**3.3.3 Impact of distance and height on reconstruction goodness**

Results described previously are good, but are related to the baseline scenario. Considering this scenario, the validation has been performed above NAND receiver, which is in a good position since its baseline from the COMO master station is between the nearest and the farthest stations. In this further analysis we have considered all the measurements (ZWDs or ZTDs) available from the MisT network (8 receivers) during the entire week (excluding only the COMO receiver, see Fig. 3). Then we have excluded data (ZWDs or ZTDs) observed by one receiver per time, keeping such data as reference for the self-consistency validation purpose for that receiver. For each case we have run our tomographic reconstruction considering all the 168 Hourly ZWDs (or ZTDs) available per each station for the October week, mapping them into the slant directions and including also low elevation observations (following the procedure described in paragraph 3.3.2). The obtained 168 Wet Refractivity Maps (considering ZWDs as input to the tomography) or Total Refractivity Maps (considering ZTDs as input) have then been used to evaluate the ZWD and ZTD estimates above the reference receiver, which are compared with the ZWD and ZTD observations above that receiver. This analysis has been repeated for each receiver of the MisT network.

Root Mean Squares of ZWDs and ZTDs differences (measured-estimated after reconstruction) are then reported in function of the distance of the station from the COMO master station or in function of the height of the station. Such results are plotted in Fig. 5. The same analysis has been performed considering data taken by the MisT network extended to the two mountainous receivers during 12 October from 9:00 AM to 7:00 PM

Results related to this analysis are summarized in Table 3. They confirm the importance of the availability of low elevation measurements issued from different altitudes to improve the estimation of vertical refractivity gradients in such a tomographic approach. It has to be noted that the availability of external independent information (atmospheric models or, better, meteorological data) for modelling the SWD component of low elevation observations in the outer volume seems to be necessary in this case. Because of the MisT network design (receivers not homogeneously distributed in the inner volume), the internal procedure based on the coarse tomographic reconstruction (case a)) is not very effective.

Table 3. Self-consistency results considering SWD derived by low elevation observations (taken during the October week) after the application of the outer volume wet refractivity modelling strategies a., b. and c.. Results are relative to the statistics of ZWD errors

(measured-estimated after reconstruction) over the NAND reference station. Results related to the baseline scenario are reported in the first column as a reference. In the last column the

Results described previously are good, but are related to the baseline scenario. Considering this scenario, the validation has been performed above NAND receiver, which is in a good position since its baseline from the COMO master station is between the nearest and the farthest stations. In this further analysis we have considered all the measurements (ZWDs or ZTDs) available from the MisT network (8 receivers) during the entire week (excluding only the COMO receiver, see Fig. 3). Then we have excluded data (ZWDs or ZTDs) observed by one receiver per time, keeping such data as reference for the self-consistency validation purpose for that receiver. For each case we have run our tomographic reconstruction considering all the 168 Hourly ZWDs (or ZTDs) available per each station for the October week, mapping them into the slant directions and including also low elevation observations (following the procedure described in paragraph 3.3.2). The obtained 168 Wet Refractivity Maps (considering ZWDs as input to the tomography) or Total Refractivity Maps (considering ZTDs as input) have then been used to evaluate the ZWD and ZTD estimates above the reference receiver, which are compared with the ZWD and ZTD observations above that receiver. This analysis has been repeated for each

Root Mean Squares of ZWDs and ZTDs differences (measured-estimated after reconstruction) are then reported in function of the distance of the station from the COMO master station or in function of the height of the station. Such results are plotted in Fig. 5. The same analysis has been performed considering data taken by the MisT network extended to the two mountainous receivers during 12 October from 9:00 AM to 7:00 PM

evident outliers due to measurements (see blue dots in Fig. 4) were removed.

**3.3.3 Impact of distance and height on reconstruction goodness** 

receiver of the MisT network.

interpolation of the meteorological data).

outer volume (this was done by a bilinear space interpolation and a linear time

(only 10 Hourly averaged ZWDs or ZTDs observations are contemporaneously available to any receivers of the "extended" network). In this case results are shown in Fig. 6.

