Approach to leveraging real-time GNSS tomography usage

Abstract

The signal of the GNSS satellites can be used to estimate the amount of water vapor in the atmosphere. For this reason, GNSS observations are nowadays routinely used by several meteorological institutes (e.g., MetOffice, Meteo France) to monitor weather events and to improve their weather forecasts quality. The analysis of a whole network of GNSS stations to estimate a full three-dimensional model of the water vapor content is a challenging and computationally demanding task. For this purpose, a tomographic system SEGAL GNSS Water Vapour Reconstruction Image Software (SWART) was developed and tested. The new method makes use of parallelized algebraic reconstruction techniques (ARTs) and supersedes other implementations in terms of speed by at least 50% for small networks. For SWART, the computation time grows linearly with the number of observations. As a result, the new method makes possible to estimate the water vapor for larger GNSS networks and can be used for near-real-time weather predictions. To show its potential, data from 26 stations in Poland were analyzed using data from a period of 56 days. Good agreement in the estimated water vapor between SWART and radiosondes solutions was obtained, with a mean RMS of 1.5 g/m³ for the lower layers and an overall improvement of 5% until the layer 6750 m when compared with the atmospheric model (WRF). Furthermore, rapid and strong variations observed by radiosondes were not modeled by the WRF but were detected by GPS tomography.

Appendices

Appendix 1

1.1 Algebraic reconstruction techniques

Algebraic reconstruction technique is an iterative approach for imaging reconstruction using data obtained from a series of projections such as those obtained from electron microscopy, X-ray photography and in medical imaging like in computed axial tomography—CAT scans (Kak and Slaney 1988). The projection data are usually obtained by a small-angle rotation around the object. It is not possible to measure a large number of projections in every situation, and it may also happen that they are not uniformly distributed on complete angular distance, i.e., on 180° or 360° (used in 3D), which is the case of GNSS tomography (Nirvikar 2012).

There are two different techniques for algebraic reconstruction, namely algebraic reconstruction techniques (ARTs) and simultaneous iterative reconstruction techniques (SIRTs).

1.2 ART methods

The algebraic reconstruction techniques (ARTs) are row action methods that treat the equations one at the time. This means that in each iteration, each equation is solved individually. The updates in each ith iteration are made using Eq. 14 plus a relaxation parameter ${\uplambda }_{{\text{k}}}$ described by:

$$ {\mathbf{\mathop{f}\limits^{\rightharpoonup} }}^{{\left( {\mathbf{i}} \right)}} = {\mathbf{\mathop{f}\limits^{\rightharpoonup} }}^{{\left( {{\mathbf{i}} - 1} \right)}} + {{\varvec{\uplambda}}}_{{\mathbf{k}}} \frac{{{\mathbf{p}}_{{\mathbf{i}}} - {\mathbf{\mathop{f}\limits^{\rightharpoonup} }}^{{\left( {{\mathbf{i}} - 1} \right)}} .{\mathbf{\mathop{w}\limits^{\rightharpoonup} }}_{{\mathbf{i}}} }}{{{\mathbf{\mathop{w}\limits^{\rightharpoonup} }}_{{\mathbf{i}}} .{\mathbf{\mathop{w}\limits^{\rightharpoonup} }}_{{\mathbf{i}}} }}{\mathbf{\mathop{w}\limits^{\rightharpoonup} }}_{{\mathbf{i}}} \user2{ }. $$

(14)

What distinguishes the various methods is the order in each row is processed (Hansen and Saxild-Hansen 2012). The ART reconstructions usually suffer from salt-and-pepper noise,¹ which is caused by the inconsistencies introduced in the set of equations by the ${\text{w}}_{{{\text{ij}}}}$ approximations as they are usually measured by experiments (Kak and Slaney 1988).

1.3 SIRT methods

The simultaneous iterative reconstruction techniques (SIRTs) are named “simultaneous” as all equations are solved at the same time in one iteration, based on matrix multiplications (Hansen and Saxild-Hansen 2012). Usually, these methods converge slower to the solution than the ART methods; however, they result in better looking images (Kak and Slaney 1988). The SIRT methods work by first solving all the equations, without changing the cell values, and then only at the end of each iteration, the cell values are changed/updated, being the average value of all computed changes for that cell.

