Atmospheric correction for interferometric synthetic aperture radar technique

Atmospheric Correction for InSAR Technique

Atmospheric Correction for InSAR Technique is a set of methods to mitigate atmospheric effects on interferograms using algorithms or external data. Geological hazards, such as earthquakes, volcanoes, and landslides, cause significant threats to human life and property all over the world. Interferometric synthetic aperture radar (InSAR) potentially delivers accurate (millimeter level) fields of ground displacement over large areas (hundreds of kilometers) . The accuracy of the InSAR measurements is affected by phase decorrelation, orbital errors, topographic residuals, phase-unwrapping errors, imperfect simulation of the imaging geometry, and extra path delay due to the propagation of the microwave signal through the atmosphere (Eq. (1)).

Δφ = φ_orb + φ_topo + φ_def + φ_atm(φ_ion + φ_trop) + φ_noise

However, several factors affect the reliability and accuracy of its applications. Among them, atmospheric artifacts φ_atm due to spatial and temporal variations of atmosphere state often pose noise to interferograms . The phase delay can lead to 10-14 cm errors in deformation products if there is a 20% difference in the relative humidity . Therefore, atmospheric artifacts mitigation remains one of the biggest challenges to be addressed in the InSAR community and it can mask smaller surface displacements due to interseismic slip, subduction zone slow slip events, and creep . On the other side, atmospheric noise can be regarded as a useful option for Meteorology. The main advantage of InSAR maps of Precipitable Water Vapor (PWV) is their high spatial resolution (up to 20 m for C-band sensors) if compared with GNSS and other spaceborne passive sensors. The low temporal resolution of InSAR PWV times series is the main disadvantage of this technique when seen from the side of meteorologists. However, these characteristics made the assimilation of InSAR maps of PWV into NWMs interesting for meteorologists only when studying troposphere’s properties at a local scale .

Atmospheric noise

Propagation of the microwave signal through the atmosphere is affected by (a) the free electrons in the ionosphere and (b) water vapor in the troposphere.

Ionosphere

Ionospheric phase noise is caused by variations in density of free electrons (between 100 and 1000 km altitude) along the travel path, resulting in a phase advance of the radar signal that becomes more significant for larger wavelengths, such as for P and L-band SAR. The ionospheric contributions impact on the C band SAR data is small and usually is negligible . In other words, electromagnetic waves are refracted by entering into the ionosphere which can be described by the refraction index. However, the dominant long-wavelength component can be removed by low order polynomial models due to it spatially correlated nature .

Troposphere

The troposphere, which is the lower layer of the atmosphere and where convective processes dominate over radiative processes, contains the majority of the water vapor mass (around 99% is contained in the troposphere, 50% of which is between the earth surface up to about a height of 1.4 km, clouds, precipitation, and others; most of the world’s weather takes place in this layer, both on the global and local scales . The tropospheric path delays caused by differences in temperature, atmospheric pressure, and foremost the water vapor in the lower part of the atmosphere. According to the physical properties of the atmosphere, microwave atmospheric delays are generally considered as the sum of the turbulent component and the vertical stratified component . The turbulent component of atmospheric effects affect both flat and mountainous terrains, and the vertical stratified component, which is highly correlated with topography, only influences the hilly or mountainous terrain . As tropospheric delays are integrated along the radar traveled path (Line of Side), the introduced noise can exhibit a strong topography-correlated component as well. Different methods exist that either estimate the atmospheric signal based on the interferometric phase or by using auxiliary data which will be explained in section 3. Although tropospheric effect is regarded as noise in the InSAR community, it has great advantages in meteorology and enables scientists to predict and model water vapor in the troposphere. Propagation delay of electromagnetic wave of radar satellites in troposphere will be explained in the following in terms of meteorology and InSAR community.

Layers of troposphere and amount of water vapor in each altitude

Layers of troposphere and amount of water vapor in each altitude

In terms of Meteorology

Schematic resolution comparison of InSAR and GPS for a large area

Schematic resolution comparison of InSAR and GPS for a large area

In terms of Metoorology tropospheric delay interestingly can be regarded as a useful tool for Meteorology purposes. Traditional methods for measuring water vapor include using a) Meteorological stations b) Radiosonde c) Spectrometer and d) GPS. Recently, InSAR has been recognized as a promising tool to generate high-resolution maps of atmospherical precipitable water vapor temporal changes (ΔPWV) from the propagation delay of radar signal in the atmosphere . Although, InSAR temporal sampling of atmosphere properties is coarser than the GPS, the main advantage of InSAR PWV maps is their high spatial resolution of up to a few meters . Therefore, scientists investigate a new methodology to increase the temporal and spatial resolution based on the synergic use of GPS and InSAR .

