The tropospheric delay may be decomposed into two components: the dry (or hydrostatic) delay and the wet (or non-hydrostatic) delay. The dry component usually consists of 90% of the total delay and varies with the local temperature and atmospheric pressure [122] whereas the wet component is generally smaller (from a few millimeters in arctic regions to forty centimeters in tropical regions) but more variable.
The Slant Tropospheric Delay (STD) for a satellite signal received at any elevation angle can be computed by using following equation defined in [61] and [62]:
𝑆𝑇𝐷 = 10−6∫ 𝑁𝑑𝑠 Equation 147-STD
Where N defines the refractive index and ds is the differential form of path length over which the integration is performed. The integration is over the entire signal path length, although the tropospheric delay is understood to be zero for the path outside the troposphere. Equation 147 can be decomposed into 2 components the dry part Slant Hydrostatic (or dry) Delay denoted 𝑆𝐻𝐷 and Slant Wet Delay denoted 𝑆𝑊𝐷 as:
𝑆𝑇𝐷 = 𝑆𝐻𝐷 + 𝑆𝑊𝐷 = 10−6∫ 𝑁
𝑑𝑑𝑠 + 10−6∫ 𝑁𝑤𝑑𝑠
Equation 148 - Decomposition of STD
Where 𝑁𝑑 and 𝑁𝑤 are respectively the dry and the wet refractivities defined by the following equations [123]: 𝑁𝑤= (
𝑘2𝑒
𝑇 +
𝑘3𝑒
𝑇2) × 𝑍𝑤−1
𝑁𝑑= 𝑘1𝑃
𝑇 × 𝑍𝑑−1
Equation 150 - Dry refractivity
Where 𝑘1, 𝑘2and 𝑘3are constants, 𝑃 is the partial pressure of the dry air, 𝑇 is the temperature and 𝑒 is the partial pressure of water vapor, 𝑍𝑑 and 𝑍𝑤 are the compressibility factors for dry air and water vapor respectively. Figure 70 considers a troposphere split into layers (as defined in NWMs for example).
Figure 70-Illustration of a troposphere split into layers
Therefore, by applying this previous decomposition, an approximation of the STD may then be formed from a summation of the refractive indices at these 𝐿 levels (e.g. 10 levels shown in Figure 58).
𝑆𝑇𝐷 ≈ 10−6𝑂(𝜃)𝑑 ∑ 𝑁
𝑖 𝐿 𝑖=0 Equation 151 - STD over layers
Where the refractivity index 𝑁𝑖 is a function of the atmospheric conditions at that level 𝑇𝑖, 𝑅𝐻𝑖 and 𝑃𝑖, 𝑂(𝜃) is the obliquity factor as a function of the elevation 𝜃 (as defined in Equation 13) and 𝑑 is the height between layers. In the event of perfect horizontal spatial correlation such that the atmospheric parameters 𝑇𝑖, 𝑅𝐻𝑖 and 𝑃𝑖 and thus the refractive indices 𝑁𝑖 are independent of 𝜃, then the Zenith Tropospheric Delay (ZTD) is given as.
𝑍𝑇𝐷 ≈ 10−6𝑑 ∑ 𝑁 𝑖 𝐿 𝑖=0
Equation 152 –ZTD over layers
The relationship between the surface parameters 𝑇𝑠, 𝑅𝐻𝑠 and 𝑃𝑠 and the atmospheric parameters throughout the troposphere can be approximated through the use of vertical profile which enables to get for example 𝑇𝑖 from 𝑇𝑠. Alternatively the total refractivity along the path can be approximated using an empirical model in terms of the atmospheric parameters at the receiver (often at the surface).
𝑍𝑇𝐷 ≈ 𝑓(𝑇0, 𝑅𝐻0, 𝑃0)
Equation 153 -Empirical Model for ZTD
Based on these notions, the following subsections outline different methodologies for determining the STD which have been used during this research, firstly taking into account the surface atmospheric data and then NWM data over layers.
5.3.2 2D Empirical Model
This methodology is the simplest for computing STD. It consists at first of estimating the Zenith Tropospheric Delay (ZTD) before applying the mapping function, for example as found in [16]:
𝑆𝑇𝐷 = 𝑂(𝜃) × 𝑍𝑇𝐷
Equation 154 - Relation between STD and ZTD
The ZTD is obtained by using surface atmospheric parameters of Pressure, Relative Humidity, and Temperature. Vertical profiles introduced in 5.3.1 are assumed for deriving these three parameters. Different empirical models have been derived [68] [62] [122] [61] [124] [125] [115], also based on splitting the delay into the Zenith Wet Delay (ZWD) and Zenith Hydrostatic (dry) Delay (ZHD) components. ZHD can be computed through one such model, the Saastamoinen model [66]:
𝑍𝑇𝐷𝑠𝑎𝑎𝑠 = 0.002277 × [𝑃 + ( 1255
𝑇 + 0.05) × 𝑒]
Equation 155 - Saastmoinen
Where 𝑃 is the Pressure, 𝑇 is the Temperature and 𝑒 is the partial pressure of water vapor related to the relative humidity by the Clausius-Clapeyron equation [61]:
𝑒 = 𝑒𝑠× 𝑅𝐻 100
where is the saturation water vapor pressure. These parameters are often taken at the surface, for a receiver located at or close to the surface.
