Fast and Robust Estimation of Atmospheric Phase Screens Using C-Band Spaceborne SAR and GNSS Calibration

Abstract

Over the last few years, a growing interest has been observed in the field of Interferometric Synthetic Aperture Radar (InSAR) meteorology. The atmosphere has always been seen as a disturbance in interpreting interferograms (the output product of InSAR processing). A space-borne radar, however, can sense the refractive index of the medium it travels. The refractive index, in turn, is sensitive to pressure, temperature, and humidity of the air. Therefore, SAR data contains information about the atmosphere’s status and can be exploited by Numerical Weather Prediction Models (NWPM) as additional information to improve weather forecasts. This chapter investigates a fast and robust method for generating the so-called Atmospheric Phase Screens (APS) from InSAR data. The method exploits both Permanent Scatterers (PS) and Distributed Scatterers (DS) in an optimal way leading to wide and dense APS maps. When operating at large scales, it is also mandatory to calibrate the data using a network of Global Navigation Satellite System (GNSS) receivers. The calibration can remove the so-called Orbital Phase Screens (OPS) that otherwise severely corrupt the atmospheric measurements. Results using real data acquired by the European Sentinel-1 mission show the potential of InSAR meteorology to provide valuable data to improve weather forecasts.

1 Introduction

A precise weather forecast is crucial for several applications, such as agricultural planning, prevention of natural disasters, and prediction of solar energy production. Numerical Weather Prediction Models (NWPM) rely on abundant, continuous, and accurate input data to generate weather forecasts. These data can be provided by several different sensors such as radiosonde, ground-based, or space-borne radars and in-situ weather stations. Unluckily, the number of measures in sub-Saharan Africa is insufficient to provide an accurate weather forecast. In such a vast continent is indeed very complicated to install and maintain in-situ weather stations. The solution within the TWIGA¹ H2020 project (Transforming Water, weather, and climate Information through in situ observations for Geo-Services in Africa) comes from the exploitation of remotely sensed data and in particular from space-borne Synthetic Aperture Radar (SAR) data. While in passive optical systems, the sensor measures the reflection of the sunlight on an object, in a SAR system, an antenna brings the illumination to the scene, and the echo is recorded [2].

The most important feature of an active and coherent remote sensing system like a SAR is the capability to exploit the phase information of the echo signal [4]. The mentioned phase contains information about the optical path between the sensing platform and the targets on the ground. The optical path includes the geometrical distance between sensor and target and the refractive index of the medium (i.e. atmospheric conditions). The latter is the measure of interest for NWPM. The refractive index is related to the water vapor content in the atmosphere, thus it can be a valuable product for the ingestion [5].

A meteo-hydrological product must respect some requirements in terms of horizontal and temporal resolution to be helpful in the ingestion process. The Observing Systems Capability Analysis and Review (OSCAR) tool [1], developed by the World Meteorological Organization (WMO) sets the minimum horizontal resolution for Integrated Water Vapor (IWV) maps in high-resolution NWPMs at 20 km with a goal, over which further improvements are not necessary, of 0.5 km. Concerning the temporal resolution, the minimum requirement is 6 h with a goal at every 15 min. The requirements are summarized in the Table 1.

Table 1 Requirements for integrated water vapor products into NWPM

A SAR system is generally able to respect the spatial resolution required by a NWPM. The field of view of a space-borne SAR is usually several thousands of km² with a few meters of resolution. Even if spatial filtering is employed, the final resolution will still be better than the goal one (below 500 m). It is not the same for the temporal resolution that, in the case of SAR systems, is much coarser than the threshold one. A temporal revisit of 6 days is the best we can achieve thanks to the Sentinel-1 constellation, and it is nowhere near the threshold of 6 h defined by OSCAR.

A Global Navigation Satellite System (GNSS) network is the opposite: it can provide continuous measurement in time; however, it is just a point-wise measurement in space. Nevertheless, the two can collaborate, and in particular, the GNSS can provide valuable additional information to calibrate SAR Atmospheric Phase Screen (APS) maps. In the following section, we will review the effects of the atmosphere on the SAR measurements, and then we will proceed in detailing the processing workflow for APS maps estimation.

