4.1. Synthetic Attenuation Field
The synthetic cloud data used in this paper were produced from an idealized simulation of a mesoscale convective system using the WRF model. More details about the simulation can be found in Ref. [
12]. This cumulus type cloud has very large vertical development, the cloud base is approximately 5 km in height, with the top around 10 km. One vertical slice of the cloud field that covers 80 km in width and 0–12.5 km in height was selected as the reference field. This slice contains a very high content of water, which mainly consists of supercooled cloud water. Some areas can produce specific attenuation up to 2.35 dB/km, while there are other areas with relatively thin clouds. Very few rainwater droplets are present in this slice. A 320 × 50 grid is used to define the field with each unit square being 250 m by 250 m, i.e., both the horizontal and vertical resolutions are 250 m.
In the cloud LWC field generated by WRF, each unit square has a uniform cloud water mixing ratio given in g/g, which is defined as
where
and
are the liquid water and dry air density in g/m
3, respectively. Note that other particles such as ice crystals and snow are not considered as they have minimal effects on the attenuation of microwave signals around 30 GHz [
8].
Dry air density is obtained by [
23]
in which
is the cloud liquid water temperature in Kelvin and
is the dry air pressure in kPa. We use the barometric law [
24] to calculate
:
where
is the altitude above sea level,
is static pressure at sea level (101.325 kPa),
is gravitational acceleration (9.80655 m/s
2),
is molar mass of Earth’s air (0.0289644 kg/mol) and
is universal gas constant (8.3144598 J/(mol·K)). For each unit square in the cloud field, temperature
is calculated according to the altitude of the center of the square, assuming that the temperature lapse rate
is −0.0065 K/m from the sea level temperature
of 288 K. Note that the constant lapse rate is an approximation for the cloudy atmosphere and will introduce an inaccuracy in the temperature profile. Nevertheless, it only serves as a means to translate the WRF-generated LWC field into the reference attenuation field, and using it does not affect the validation of the retrieval method per se.
In order to generate the reference attenuation field, we use the equations provided by the International Telecommunications Union (ITU) [
25], where the specific attenuation within a cloud is obtained by
where
is the cloud liquid water specific attenuation coefficient (details can be found on Page 2 in [
25]) in (dB/km)/(g/m
3),
is the signal frequency in GHz. With the temperature and pressure profile ready for each unit square,
can be calculated using Equations (30) and (31) and the specific attenuation
can then be calculated by Equation (33) with the signal frequency
GHz. The reference attenuation field is shown in
Figure 3a.
4.3. Retrieval Using Rectangular Basis Functions
In this section, we use rectangular basis functions obtained by Equation (27) to retrieve the attenuation field. Here, is assumed to be 320 × 50 and width for each function to be 250 m. In other words, the retrieved attenuation field is defined by a grid exactly the same as that of the reference field.
Theoretically, parameter
for the differential approach in Equation (21) needs to be small so that the assumption that
stays static can be met. However, simulations show that
will generally make the differential distances (
) very close to zero and thus make the algorithm sensitive to noise and estimation errors. After some experiments, parameter
is set at 3, which seems to produce the best retrieval outcomes. After calculating the differential matrix
, a constrained least-squares algorithm using an interior-point method (lsqlin in Matlab, see [
26,
27]) is employed. The upper and lower bounds for
are set to 5 and 0, respectively. In addition, we assume that there being no cloud in the bottom 320 × 20 unit squares is prior knowledge. In all of the following simulations, therefore, the weights for the basis functions below 5 km are considered zero and only the weights for the basis functions above 5 km (320 × 30 basis functions in this case) are retrieved.
Figure 3b shows the retrieved attenuation field obtained by the least-squares algorithm. Note that the retrieved specific attenuation is greater than the reference value (overestimation) for many of the unit squares. This is because, in the first iteration of the algorithm, by assuming
in Equation (13) to be zero, the impact of
is included in the path-integrated attenuation
and makes it effectively larger. The retrieved attenuation field can be used to update
and therefore enable the second iteration of the least-squares. The second retrieved field will be underestimated because
is overestimated, as the attenuation field is overestimated in the first iteration. As a result, the retrieved fields will oscillate between iterations, but averaging the attenuation fields of two consecutive iterations will yield satisfactory results for rain field estimation [
6].