The methodology for computing accurately ZWD by using surface data and a novel vertical profile approximation is detailed in [68].
Of the tropospheric delay models referenced above, the Modified Hopfield Model [125] is a practical choice for computations at the lower part of the atmosphere including both hydrostatic (dry) and wet components [7] [61] whilst the Saastamoinen model provides a good estimate for the upper dry atmosphere. These two models are thus used where appropriate, in particular, the MHM for the weather wall model approximation explained in section5.2.1 by Figure 54.
In this section, tropospheric delays were derived using only surface atmospheric data. When the atmospheric parameters are available at both the surface and at multiple layer it should be possible to use another model to more accurately estimate this tropospheric delay. The following section describes such a methodology.
5.3.3 3D layered Model
As already mentioned, in the previous section 5.3.2 only surface atmospheric data was used, whilst in NWMs the atmospheric parameters are available at both the surface and at multiple layers. It is therefore possible to use this 3D data model to more accurately determine the total tropospheric delay. An integration of the refractive index as suggested in subsection 5.3.1 is then possible (Figure 70), either directly in the slant domain (through ray tracing) or in the zenith before application of the mapping function. The Saastamoinen model [66] (Equation 8) is used for the near-negligible component above the highest NWM layer.
The numerical integration process uses a variable path increment length as the variation of the atmospheric parameters at higher altitudes is smoother and more predictable [126]. In fact, the NMW layers are also more tightly spaced near the ground to capture this greater variability. Interpolation is used to obtain the values of the atmospheric parameters at points along the signal path. Interpolation in the vertical domain is linear in the case of temperature and relative humidity and exponential for the pressure. Interpolation in the horizontal domain is performed through a spherical distance weighted averaged. Figure 71(a) shows vertical interpolation of atmospheric parameter A between layers i and i+1 for a NWM grid point which does not require horizontal interpolation. Figure 71(b) illustrates the additional horizontal interpolation required to obtain the atmospheric parameters between NMW layers at an arbitrary point.
Figure 71-(a) -Vertical Interpolation - (b) Additional Horizontal Interpolation
The numerical integration process differs slightly between the ZTD and STD cases [61] .The path increment is scaled according to the elevation so to have approximately the same number of integration points per layer and avoid excessive computation time. Another methodology is to find ZTD numerically by integration and then apply the mapping function as per Equation 154.
By using the Arome NWM data, each of these parameters (Height on Figure 72, Pressure on Figure 73, Temperature on Figure 74 and RH on Figure 75) are represented for illustration purpose in a plane format as a function of longitude (for a fixed latitude) over the path length for a satellite elevation angle of 5°:
Figure 73-Interpolation of Pressures
Figure 75-Interpolation of RH
Observing these figures, note that the number of integration points is greater at the beginning of the “ray- tracing” algorithm since the path increment length varies. Also, as it could be expected, Height parameters in Figure 72 and Pressure parameters in Figure seems to “linearly” increase and decrease respectively during the ray-tracing algorithm. Furthermore, the exponential vertical profile of the Pressure can also be remarked through this Figure 73. The temperature in Figure 74 decrease until a highest level, decrease a little bit and seems constant at the end of the interpolation. This observation is in accordance with the behavior of the Height level temperatures (linear vertical profile of the temperature). By seeing at the Figure 75, it can be noted that this relative humidity parameters appear to be the less predictable and stable parameters during the ray tracing algorithm. Therefore by viewing the Equation 149, the dependency of the wet component of the troposphere to the relative humidity, explains the fact that this wet component is also the more difficult part of the tropospheric delay to model.
Once each parameter is obtained, wet and dry refractivities can be computed over the path length as explained in 5.3.1 through Equation 149 and Equation 150. These values are represented in the following figure.
Figure 76-Interpolation of Refractivities
These refractivities parameters are used to compute the tropospheric delays as explained through the Equation 148. These single delays can then be combined to estimate the differential range tropospheric delays over paths 2 and 3 as presented in Figure 1. Also because in this Chapter 5, the focus is made on the worst case conditions, the following section details the research and the methodology to obtain the worst differential range Tropospheric Delay.