2 The Essence of SAR and InSAR

The complex radar image (also called Single Look Complex, SLC) contains information both about the intensity of the reflection and about the sensor-target travel time [7]. The former is encoded in the image amplitude, while the latter in the image phase. The phase of a single image is, however, not exploitable since it is the result of the coherent superposition of many elementary scatterers inside the resolution cell. Information can be extracted through a process called interferometry. An interferogram is formed by coherently combining two SAR images:

$$\begin{aligned} I = P_1P_2^* \propto \exp {\{-j(\phi _1 - \phi _2)\}} = \exp {\{-j\psi \}} \end{aligned}$$

(1)

where $P_1$ is the first image, $P_2$ is the second image, $\phi _1$ and $\phi _2$ are their respective phases, $\psi$ is called interferometric phase and $^*$ indicates the complex conjugate. The interferometric phase can be modeled as the sum of several terms:

$$\begin{aligned} \psi = \psi _{\textrm{flat}} + \psi _{\textrm{topo}} + \psi _{\textrm{defo}} + \psi _{\textrm{atmo}} + \psi _{\textrm{orbit}} + \psi _{\textrm{scat}} + \psi _{\textrm{noise}} \end{aligned}$$

(2)

where $\psi _{\textrm{flat}}$ is a reference phase, $\psi _{\textrm{topo}}$ is a component linearly related with the topography of the scene, $\psi _{\textrm{defo}}$ is proportional to the deformation of the scene between the two acquisitions, $\psi _{\textrm{atmo}}$ is the APS we are interest into, $\psi _{\textrm{orbit}}$ is the orbital phase screen induced by errors in the knowledge of the satellites orbits during acquisitions, $\psi _{\textrm{scat}}$ is due to the difference of scattering mechanism between the two acquisitions and $\psi _{\textrm{noise}}$ is the always-present noise in the measure. Each component may or may not be useful for a particular purpose. If, as in our case, the atmospheric conditions are the objective of the research, all the other terms except for $\psi _{\textrm{atmo}}$ are considered as noise and must be removed from the interferometric phase.

3 Tropospheric Effects on SAR Data

When the radar signal travels through the atmosphere, it is delayed by a quantity called propagation delay. The delay depends on the refractive index of the medium itself. If the refractive index is not unitary, the optical distance R between the target and the satellite can be approximated as the sum of the geometrical distance $R_g$ and the equivalent excess path generated by the non-unitary refractive index $R_a$. A common value for $R_a$ in the vertical (zenith) direction is between 2.2 and 2.7 m [10].

The excess path delay can be further divided into a spatially and temporally smooth component called dry (or hydrostatic) delay and a spatially turbulent one called wet delay. The latter is the one that is directly proportional to the water-vapor density in the medium. The dry delay is usually in the order of a couple of meters, while the wet one is no more than 0.3 m [10].

The wet delay is the main responsible for the so-called atmospheric artifacts in any interferometric SAR processing such as Digital Elevation Model (DEM) estimation or deformation monitoring. The fluctuations of this term are significant both in space and time, and we can refer to them as atmospheric turbulence. The reduced spatial scale and high temporal variability make it difficult to estimate the wet delay using external measures in an accurate and high-resolution fashion. Nevertheless, this contribution aims to estimate this delay that still contains valuable information about the water vapor content in the troposphere. In the following section, the complete processing scheme is described in detail.

4 Atmospheric Phase Screen Estimation

The complete workflow for the estimation of the APS maps is depicted in Fig. 1. Each step will be described in this section.

Fig. 1

Scheme of the complete workflow for the estimation of atmospheric phase screens

4.1 Data Pre-processing

We start from a set of freely available images from the European SAR constellation Sentinel-1. The temporal baseline between two consecutive images is in the order of 6/12 days. This particular temporal baseline is guaranteed by the revisit frequency of the constellation. First of all the images must be properly coregistered [9] and the topographic component of the interferometric phase must be compensate using a DEM. Both steps can be executed using the open source software SNAP provided by ESA.