The general form of these methods is as follows:

$$ {\mathbf{\mathop{f}\limits^{\rightharpoonup} }}^{{\left( {\mathbf{k}} \right)}} = {\mathbf{\mathop{f}\limits^{\rightharpoonup} }}^{{\left( {{\mathbf{k}} - 1} \right)}} + {{\varvec{\uplambda}}}_{{\mathbf{k}}} {\mathbf{TA}}^{{\mathbf{T}}} {\mathbf{M}}\left( {{\mathbf{p}} - {\mathbf{A}} {\mathbf{\mathop{f}\limits^{\rightharpoonup} }}^{{\left( {{\mathbf{k}} - 1} \right)}} } \right), {\mathbf{k}} = 0, 1, 2, \ldots $$

(15)

where A is the matrix with the various projections, ${\uplambda }_{{\text{k}}}$ is a relaxation parameter and the matrices M and T are diagonal and symmetric positive definite. The various SIRT methods depend on these matrices. ${\uplambda }_{{\text{k}}}$ is defined as $\frac{2}{{{\uprho }\left( {{\text{TA}}^{{\text{T}}} {\text{MA}}} \right)}} - {\upvarepsilon }$, where ${{ \rho }}$ is the spectral radius and ${\upvarepsilon }$ the machine´s epsilon.

1.4 Simultaneous algebraic reconstruction technique (SART)

SART is a kind of variation on the algebraic approaches already discussed (ART and SIRT) that combine the best of both. This technique was first reported in Andersen and Kak (1982). The main features can be described as: First, to reduce errors in the approximation of ray integrals of a smooth image by finite sums, the traditional pixel basis is abandoned in favor of bilinear elements. To further reduce the noise resulting from the unavoidable, but now presumably considerably smaller inconsistencies with real data projection, the correction terms are simultaneously applied for all the rays in one projection; this contrasts with the ray-by-ray updates in ART. In addition, a heuristic procedure is used to improve the quality of reconstructions: A longitudinal Hamming window² is used to emphasize the corrections applied near the middle of a ray relative to those applied near its ends.

Appendix 2

Statistics for all levels and both 1-h and 24-h solutions with respective Student’s p values. The p values were calculated as a two-sided test results for mean (bias) of residuals between SWART and RS as well as WRF and RS.

See Tables 7 and 8.

Appendix 3

Number of examples selected from all 56 days of Case 2, containing inversions in vertical profiles.

See Figs.

Fig. 8

Tomographic retrieval for a specific day (May 11, 2013) at 12h00 with WRF initialization every hour (left graph) and with WRF initialization each 24 h (right graph). RAOB-HF = radiosonde high frequency, RAOB = radiosonde, TOMO2 = GNSS derived and WRF = numerical model

8,

Fig. 9

Tomographic retrieval for a specific day (May 08, 2013) at 12h00 with WRF initialization every hour (left graph) and with WRF initialization each 24 h (right graph). RAOB-HF = radiosonde high frequency, RAOB = radiosonde, TOMO2 = GNSS derived and WRF = numerical model

9,

Fig. 10

Tomographic retrieval for a specific day (May 14, 2013) at 12h00 with WRF initialization every hour (left graph) and with WRF initialization each 24 h (right graph). RAOB-HF = radiosonde high frequency, RAOB = radiosonde, TOMO2 = GNSS derived and WRF = numerical model

10,

Fig. 11

Tomographic retrieval for a specific day (May 15, 2013) at 12h00 with WRF initialization every hour (left graph) and with WRF initialization each 24 h (right graph). RAOB-HF = radiosonde high frequency, RAOB = radiosonde, TOMO2 = GNSS derived and WRF = numerical model

11,

Fig. 12

Tomographic retrieval for a specific day (May 20, 2013) at 12h00 with WRF initialization every hour (left graph) and with WRF initialization each 24 h (right graph). RAOB-HF = radiosonde high frequency, RAOB = radiosonde, TOMO2 = GNSS derived and WRF = numerical model

12,

Fig. 13

Tomographic retrieval for a specific day (May 24, 2013) at 12h00 with WRF initialization every hour (left graph) and with WRF initialization each 24 h (right graph). RAOB-HF = radiosonde high frequency, RAOB = radiosonde, TOMO2 = GNSS derived and WRF = numerical model

13,

Fig. 14

Tomographic retrieval for a specific day (June 17, 2013) at 12h00 with WRF initialization every hour (left graph) and with WRF initialization each 24 h (right graph). RAOB-HF = radiosonde high frequency, RAOB = radiosonde, TOMO = GNSS derived and WRF = numerical model

14 and

Fig. 15

Tomographic retrieval for a specific day (June 22, 2013) at 12h00 with WRF initialization every hour (left graph) and with WRF initialization each 24 h (right graph). RAOB-HF = radiosonde high frequency, RAOB = radiosonde, TOMO2 = GNSS derived and WRF = numerical model

15.