Figure 3c shows the final retrieved field by averaging the third and fourth iterations. It can be seen that, for both heavy cloud areas where there is large vertical development and thin cloud areas, the retrieved field and the reference field have good agreement.
Assuming that the ground receiver antenna has high directivity, we can simplify the relation between the sky noise and the path-integrated attenuation by assuming that the sky temperature picked up by the antenna is from one direction only, i.e.,
As a result, from Equations (8) and (34), we can conclude that
increases with
in a nonlinear manner. The nonlinear relation can be observed in
Figure 4, where
(red line),
(blue dashed line) and
(black line) are plotted over the entire course of the overpass. The left panel is for the receiver located at 40 km and the right panel is for the receiver located at 60 km. The ratio of
is large for small
(e.g.,
when
dB) and is small for large
(e.g.,
when
dB).
For the iterative approach that generates
Figure 3c, the least-squares computing needs to be run four times, and each time it deals with
equations. The relation between
and
in the above can be utilized to reduce the number of the least-squares computing to one time only, thereby reduce the computing complexity of the retrieval process. We propose the following piecewise linear (PL) function to process the retrieved
after the first least-squares computing:
where
The value of
for each interval is chosen empirically from multiple simulations to achieve the best retrieval outcome. The retrieved attenuation field by applying
on Equation (15) is shown in
Figure 3d, which is very similar to the retrieved field by averaging the third and fourth iterations.
In reality, the basis functions can only be an approximation of the actual field. Different types of basis functions will introduce different levels of errors into the retrieval model. To investigate the effects of larger rectangle basis functions, we consider the retrieved attenuation field represented by 160 × 25 rectangular basis functions for the second set of simulations. The width of each function (
) is 500 m, i.e., each unit square is now four times as big as before in area. We use the same PL function to process the retrieved
. The final retrieved attenuation field is shown in
Figure 3e. It can be seen that similar results are achieved but with poor spatial resolutions.
As a global index of the performance for the retrieval, the root-mean-square error (RMSE) between the retrieved attenuation field and the reference field has been considered, which is defined by
where
denotes the total number of comparison points,
and
are the specific attenuations at a particular comparison point for the reference field and the retrieved field, respectively. For the following results, the comparison points are assigned to the centers of the top 320 × 30-unit squares in the reference field. The RMSE for the retrieved attenuation field after the first iteration (
Figure 3b) is 0.323 dB/km. Averaging the third and four iterations (results shown in
Figure 3c) reduces the RMSE to 0.101 dB/km and the PL function (results shown in
Figure 3d) also brings the RMSE down to 0.107 dB/km. The RMSE for the retrieved attenuation field with enlarged rectangular basis functions (
Figure 3e) is 0.215 dB/km.
4.4. Retrieval Using Gaussian Basis Functions
In this set of simulations, we replace the 160 × 25 rectangular basis functions in
Section 4.2 with the same number of Gaussian basis functions, with the center location of each function unchanged. The width of the Gaussian functions (
) is set to 1.5 km.
To compute the integral in Equation (17), let us divide the sample link
into a large number (
) of segments of the same length
. Each segment is so small in relation to the width of the Gaussian function that
can be regarded constant for anywhere on a segment. Consequently, Equation (17) becomes
where
is the coordinates of the center of the
i-th segment of sample link
.
After the differential matrix
is calculated, the same least-squares algorithm is employed to retrieve
. However, intuitively, a different PL function
needs to be adopted because of the different nature of the Gaussian basis function, compared with the rectangular basis function. Multiple simulations suggest that a simple PL function such as the following would generate satisfactory results:
A contour map of the final retrieved field after the PL function is applied is shown in
Figure 5a. It has good agreement with the reference field. The RMSE of this field is 0.207 dB/km, which is smaller than the rectangular case in
Figure 3e.