In terms of Interferometry (InSAR)

From InSAR side signal delay caused by variation of tropospheric properties in space and time is source of major challenges for interferometry. Tropospheric perturbation, caused by the differences in relative humidity, temperature, and pressure in the lower part of the troposphere between two acquisitions may generate additional fringes of up to 15–20 cm on differential interferograms . The atmospheric effects on InSAR measurements span a broad spectrum from long-wavelength to short-wavelength. The long-wavelength signal, which represents a ramp on the interferogram, is caused by, for instance, a slow-moving weather system. This is the most dominant error sources in interferograms with a large spatial extent and may indistinguishable from other long-wavelength errors such as the orbital ramps , ocean tide loading, and solid earth tides. The short wavelength signal results from, for example, the combination of the rapidly changing local atmospheric turbulence and the stratified component related to local topography.

Tropospheric correction methods

A phase delay through the troposphere can be characterized by the refractivity. This delay can be divided into two components: Dry delay and Wet delay. The dry delay, which is related to the pressure of dry air and temperature, and the wet delay, which is mainly related to the water vapor content present in the troposphere .

$\Delta _{atm}=10^{-}6\int N(z)dz$

Figure 4. Schematic representation of the electromagnetic waves delays in tropospheric based on dry and wet components. Zenith Total Delay (ZTD) correspond to total delay, Zenith Wet Delay (ZWD) corresponds to Wet delay and Zenith Hydrostatic Delay (ZHD) is related to Dry delay of troposphere.

Figure 4. Schematic representation of the electromagnetic waves delays in tropospheric based on dry and wet components. Zenith Total Delay (ZTD) correspond to total delay, Zenith Wet Delay (ZWD) corresponds to Wet delay and Zenith Hydrostatic Delay (ZHD) is related to Dry delay of troposphere.

where N is the atmosphere refractivity and can be calculated using the atmospheric parameters as follows:

$N_{tropo}={\Bigl (}k_{1}{\frac {P}{T}}{\Bigr )}_{hydr}+{\Bigl (}K'_{2}{\frac {e}{T}}+k_{3}{\frac {e}{T^{2}}}{\Bigr )}_{wet}$

where P indicates total atmospheric pressure, T is the temperature and e is the partial pressure of water vapor. The coefficients k₁ =77.689 ± 0.009 K/hPa, k₂ = 71.2952 ± 1.3 K/hPa, and k₃ = 3.75463 × 105 ± 0.0076 × 105 K₂/hPa are called refractivity constants .

$d_{tropo}=10^{-6}(cos\theta )^{-1}\int _{h=h1}^{h=h_{top}}N_{tropo}(h)dh$

$\phi _{tropo}={\frac {-4\pi }{\lambda }}d_{tropo}$

where d_tropo is the tropospheric delay (1-way), θ the incidence angle, λ the radar wavelength. The dry component is correlated with the topography due to the pressure variation with height, and it can reach 2.3–2.4 m in the zenith direction (zenith hydrostatic delay (ZHD)). Due to the temporal and spatial stability of the dry constituents of the atmosphere, we can ignore the dry component. The wet component (zenith wet delay (ZWD)) represents about 10% of the total delay (varying approximately between 20 and 300 mm) regards the water content motion and turbulent mixing.

The total LOS single path tropospheric delay dL_LOS (z) at an elevation z is the integral of the air refractivity between the surface elevation z and an elevation of reference z_ref and is modeled as  :

$\delta L_{LOS}^{s}(z)={\frac {10^{-6}}{cos\theta }}\{{\frac {k_{1}R_{d}}{g_{m}}}{\bigl (}P(z)-P(z_{r}ef))+\int _{z}^{z_{r}e}{\Bigl (}{\Bigl (}k_{2}-{\frac {R_{d}}{R_{v}}}k_{1}{\Bigr )}{\frac {e}{T}}+k_{3}{\frac {e}{T^{2}}}{\Bigr )}dz\}$

where θ is the local incidence angle, Rd = 287.05 J.kg⁻¹.K⁻¹ and Rv = 461.495 J.kg⁻¹.K⁻¹ are respectively the dry air and water vapor specific gas constants, g_m is a weighted average of the gravity acceleration between z and z_ref, P is the dry air partial pressure in Pa, e is the water vapor partial pressure in Pa, and T is the temperature in K.

Since there is no accurate technique that could determine the total refractivity on the same spatial scale and temporal sampling as the interferogram itself, various methods have been developed to account for the atmospheric contribution within interferograms.