Also the GNSS data must be processed at this stage to obtain an atmospheric product called Zenith Total Delay (ZTD) that is the instantaneous atmospheric delay as sensed by a GNSS station. The processing of GNSS data is done using the free and open source software GoGPS [8]. The details about the processing of GNSS data for the extraction of atmospheric products are out of the scope of this contribution.

4.2 Phase Linking for APS Estimation

Once the images have been coregistered and compensated for topography, the phase estimation can take place. The workflow implements the so-called Phase Linking [3] algorithm. With N images in the stack, we can form $N(N-1)/2$ different interferograms. Phase Linking tries to justify all these interferograms with $N-1$ equivalent (i.e., best-fitting) interferograms.

We can put the algorithm in a mathematical framework by saying that, if the temporal covariance matrix of the data contains all the possible interferograms between couples of images, Phase Linking finds the best Rank-1 approximation of the covariance matrix. The advantage of this method is that, by exploiting the whole covariance matrix, it automatically reaches the best possible phase estimation (in terms of variance) both in the presence of a PS and a DS.

In Fig. 2 we depicted three covariance matrices, and in yellow, we highlighted the portion of the matrix exploited by different phase estimation algorithms.

Fig. 2

Each algorithm exploits a different portion of the coherence matrix. Phase Linking, for example, utilizes all the available interferograms, the AR(1) just the interferograms at the shortest temporal baseline, while the standard DInSAR method exploits all the interferograms at varying temporal baseline

The Phase Linking exploits all the possible interferograms leading to optimal estimation of the interferometric phases. It is not the case for the AR(1) algorithm that foresees interferogram formation at a short temporal baseline followed by integration. The disadvantage is that during integration, the noise is accumulated; thus, the solution is not optimal. Another technique called DInSAR foresees the interferogram formation at varying temporal baselines. This way of estimating the InSAR phase is particularly noisy for decorrelating targets at long baselines.

The Phase Linking algorithm is a generic estimator for interferometric phases. Some practical precautions are needed to apply this algorithm for atmospheric phase screen retrieval:

• 1. The total temporal extent of the stack must be short. The first consequence of reducing the number of images in the stack is to reduce the effect of terrain deformations. In the model of the interferometric phase (Eq. 2), a term due to the presence of ground deformation is present. With an average subsidence rate of 10–20 mm/year, if the stack is kept short, the impact of deformation on the interferometric phase is minimal. Considering 8 images with a total temporal extent of about 50 days (in the case of Sentinel-1), the error will be less than 2 mm. This bias in the estimate is tolerable for our purposes. Notice that these considerations are valid just in the presence of a small-medium subsidence rate, not for extreme deformations or earthquakes where the subsidence can be in the order of several tens of cm.

  The stack temporal extent also needs to consider the average “life” of a distributed scatterer. As already pointed out, we exploit all possible interferograms with N images with Phase Linking. If the coherence of the interferograms with a very long temporal baseline is very low, they will bring noise into the final estimate.

• 2. The number of independent samples used to estimate the covariance matrix must be carefully selected. The interferometric phase’s quality is tightly related to the number of independent samples used to estimate the covariance matrix.

  We found that with roughly 600 independent samples, the standard deviation of the phase always remains below 1 mm. Sentinel-1 shows a nominal resolution of $5 \times 20$ m² in IW acquisition mode in range and azimuth, respectively. The images are over-sampled at about $2.5 \times 15\,{\textrm{m}}^2$. To obtain a number of independent looks equal to 600, we need roughly 3000 pixels. We choose a $149 \times 25$ window spanning $375 \times 375$ m². This last figure is the resolution of our APS maps which is well below the OSCAR requirements.

Once the interferometric phases have been derived, they are unwrapped: phase unwrapping is a well-known technique and will not be discussed in this contribution.

4.3 GNSS Calibration

In the interferometric phase model described in Sect. 2, the term $\psi _{\textrm{orbit}}$ represents the so-called Orbital Phase Screen (OPS). This component arises due to the inaccurate knowledge of the satellite’s orbit during the acquisition of each image. The OPS manifests as a phase plane (i.e., a trend) in the interferogram. The most common solution in literature is simply fitting and removing a plane in the interferometric phase. However, this process could corrupt the estimated APS’s low-frequency components.