Examining the change in specific attenuation along a vertical line shows in detail the performance of different retrieval methods. For instance,
Figure 5b is for a vertical line located at
km (shown by a black dashed line in
Figure 5a) where the cloud is relatively thin. The green line shows how the specific attenuation changes with height in the reference field. The blue and red lines are for the retrieved fields using the PL function with respectively small and large rectangular basis functions. For the retrieved field with Gaussian basis functions (
Figure 5a), the specific attenuation is plotted in black. The same plots are done for heavier clouds (vertical line located at
km, marked by a black dashed line in
Figure 5a), shown in
Figure 5c. For all three types of basis functions, the retrieved lines are able to closely follow the reference line and the level of error is the lowest for small rectangle basis functions. When the Gaussian basis functions are employed, as they are continuous, continuity is guaranteed in the retrieved field. The peak of the Gaussian retrieved line is generally lower than that of the reference line. This is partly due to the larger value of
chosen to achieve a lower overall RMSE and partly due to the number of basis functions of the retrieved field being less than the number of grids in the reference field.
In addition to the RMSE (Equation (37)), the Pearson correlation coefficient (PCC) is also employed to compare the retrieval results across different methods. PCC is defined by
where
is the mean of
and
is the mean of
.
The scatter plots of the retrieved specific attenuations against the reference values at the comparison points can be found in
Figure 6. Panel (a) is for the retrieved field by the averaging method (the result in
Figure 3c), and the PCC is 0.986. Panel (b) is for the retrieved field by the PL function (the result in
Figure 3d), with the PCC being 0.973. There are some horizontal gaps in the plot, indicating that discontinuity is present in the value range of the retrieved field (
). This is because the retrieved
after the first least-squares computing was reduced unevenly by the PL function. Panel (c) shows the scatter plot for the retrieved field in
Figure 3e, and the PCC is 0.883. The scatter plot for the retrieved field with Gaussian basis functions (
Figure 5a) is in panel (d), and the PCC is 0.892.
4.5. Effects of Larger Spacing among Receivers
Simulations were also carried out for more sparsely distributed receivers. For instance,
Figure 7a shows the retrieval result when there are 37 receivers and the distance between any adjacent two is 2.5 km. The same rectangular basis functions and the same PL function for
Figure 3d are employed. The number of equations will not be sufficient for the least-squares algorithm to function correctly if the number of signal samples stays at 400. Hence,
is increased to 800 for each receiver. Note that this only uses
of the satellite signals. The RMSE and the PCC are 0.135 dB/km and 0.956, respectively. While it can be seen that the retrieval outcome is not as good as that in
Figure 3d with 91 receivers, key features of the cloud can still be identified.
When the number of receivers is reduced to 19, which means that two adjacent ones are 5 km apart, the number of signal samples for each receiver is increased to 1600 in order to obtain a sufficient number of equations. Note that this only uses
of the satellite signals. The retrieval result (
Figure 7b) further degrades in spite of the increase of samples. The RMSE and the PCC are 0.210 dB/km and 0.891, respectively. For both simulations with 37 and 19 receivers, further increasing the number of samples does not improve the performance of the retrieval. This indicates that the density of the receivers plays a more important role than the number of samples in the accuracy of the retrieval results.
4.6. Partial Retrieval
The variations in cloud horizontal scale are often quite significant [
28]. Some cloud covers stretch over hundreds of kilometers and it may be difficult to measure the entire field at the same time. It would be of interest to see how the retrieval method would perform if only part of the cloud is reachable by the signal links. Thus, simulations were carried out for scenarios where ground receivers are restricted to smaller areas. For example, 11 receivers placed from
km to
km (marked by the green bar at the bottom of
Figure 8a) are used for retrieving the attenuation field. We firstly identify the targeted area (marked by the red dashed line in
Figure 8a) and define it using 120 × 30 (
m) rectangular basis functions. The final retrieved field after the PL function is shown in
Figure 8b. It can be seen that the middle section directly above the receivers has been retrieved successfully while for other areas the retrieved field is erroneous. The limited number of receivers with restricted locations cannot provide enough information to retrieve the left and right sections of the targeted area. Secondly, we redefine the target area (marked by the purple dashed line in
Figure 8a) with 30 × 15 Gaussian basis functions. The final retrieved field after the PL function is shown in
Figure 8c. The retrieved field in the middle section also agrees with its counterpart in the reference field, but the left area outside the 35 km mark and the right area outside the 45 km mark have a high level of errors.
The results above indicate that partial retrieval is possible when a limited number of receivers or signal links are available. Because most of the signal links have high elevation angles, the best retrieval performance is found in the area directly above the receivers. It is also implied that tomographic retrieval can be carried out in a sectioned manner. For a targeted area within the attenuation field, only the signal links relevant to it are needed for its retrieval.