In this work, the effect of the observing geometry on the tomographic reconstruction quality of both a regularized least squares (LSQ) approach and a compressive sensing (CS) approach for water vapor tomography is compared based on synthetic Global Navigation Satellite System (GNSS) slant wet delay (SWD) estimates. In this context, the term “observing geometry” mainly refers to the number of GNSS sites situated within a specific study area subdivided into a certain number of volumetric pixels (voxels) and to the number of signal directions available at each GNSS site.
The novelties of this research are (1) the comparison of the observing geometry's effects on the tomographic reconstruction accuracy when using LSQ or CS for the solution of the tomographic system and (2) the investigation of the effect of the signal directions' variability on the tomographic reconstruction.
The tomographic reconstruction is performed based on synthetic SWD data sets generated, for many samples of various observing geometry settings, based on wet refractivity information from the Weather Research and Forecasting (WRF) model. The validation of the achieved results focuses on a comparison of the refractivity estimates with the input WRF refractivities.
The results show that the recommendation of

In this paper, we intend to determine the three-dimensional (3-D) atmospheric water vapor distribution for each point in time. This adds further essential information to the spatio-temporal analyses of two-dimensional (2-D) water vapor fields commonly used in weather forecasting and climate research. In addition, atmospheric water vapor delays the microwave signal propagation within the atmosphere and thus represents an error source in, e.g., Global Navigation Satellite System (GNSS) and Interferometric Synthetic Aperture Radar (InSAR) observations. Therefore, a precise knowledge of the water vapor field, for example, is required for accurate deformation monitoring using InSAR (

One of the main limiting factors in water vapor tomographies consists of the point-wise GNSS observing geometry, which causes an ill-conditioned inverse tomographic model that needs to be regularized. Yet, even after regularization, the observing geometry composed, e.g., of the number and the geographic distribution of the GNSS sites, the SWD signal directions, and the voxel discretization still affect the quality of the tomographic solution. This work therefore meets the challenge of comparing the observing geometry's effect on a GNSS-based water vapor tomography solved by means of least squares (LSQ) or by means of compressive sensing (CS). By investigating the observing geometry's effect on the LSQ and CS solution strategies, the differences between the LSQ solution and a CS solution approach benefiting from the signal's sparsity in an appropriate transform domain for regularization are better understood, and recommendations can be given for future water vapor tomography campaigns and the processing of their measurements. Based on synthetic data sets deduced from the Weather Research and Forecasting (WRF) model described in

Current water vapor tomographies can be distinguished, e.g., based on the methodology and the data sets applied for solving the tomographic model.
The tomographic model is commonly established based on the directions along which space-geodetic SWD estimates are acquired and based on a discretization of the investigated atmospheric volume into volumetric pixels (voxels), e.g., of constant refractivity. The existing tomography solution approaches applied to such a discretized atmosphere are subdivided into iterative and non-iterative techniques.

In addition to slant wet delay estimates from the GNSS,

Independently of the reconstruction strategy, due to the point-wise GNSS observing geometry, the tomographic system of equations is usually ill posed and needs to be regularized, e.g., (i) by constraining the tomographic system by means of pseudo observations, (ii) by introducing additional observations from models, from simulations, or from other sensors, or (iii) by decreasing the number of voxels crossed by no rays at all.

Both

However, compressive sensing only yields encouraging results if the input data acquisition – corresponding, in water vapor tomography, to the determination of SWD estimates – fulfills certain prerequisites. For general applications of CS,

For LSQ,

In order to analyze the observing geometry's effect on the quality of the LSQ and CS solutions to water vapor tomography, different observing geometry settings are defined. Based on synthetic SWD estimates derived from WRF, 3-D water vapor distributions are reconstructed for each of the defined observing geometry settings using both LSQ and CS. The quality of the LSQ and CS solutions to water vapor tomography is then compared with respect to the respective observing geometry settings.

For tomography using GNSS SWDs,

As in

The LSQ solution to Eq. (

Moreover,

When aiming at a tomographic reconstruction of atmospheric water vapor by means of compressive sensing, the parameters

In Sect.

The (ii) minimum number of seven sites originates from the real GNSS Upper Rhine Graben (URG) network site distribution within the analyzed study area. The maximum number of sites is chosen such that the rule of thumb of

For each of the described observing geometry settings, synthetic SWD observations as input for the tomographic system are deduced from one single WRF simulation covering an about 200 km

Schematic illustration of the generation of synthetic GNSS SWDs according to

The horizontal distribution of the synthetic GNSS sites within the URG study area is shown in Fig.

Distribution of the 7 GNSS permanent sites (blue squares) as well as of the 5 to 25 additional, synthetic sites (black symbols) within the URG study area. The additional, synthetic sites are distributed within a grid that uniformly covers the study area. Triangles, pentagons, hexagons, diamonds, and circles represent the first, second, third, fourth, and fifth group of five additional sites each.

From WRF, simulations of water vapor mixing ratio, temperature, pressure, and geopotential height are available at a 900 m horizontal resolution for generating the synthetic GNSS SWDs within the

For the most humid acquisition date (27 June 2005) for which WRF simulations were provided for this research and for an exemplary voxel in the lower middle of the lowest voxel layer, Fig.