In this contribution, we developed a novel technique based on a network of GNSS stations. The workflow extract from SAR data the value of the APS over each station. The ZTD provided by each station is differentiated in time to form the differential ZTD (dZTD) that is then projected in the slant range using the SAR looks angle, obtaining a GNSS-derived equivalent APS. We recall now that the SAR interferogram contains both the APS and the OPS. If we remove from the SAR interferogram the GNSS-based APS, what is left is just the orbital plane.

Only then it is possible to fit a plane and estimate the OPS without the risk of filtering the signal of interest. Moreover, if the plane is calculated in radar coordinates, the parameters of the plane equation are related to physical quantities of the SAR acquisition, namely the normal baseline error and the derivative of the parallel baseline error. Once the parameters have been estimated, the forward problem can be computed for each pixel in the interferogram grid leading to a set of calibrated APS.

4.4 Transforming the Maps from Differential to Absolute

The last step before the ingestion into NWPM in the so-called absolutization process [6]. All the maps are differential with respect to a common but unknown reference atmosphere. The simplest and most effective solution is to provide an a-priori map to be summed to each interferogram. One of the sources of the prior could be a NWPM itself. It is sufficient to sum a single absolute ZTD map to all the phase-linked interferograms to obtain a stack of absolute ZTDs (see Eq. 1). We remark again that a NWPM or a GNSS network can provide this prior. In this latter case, the absolute ZTD must be interpolated over the whole SAR grid.

5 Results with Sentinel-1 Data

In this section we provide some results using real data acquired with the constellation Sentinel-1 (S1) over South Africa. The data, composed by five S1 frames covers an area of $250 \times 850$ km². The footprint is depicted in Fig. 3. The area shows also a very poor mean coherence meaning that the scene is very unstable due to temporal decorrelation making it hard to obtain a reliable estimate with traditional InSAR phase estimators.

Fig. 3

The five Sentinel-1 frames used lead to a $250 \times 850$ km² interferogram

In Fig. 3 we depicted also a single APS maps. Five images form the stack, but for brevity, we decided to depict just one of them. The visible holes are areas where the phase estimation is unreliable such as water bodies or dense forests with severe temporal decorrelation. Spatial variograms and spatial spectra have been computed to characterize the maps derived statistically.

In Fig. 4a the spatial variogram is depicted. In black dashed lines, all the variograms for the five images are represented, in blue the average variogram, while in red and green the 2/3 power law and 5/3 power laws. From the figure, it is evident that the data follow on average the theoretical model proving its ability to capture the APS dynamics. The same fit can be found for radially average spectra. In Fig. 4b the results are shown: the derived spectra fit well the theoretical model provided for the turbulent part of the APS. The model is a $1/f^\alpha$ power law where $\alpha = 5/3$ for radially averaged power spectra [4].

Fig. 4

a Variogram of the APS in the South African case study. The data follows the theoretical model. b Average power spectra of the data. Also in this case the theoretical model is respected

Fig. 5

Comparison between GACOS and one of the estimated APS maps. The bias between the two is roughly 1 cm, with a standard deviation around 2 cm

We also validated the maps with an external source. In our case, we used freely available data from the Generic Atmospheric Correction Online Service (GACOS). It can provide absolute ZTD and differential APS derived by NWPM to correct atmospheric artifacts in InSAR products. In our study we did not use these datasets to correct the APS, but to validate it. In Fig. 5 one comparison is depicted where we can see from the histogram that the error is slightly more than 1 cm in the mean and with a standard deviation of roughly 2 cm.

6 Conclusion

In this contribution, we presented a novel way to estimate Atmospheric Phase Screens (APS) from a stack of SAR data. The method exploits both Permanent and Distributed Scatterers (PS,DS): this feature allows the generation of wide and dense maps that can be ingested into NWPM to improve weather forecast. The algorithm has been extensively described in this chapter, its working principle is detailed, and results using real data acquired by the European constellation Sentinel-1 are presented. The validation has been carried out using a statistical and theoretical analysis of the variograms and spectra of the maps and by comparing them with a NWPM (GACOS). In both cases, the accordance is very high, showing the capability of the proposed method to capture most of the dynamics of the atmosphere.