GPSEST 3.1”) are
residuals divided by the square root of the diagonal element of the cofactor matrix of the
residuals
vnorm(i) = v(i)
. (7.15)
Dii(v)
The cofactor matrix of the residuals is the difference between the inverse weighting matrix
of the actual observations and the cofactor matrix of the adjusted observations
with
D(v) = P −1 − D(y) (7.16)
D(y) = A(A ⊤ P A) −1 A ⊤ . (7.17)
In contrast to real residuals, normalized (phase as well as code) residuals are always converted
to one-way L1 carrier phase residuals, i.e., if you divide these residuals by the a
posteriori sigma of unit weight, you should get random variables with a standard deviation
of 1. For reliable outlier detection using program RESRMS when analyzing low-elevation
data, applying an elevation-dependent observation weighting model or introducing stationspecific
weights, it is recommended to save normalized residuals. Be aware that, e.g., a real
double-difference L3 carrier phase residual of 36 mm corresponds to a normalized (one-way
L1-)residual of 6 mm – assuming that no elevation- or station-specific weighting took place.
A special option NORM APRIORI is available for conversion of real residuals into normalized
residuals using the a priori variance of observations
D(v) ≃ P −1
. (7.18)
This option may be useful if epoch parameters are present which are pre-eliminated epochwise
and back-substituted in a final step (see Sections 7.5.5 and 7.5.6). If the option “Varcovar
wrt epoch parameters” is set to SIMPLIFIED to speed up the back-substitution the matrix
A ⊤ P A in Eqn. (7.17) only refers to the epoch parameters. As a consequence, residuals of
observations not contributing to epoch parameters are normalized differently than those of
observations that do contribute if option NORMALIZED is used and the residuals would not
be comparable.
If residuals are requested by specifying a residual output file in panel “GPSEST 2.1: Output
Files 1” the observation equations, i.e., the design matrix A and the observation vector
y are saved to a binary scratch file while processing each observation and accumulating
the normal equation matrix A ⊤ PA. After solving the normal equation the scratch file is
read and the adjusted residuals are computed according to Eqn. (7.2). As a matter of fact,
residuals can only be computed if all parameters are solved for. No parameter, not even
ambiguity parameters may be pre-eliminated before residuals are computed. The exception
are epoch parameters such as clock corrections or kinematic coordinates which are handled
in a special way (see Section 7.5.3). Parameters may, however, be pre-eliminated using the
option PRIOR TO NEQ SAVING (see Section 7.5.5) if they should not appear in the normal
equation saved for later use in ADDNEQ2 (see Chapter 9). This is particularly important
for ambiguity parameters which are not supported by ADDNEQ2: Saving a residual file
and a normal equation file in the same run is only possible if ambiguity parameters are
pre-eliminated with option PRIOR TO NEQ SAVING.
Bernese GPS Software Version 5.0 Page 145