Absolute differences between estimated refractivity and the WRF refractivity in ppm for an exemplary voxel in the lower middle of the lowest voxel layer for the 48 samples of each investigated observing geometry setting. The two left columns use an ordinate ranging from

As expected, an increased number of sites and an increased number of signal directions per site, in general, decrease the mean of the absolute difference (called mean difference in the following) and the standard deviation of the difference between estimated refractivities and WRF refractivities. Yet, as shown in Fig.

Averaged over all voxels, this figure shows the mean of the absolute difference and the standard deviation (SD) of the difference between estimated refractivities and WRF refractivities in ppm, deduced from 48 samples of each investigated observing geometry setting composed of a certain number of synthetic GNSS sites and various numbers of signal directions per site. The dashed and dotted lines serve for better following the variation of the represented quantities with the number of sites, but only the discrete values indicated by the markers should be evaluated. The legend in the upper left subplot holds for all the subplots. In each subplot, the improvement by introducing 32 sites instead of 7 sites is given in red and blue. Degradations are given with a minus sign.

A more in-depth analysis does not show any significant differences among the individual voxels. Moreover, no significant systematics between the LSQ and CS solutions appeared, neither at the boundary layer nor at the top of the atmosphere. For most of the tested scenarios, LSQ provides better results than CS. However, if there are many sites and many different signal directions available, CS yields more accurate and more precise results than LSQ.

When investigating the standard deviation of the differences between estimated refractivities and WRF refractivities for the CS case, considering an increased number of synthetic GNSS sites only while keeping a constant number of five signal directions per site is not advantageous. However, as of 15 different signal directions per site, a clear improvement in standard deviation is visible when increasing the number of sites in the tomographic setting solved by means of CS. Independently of the number of sites, for realistic GPS-like observing geometry settings with 5 to 10 signal directions per site, the LSQ refractivity estimates are more precise than the CS refractivity estimates. In contrast, as of 15 signal directions per site, the CS solution yields more accurate and more precise refractivity estimates than the LSQ solution if at least 22 sites are available. That is, this study shows that LSQ is less sensitive to the number of signal directions than CS. Therefore, we recommend using LSQ for water vapor tomography with GPS-only observations and CS for water vapor tomography with multi-GNSS SWD estimates.

In the case of the maximum number of sites and the maximum number of signal directions per site (32 sites and 20 signal directions per site), when averaged over the 48 considered samples per observing geometry, the mean difference and the standard deviation of the LSQ or CS reconstruction attain values of about 0.3 ppm or 0.0 ppm. Therefore, the number of sites and the number of signal directions per site are of particular interest when aiming at a very accurate and very precise tomographic reconstruction using CS.

For the given

Consequently, the following three main results are summarized from this study.

The rule of thumb of

Based on site distributions obeying the rule of thumb of

LSQ seems to be less sensitive to the number of signal directions than CS. Therefore, CS should only be used in the case of multi-GNSS SWD estimates yielding a variety of at least 15 signal directions per site.

Section

Moreover, as the presented approach only relies on a synthetic data set deduced from WRF, the synthetic SWDs introduced within the tomographic system in this research are too optimistic when compared to real GNSS SWD estimates. Therefore, the conclusions drawn in Sect.

Furthermore, Sect.

In addition, a low or high number of signal directions chosen from a synthetic multi-GNSS constellation for the recommendation of LSQ or CS for GPS-only or multi-GNSS water vapor tomography applications should not be set equal to considering a real GPS-only or real multi-GNSS setting. Choosing a small number of signal directions from a multi-GNSS constellation yields a higher variability in the signal directions than choosing the same small number of signal directions from a GPS-only constellation. Since a high number of signal directions proved to be of particular importance in the case of a CS solution, the quality of the refractivity estimates deduced using CS may decrease if real GPS-only signal directions are chosen.

Finally, future research should analyze in more detail which signal directions are necessary in an LSQ- or CS-based water vapor tomography in order to reconstruct well the refractivities of as many voxels as possible. A two-step CS LSQ may then first yield accurate refractivity estimates for most voxels by means of CS and then use the geometric smoothing constraints applied in LSQ to improve the refractivity estimates of those voxels in which CS yields inaccurate refractivity estimates even if a high number of sites and a high number of signal directions per site are available.

As the input data result from a cooperation and were exclusively provided for this research, they cannot be accessed online. For more information on the input data, please contact the Remote Sensing Technology Institute of the German Aerospace Center.

MH developed the theory and performed the computations. PEB and SH verified the analytical methods and supervised the findings of MH's work.

Within the last years, Marion Heublein and Stefan Hinz collaborated with the Signal Processing in Earth Observation (SiPEO) team of the German Aerospace Center and with former collaborators Giovanni Nico (Consiglio Nazionale delle Ricerche Bari, Italy) and Pedro Benevides (University of Lisbon, Portugal).

Thanks to Franz Ulmer (formerly at the German Aerospace Center at the Remote Sensing Technology Institute) for providing WRF data.

The first author was supported by a scholarship of the Deutsche Telekom Stiftung. This publication of this research has been supported by the KIT-Publication Fund of the Karlsruhe Institute of Technology (grant application no. 1189).The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.

This paper was edited by Gunter Stober and reviewed by two anonymous referees.