The development of the model involves the harmonization of two parallel input components: the value of the geodetic deviation of the user’s position caused by the tropospheric error and the temporal-spatial harmonization of the corresponding meteorological input data.
2.4.1. Determination of the Size of the Tropospheric Error
Basically, the determination of the tropospheric error by measuring the pseudorange can be conducted in two ways:
(i) by determining and removing all systematic and random errors; and
(ii) by determining the deviation from the known position using a preselected tropospheric model and estimating the deviation residuals as unknown parameters [
25].
In the first method, the pseudorange is calculated according to the general formula [
10]:
where
is the pseudorange from position
A to satellite
i;
is the geometric distance;
is l speed of light;
is ionospheric and tropospheric delay;
is the satellite orbit and clock error;
is the multipath trajectories error;
is relativistic error; and
is the random error.
The multipath error
can be determined programmatically, usually based on the receiver and antenna equipment manufacturer. Sources of ionospheric delay, relativistic errors, satellite orbit errors, and clock errors are included in the navigation message or, in the case of post-processing, from the appropriate ground truth data. In any case, it is necessary to perform software processing to ensure appropriate data sources are used for additional corrections. For additional verification, the values of the isolated and determined tropospheric delays of a known position were compared with the values determined using the radiosonde signals or another system [
10].
The model development was based on the determination of the user’s position in accordance with the selected modeling approach and its initial limitations. It was obtained by determining the pseudoranges and isolating the accuracy deviation from the user’s position caused by the tropospheric error within the total positioning error. A position determined in this way can be defined in a simpler form as the difference between the signal reception time which is determined based on its clock and time (determined by the satellite clock), which can be represented with the following expression [
4,
44]:
where
is the pseudorange of the i-th satellite;
is the time of the signal reception determined by the receiver clock (s); and
is the time of signal transmission determined by the satellite clock (s).
The misalignment of the satellite and receiver clocks (with input data contained in the navigation message), the ionospheric, tropospheric, and measurement error, and the expression (2) takes the form (3) by introducing the parameters of the geometric distance between the antennas of the satellite and the receiver:
where
is the geometric distance between the satellite and receiver antenna;
is the receiver and satellite clock offset;
is the ionospheric error;
is the tropospheric error; and
is the measurement error.
The conversion of the geodetic position into a data set in ECEF ETRS89 format with values expressed in meters is expressed according to the following expressions:
where
is the geodetic longitude;
is the geodetic latitude;
is the ellipsoidal height;
is the semi-major axis of the reference Earth ellipsoid (6,378,137 m)
is the first numerical eccentricity of the ellipsoid; and
is the flattening coefficient of the reference Earth ellipsoid.
The Saastamoinen model of tropospheric zenith correction was used, and the present parameters were transmitted in the navigation message as a common source of input program corrections for all measuring stations and periods and for standardizing other program settings. The input program parameters for pressure, absolute temperature, and partial pressure are determined by the expressions of the standard atmosphere model [
44]:
where
is the total air pressure (hPa);
is the temperature in degrees Kelvin;
is the partial atmospheric pressure (hPa);
is the geodetic altitude above sea level; and
is the relative humidity. The applied Saastamoinen model uses a constant relative humidity value of 70%.
The tropospheric correction in this configuration was calculated using the following algorithms [
45,
46]. For the mapping function in the selected program setting, the Niell model is calculated according to the expression [
47,
48,
49]:
where
is the signal elevation angle;
is the signal zenith angle;
is the tropospheric gradient in the north direction;
is the tropospheric gradient in the east direction; and
is the non-hydrostatic mapping function of the non-hydrostatic Niell (New) Mapping Functions (NMF).
The total tropospheric delay was calculated according to the following expression:
where
is the total tropospheric delay;
is the hydrostatic component of zenith tropospheric delay (in meters and determined by the Saastamoinen model);
is the tropospheric total zenithal retardation; and
is the (NMF) hydrostatic mapping function. The parameters of tropospheric gradients and total tropospheric delay in zenith direction were estimated using the extended Kalman filter (EKF) [
45]. The parameters for the correction of the ionospheric delay (A—the numerical coefficient of the maximum total free electron content in the ionospheric layer F2; F—the index of solar activity; and Ap—the daily index of geomagnetic activity) included in the GLONASS navigation message (broadcast ionosphere model) [
47] have the following form:
The determination of ionospheric delay
^ (m) was performed using Klobuchar’s model due to single-frequency computing. Despite the efficiency of Klobuchar’s model, more effective reducing of the ionospheric delay would be achieved using an iono-free combination or dual-frequency receiver. However, considering the format of available input data, a single-frequency receiver with the Klobuchar model was used. Input data regarding clock parameters and ephemerides were sent in the navigation message (broadcast ephemerides and clock parameters) and used in calculations in the following form [
44,
46,
50]:
In addition, the initial software setup included a single positioning mode and a 3° elevation mask value (due to the scope of applicability of the NMF mapping function) to isolate the deviation in the geodetic accuracy of the user’s position caused by the tropospheric error component.
2.4.2. Statistical Analysis and Model Development
The remaining value of the tropospheric error was determined based on the difference in the geodetic accuracy of the position determined both with and without tropospheric correction (using the Saastamoinen model). It is important to emphasize that, in addition to the tropospheric component, such determined deviations still contain several unmodeled systematic and random errors, including the residual ionospheric error, the satellite position and clock error, errors related to multipath, and solid tides. However, the only input difference when comparing the accuracy of the geodetic position was the application of the Saastamoinen model of tropospheric correction (as the other applied algorithms were identical); therefore, the effect of the resulting final difference between the two models can be deterministically attributed to the tropospheric component within the total geodetic position error.
The difference in geodetic accuracy of the user’s position can be analyzed as a function of the influence of the applied Saastamoinen model of tropospheric delay correction since all other parameters were set identically in both cases. This provided the theoretical basis for quantifying the effect of the Saastamoinen model on improving the geodetic accuracy of the observed GNSS positions.
The resulting RTE value of the known position caused by the non-modeled part of the tropospheric delay was subjected to a statistical regression procedure to determine the correlation with real meteorological surface parameters. Statistical correlation with its positive parameters are the basis for developing a mathematical model to reduce RTE as a function of surface meteorological parameters (as independent input variables).
The stages in the creation of the proposed model are shown in
Figure 2.
Meteorological independent input variables are as follows: is the temperature (°C); is the precipitation (mm); is the relative humidity (percentage); is the precipitation water (mm); is the pressure (hPa); and , , and are deviations along the x, y, and z axes in the proposed model (m).
The proposed model contains a mathematical expression for each axis as the observed deviation from the exact geodetic position followed the ECEF coordinate system. The final form of the proposed model is shown as follows. The amounts of the coefficients of the input predictors are the result of the regression analysis that describes the form and intensity of the mutual connection between the meteorological input predictors and the output variable RTE. For the
X axis, the model component
has the form:
For the
Y axis, the model component
has the form:
For the
Z axis, the model component
has the form:
The final form of the proposed model (
) is based on an extension of the existing Saastamoinen model and represents the sum of the corrections made by the Saastamoinen model and the proposed model for each axis:
where
is the correction value realized by the Saastamoinen model (in m).