Calculate the covariance matrix

[samperizal]

Greetings.

Someone tell me how to reconstruct the complete covariance matrix from the data obtained with RTKNAVI or RTKPOST.

Thanks in advance.

Sorry for my English.

[gpoleszuk]

For example

% (x/y/z-ecef=WGS84,Q=1:fix,2:float,3:sbas,4:dgps,5:single,6:ppp,ns=# of satellites)
%  GPST                      x-ecef(m)      y-ecef(m)      z-ecef(m)   Q  ns   sdx(m)   sdy(m)   sdz(m)  sdxy(m)  sdyz(m)  sdzx(m) age(s)  ratio
2023/01/07 07:00:00.000   ****099.5227  -****927.2761  -****538.8151   2  10   0.0082   0.0107   0.0047   0.0040   0.0033  -0.0043   0.00    0.0

Then, if the sdx, sdy and sdz are provided, you can reconstruct the main diagonal of the covariance matrix squaring each value.

Now, to reconstruct the covariances, since they are provided as sdxy, sdyz, sdzx you can set the off-diagonal ellements. Since the covariance matrix is a symmetrical matrix, you can set both off-diagonal elements in the upper- and low-triangulars as below:

Kxx = [
sdx^2 sdxy  sdzx
sdxy  sdy^2 sdyz
sdzx  sdyz sdz^2
]

Please, check if the covariance matrix (CM) reconstructed makes sense, for example:

1. checking the structure of the CM. It should be symmetrical and the main diagonal elements always positive numbers.
2. calculating the correlations from it, for example,

rho_xy = Kxx(1,2) / sqrt( Kxx(1,1) * Kxx(2,2) )
rho_yz = Kxx(2,3) / sqrt( Kxx(2,2) * Kxx(3,3) )
rho_xz = Kxx(1,3) / sqrt( Kxx(1,1) * Kxx(3,3) )

If the values of rho_xy, rho_yz, and rho_xz ​are between -1 and 1 the CM seems O.K.. Otherwise there are mistakes and you have to check what is wrong.

PS: Could you please inform which RTKLib version are you using in your experiments? I had the same need and I could notice that the information related to the "covariances" from rnx2rtkp are suspect, maybe misreported by the RTKLIB-2.4.3-b34. I hope you fixed it, ie. taking the square of all covariances and preserving the signal of those are negative. Check the source code and compare it with the theory related to covariance matrices : )