Fig. 5. rms of the differences between ZWDs (blue dots) or ZTDs (red dots) observed and estimated above each reference receiver, excluding data of that receiver from the input dataset before the reconstruction. All data observed by the MisT network during the entire week are taken into account. (Left) rms are plotted against the distance of the reference receiver from COMO master station. (Right) rms are plotted against the height of the reference receiver above WGS84. The degraded results obtained excluding BRUN receiver (which is the highest one) are highlighted.

Fig. 6. Like Fig. 5, but considering all data observed by the MisT network and by the two mountainous receivers during the 10 hours of 12th October, 2008.

First of all this analysis confirms the impact of a good height displacement of receivers in the network. Even if MisT network topography has not been optimized for the geography of the analyzed area and for tomographic applications, if we consider the impact of height in the evaluation of propagation delays, we can say that the lack of receivers placed at higher altitudes will worsen final results. In particular, considering the original MisT network, where all the receivers are more or less placed in the same layer of the map (Fig. 5) we want to highlight that, if data observed at the highest receiver (namely BISB, which is placed in another vertical layer) are not given in input to the tomography, the rms of the difference between estimated and measured zenith delays (both Wet and Total) is generally doubled.

GNSS Signals: A Powerful Source for Atmosphere and Earth's Surface Monitoring 187

Fig. 7. Time series of ZWD obtained integrating ECMWF (red) collocated and estimated with tomography (blue) wet refractivity maps. Left: October week data; right: November week observations. The black numbers shown the column "number" inside the map.

Even if our main goal was to demonstrate the effectiveness in adopting tomographic reconstruction procedures for the evaluation of propagation delays inside water vapour fields, the real water vapour vertical variability and its time evolution is also well reproduced. Fig. 8(bottom) shows the time evolution of wet refractivity vertical profiles evaluated in the map centre (voxel 11 – see Fig. 7) during the overall October week, considering data taken by all the available MisT receivers. Unfortunately, no meaningful

Fig. 8. Time evolution of wet refractivity distribution evaluated in the central column of the map (voxel 11) during the overall October week, considering data taken by all the available MisT receivers. Top: integrated wet refractivity along zenith (namely the ZWD time series). Bottom: vertical wet refractivity profile (measured in N-units) evolution (Heights are given

in meters).

Things are better if we consider the MisT network plus the mountainous receivers. In this case the worsening is not so emphasized, since it is compensated by other receivers placed at similar altitudes (see Fig. 6). In both cases it seems that the results worsening follows a (more than) linear rule. It is absolutely not clear why the effects on the evaluation of Wet delays and Total delays are inverted, considering or not considering the mountainous receivers. It has to be noted that the network solution obtained for the mountainous receivers is not as accurate as that obtained for the other receivers, since the mountainous sensor positions have not been fixed. Moreover, results reflect 10 hours of observations instead of the entire week.

As far as the impact with distance is concerned, it is quite difficult to identify a clear relationship with results. Obviously if we exclude data observed by the nearest receivers (ANZA or CAST) to the reference one (COMO), results are better (rms is halved considering both the weekly data of the original MisT network and the 10 hours data of the MisT network plus mountainous receivers) than that we can obtain excluding one of the other (farther) receivers. But for all the other cases, it seems that final results are insensitive to distance. It is a surprising result since we expected a certain error correlation with distance. But the farthest receivers (MGRA and DANI) are placed in opposite positions with respect the map center and are the southest receivers (see Fig. 3). If we take into account low elevation observations (even if such observations are averaged, since they are obtained simply mapping hourly averaged Zenith observations into slant directions), rays related to the northern receivers (all the others) anyway interest the atmospheric volume above the southest receivers (and not viceversa, given the orbital positions of GPS satellites). And this could probably compensate the "distance" effect. Anyway, also in this case, further analysis and measurements are necessary to better understand if there is a clear relationship.

