Partial Derivatives of Geometric Path Distance with Respect to the State Vector (x, y, z, x, y, zi) of the GPS Receiver The signal transmitting path is described by (cf. Eqs. 4.3 and 4.6 in Chap. 4)
where index k denotes the satellite, and the satellite coordinates are related to the signal emission time
i denotes the station, and the station coordinates are related to the signal reception time
Then one has
where the satellite state vector is related to the time
and
Partial Derivatives of Geometric Path Distance with Respect to the State Vector
of the GPS Satellite Similar to above, one has
Partial Derivatives of the Doppler Observable with Respect to the Velocity Vector of the Station
The time differentiation of the geometric signal path distance can be derived as
then one has
Partial Derivatives of Clock Errors with Respect to the Clock Parameters
If the clock errors are modelled by Eq. 5.163 (cf. Sect. 5.5)
where i and k are the indices of the clock error parameters of the receiver and satellite, then one has
If the clock errors are modelled by Eq. 5.164 (cf. Sect. 5.5)
then
The above derivatives are valid for both the code and phase observable equations. For the Doppler observable, denote (cf. Eq. 6.3)
then for the clock error model of Eq. 6.22 one has
Partial Derivatives of Tropospheric Effects with Respect to the Tropospheric Parameters
If the tropospheric effects can be modelled by (cf. Sect. 5.2)
where
is the tropospheric effect computed by using the standard tropospheric model,
are parameters of the tropospheric delay in path, zenith, azimuth directions, and F and Fc are the mapping and co-mapping functions discussed in Sect. 5.2. The derivatives with respect to the parameters
are then
Furthermore, if the tropospheric parameters are defined as a step function or first order polynomial (cf. Sect. 5.2) by
where
are the beginning and the ending times of the GPS survey, and
is usually selected by two to four hours. Then one has
The azimuth dependency may be assumed to be (cf. Eq. 5.121)
where a is the azimuth, and
are called azimuth-dependent parameters. Then one gets
If parameters
are also defined as step functions or first order polynomials like Eq. 6.30, the partial derivatives can be obtained in a similar manner to Eq. 6.31.
Partial Derivatives of the Phase Observable with Respect to the Ambiguity Parameters
Depending on which scale one prefers, there is
Partial Derivatives of Tidal Effects with Respect to the Tidal Parameters
If the Earth tide model in Eqs. 5.147 and 5.149 are used, then the tidal effects can be generally written as
where
are the coefficient functions, which are given in Sect. 5.4.2 in detail, and
are the love numbers and Shida number, respectively. Then one has
Ocean loading tide effects can be modelled as
where
is the factor of the computed ocean loading effect vector
Then one has