### **3.3.4 Validation against independent data**

In order to assess the goodness of inferred wet refractivity fields in different points of the grid considering independent data, we also did a comparison of ZWDs obtained vertically integrating wet refractivity fields derived after tomographic reconstruction along each column of retrieved maps with those derived by ECMWF analysis co-located in the same points (and times), even if the ECMWF horizontal resolution (0.25°x0.25°) and time resolution (6 h) are too coarse with respect those characterizing our final maps.

Statistical comparisons were performed considering the 168 wet refractivity maps obtained using data observed by the reduced MisT network (plus NAND receiver) collected during the October week and considering the 144 maps obtained for the November one. Results are shown in Fig. 7, where the time series of both ZWDs estimated after tomographic reconstruction (blue lines) and evaluated using ECMWF data (red lines) are plotted for each column of our volume discretization. We classified the areas accordingly to the corresponding rms values (computed for each ZWD difference time series, after the average bias removal) using green, yellow and red colors. As expected, the northern part is where the agreement is worse. In that area we had no receiver and less satellites were in view in the north direction. On the other hands, in the southern area, agreement is better even if no receivers were present, thanks to the availability of a higher number of rays. The best area is obviously the central one.

Things are better if we consider the MisT network plus the mountainous receivers. In this case the worsening is not so emphasized, since it is compensated by other receivers placed at similar altitudes (see Fig. 6). In both cases it seems that the results worsening follows a (more than) linear rule. It is absolutely not clear why the effects on the evaluation of Wet delays and Total delays are inverted, considering or not considering the mountainous receivers. It has to be noted that the network solution obtained for the mountainous receivers is not as accurate as that obtained for the other receivers, since the mountainous sensor positions have not been fixed. Moreover, results reflect 10 hours of observations

As far as the impact with distance is concerned, it is quite difficult to identify a clear relationship with results. Obviously if we exclude data observed by the nearest receivers (ANZA or CAST) to the reference one (COMO), results are better (rms is halved considering both the weekly data of the original MisT network and the 10 hours data of the MisT network plus mountainous receivers) than that we can obtain excluding one of the other (farther) receivers. But for all the other cases, it seems that final results are insensitive to distance. It is a surprising result since we expected a certain error correlation with distance. But the farthest receivers (MGRA and DANI) are placed in opposite positions with respect the map center and are the southest receivers (see Fig. 3). If we take into account low elevation observations (even if such observations are averaged, since they are obtained simply mapping hourly averaged Zenith observations into slant directions), rays related to the northern receivers (all the others) anyway interest the atmospheric volume above the southest receivers (and not viceversa, given the orbital positions of GPS satellites). And this could probably compensate the "distance" effect. Anyway, also in this case, further analysis and measurements are necessary to better

In order to assess the goodness of inferred wet refractivity fields in different points of the grid considering independent data, we also did a comparison of ZWDs obtained vertically integrating wet refractivity fields derived after tomographic reconstruction along each column of retrieved maps with those derived by ECMWF analysis co-located in the same points (and times), even if the ECMWF horizontal resolution (0.25°x0.25°) and time

Statistical comparisons were performed considering the 168 wet refractivity maps obtained using data observed by the reduced MisT network (plus NAND receiver) collected during the October week and considering the 144 maps obtained for the November one. Results are shown in Fig. 7, where the time series of both ZWDs estimated after tomographic reconstruction (blue lines) and evaluated using ECMWF data (red lines) are plotted for each column of our volume discretization. We classified the areas accordingly to the corresponding rms values (computed for each ZWD difference time series, after the average bias removal) using green, yellow and red colors. As expected, the northern part is where the agreement is worse. In that area we had no receiver and less satellites were in view in the north direction. On the other hands, in the southern area, agreement is better even if no receivers were present, thanks to the availability of a higher number of rays. The best area is

resolution (6 h) are too coarse with respect those characterizing our final maps.

instead of the entire week.

understand if there is a clear relationship.

**3.3.4 Validation against independent data** 

obviously the central one.

Fig. 7. Time series of ZWD obtained integrating ECMWF (red) collocated and estimated with tomography (blue) wet refractivity maps. Left: October week data; right: November week observations. The black numbers shown the column "number" inside the map.

Even if our main goal was to demonstrate the effectiveness in adopting tomographic reconstruction procedures for the evaluation of propagation delays inside water vapour fields, the real water vapour vertical variability and its time evolution is also well reproduced. Fig. 8(bottom) shows the time evolution of wet refractivity vertical profiles evaluated in the map centre (voxel 11 – see Fig. 7) during the overall October week, considering data taken by all the available MisT receivers. Unfortunately, no meaningful

Fig. 8. Time evolution of wet refractivity distribution evaluated in the central column of the map (voxel 11) during the overall October week, considering data taken by all the available MisT receivers. Top: integrated wet refractivity along zenith (namely the ZWD time series). Bottom: vertical wet refractivity profile (measured in N-units) evolution (Heights are given in meters).

GNSS Signals: A Powerful Source for Atmosphere and Earth's Surface Monitoring 189

• Space based: the sensor is placed on board a LEO satellite (400-800 km) with the aim of monitoring the entire Earth surface assuring a global coverage of the acquired reflections, which may be detected also very far from coastal zones (i.e. in the middle of the ocean); the Doppler shift experienced by the signal is the largest achievable among

The shape and extension of the footprint of the reflections depends on: the surface roughness, the sensor height above the Earth surface, the elevation of the reflected ray, the

The footprint must be considered lying on a plane tangent to the Earth surface in the specular reflection point. The distance of the specular reflection point from the receiver

Inside the area interested by the reflection, the smallest resolution achievable from a geometrical point of view is determined by the cells generated by the intersections of the

Iso-delay lines are determined considering the points on the surface by which the reflected signal arrives at the receiver with the same delay. Generally speaking, these points are ellipses and are determined considering a single chip of the GNSS code as relative delay

Iso-Doppler lines are determined considering the hyperbolas on the surface where reflected signals come to the receiver with the same Doppler shift. The zero Doppler line is computed as the line passing through the receiver and orthogonal to its velocity direction (Martin-

Clearly, we cannot forget the antenna footprint, which acts as a filter in delay and Doppler on the surface looks. When the surface is smooth, the total power received is almost coming from the first Fresnel zone defined around the specular scattering point (Beckmann & Spizzichino, 1987). In this case, the computation of the cross-correlation between the reflected signal and the local GPS code replica gives a waveform simply delayed respect to the cross-correlation of

When the surface is rough non-coherent reflections are expected and the use of the Fresnel zone to model the received power is ineffective. In this case, the glistening zone represents

N scattering elements contained in the glistening zone are considered in determining the

1

=

*n*

( )

ΛΔ

 Δ

=⋅ − (13)

( ) ( ) *m n <sup>i</sup> p m n m n*

where *Λ* is the triangle cross-correlation function and the m index indicates the quantities referred to the modelled signal generated with the local GPS code replica. Through this formulation Rp becomes a summation of triangle functions weighted with the amplitude of the nth element scattered field and delayed accordingly to the phase shift associated to each nth scattering element. The final correlation function shape in this case is shown in Fig. 9.

*R t Ae t t* φ φ

<sup>∞</sup> <sup>−</sup>

computation.

the three described scenarios.

iso-delay and iso-Doppler lines.

Neira, 1993).

cross-correlation function

direction of the incidence plane respect to the receiver velocity.

nadir increases when the elevation of the GNSS satellite decreases.

associated to each ellipse respect to the adjacent one (Martin-Neira, 1993).

the direct signal, but with the same triangle shape and a noise floor around.

the source of scattered power (Beckmann & Spizzichino, 1987).

Δ

shift improves the resolution on the surface by means of iso-Doppler lines

meteorological events happened during the observing period. Anyway, an increase of wet refractivity (water vapour concentration) can be evidenced between the 100th (4 am, 16th October) and the 120th (midnight, 16th October) hours, with a peak around the 110th hour (2 pm, 16th October). The increase is well reproduced in terms of integrated wet refractivity along zenith (see ZWD evolution in Fig. 8(top)). Moreover, meteorological data (not shown here) confirmed an increase of cloud covering during that time interval.