Time series (PS - SBAS)

One of the approaches used to mitigate tropospheric effects on InSAR measurements has been stacking/filtering. Interferogram stacking increases the signal-to-noise ratio of time-correlated/time-independent signals without any external information . The other advantage of this method is independent on external data and straightforward to implement.

Limitation: For time series analysis approaches such as PS and SBAS, a large dataset has to be processed and this method only can reduce turbulent part of troposphere noise and signals of interest may be incorrectly removed.

Estimated tropospheric delay from GNSS

GNSS operates in the microwave portion of the electromagnetic spectrum similar to SAR technique and both measurements are affected by the atmospheric delay. In addition, both provide geodetic measurements with comparable accuracy. Using of both techniques simultaneously can lead to improvements in the atmospheric disturbances as well as for the cross-validation or integration of displacement measurements .

Limitation: Interpolation of zenith delay measurements from ground-based GPS. Requires dense GPS network but GPS stations are still sparsely distributed or even absent in many regions as well as this method only samples troposphere in the vicinity of individual GPS sites. Additionally, some GPS datasets are still not freely available to the public.

Satellite spectrometers

Spectrometer measurements are Satellite-based observations from space allow estimating the atmospheric water vapor by band ratios in the near-infrared spectrum. This method has a relatively high spatial resolution to calculate the turbulent component of the atmospheric disturbance .

Limitation: Requires collocated sensors and cloud-free conditions and only available in daytime. Time differences between radar and PWV data can be regarded as limitation .

Numerical weather models (ERA-I, ERA5, MERRA1, MERRA2)

Advantage of this method is insensitive to the presence of clouds and Global/regional/ local coverage. By using the ERA, uncertainty in ZWD in high latitudes (> 30∘), reach 1-2 cm, while uncertainty in low latitudes (< 30∘), is about 2-6 cm. Most of studies reported varying degrees of success for using this method .

Limitation: The low spatial resolution and the original mismatch in time between the model and the SAR acquisition do not permit addressing the turbulent component that takes place at lower Spatio-temporal scales. Moreover, complex data processing can be regarded as disadvantage of this method.

Phase based empirical: (i) linear correction (ii) power-law correction

The correlation analysis between interferometric phases and/or topography solves for the topography-correlated or stratified component. In other words, the correlation between range change in the LOS direction of InSAR and topography are related to the path length traveled by the electromagnetic wave. Estimation of linear and power-law relationships between the interferometric phase and the topography can be used to remove stratified parts of the troposphere. To estimate the stratified APS more accurately, recent improvements have been made by analyzing phase-elevation relationship with a multiple-regression model .

Limitation: The main limitation of these model related methods is that other phase terms (e.g., turbulent atmospheric artifacts, deformation related phase, decorrelation noise) can influence the estimate of the coefficient that relates phase with elevation. This method ignores the spatial variability of tropospheric signals and can be easily biased by orbit and topographic errors .

Numerical weather models (ERA5)

Here, we aim to explain ERA5 data with more details because of advantages of this data relative to others methods. Freely available Global Atmospheric Models (development of the numerical weather prediction) can provide accurate and higher resolution parameters for characterizing the atmosphere state, such as ERA5 data generated using Copernicus Climate Change Service Information. Hence, these methods are promising and practical techniques for atmospheric noise mitigation. The ERA5 is a global atmospheric model calculated by the European Center for Medium‑Range Weather Forecast (ECMWF) based on the assimilation of different input datasets. It provides several meteorological parameters, including pressure, temperature, and relative humidity at hourly intervals at a grid of 70 km (0.25 degrees) spatial resolution and 37 vertical intervals from sea level up to 50 km.

Figure 5. ERA5 data in 37 pressure levels and each grid has 16 parameters (Table 1)

Figure 5. ERA5 data in 37 pressure levels and each grid has 16 parameters (Table 1)

In this method, the zenith path delay is estimated first, and then delay along the Line-of-Sight (Z-LOS) direction is obtained geometrically by considering the incidence angle . Once atmospheric phase delays are integrated along a zenithal path on the grid points using Equation (6), a cubic spline interpolation in the vertical direction and a bilinear interpolation in the horizontal one is performed to obtain the phase delays for the entire SAR scene from the sparse grid points. Once having the ZTD along the vertical direction, the cosine of the incidence angle is accounted for by back-projecting the result to the LOS direction . Thus, we calculate $\phi _{trop}$ in first image t₁ and second image t₂ by using equation 6 and then subtracts these phases to achieve .

$\Delta \phi =\Delta \phi _{tot}-{\bigl (}\phi _{trop}^{t_{2}}-\phi _{trop}^{t_{1}}{\bigr )}$

After that, by subtracting it from the original unwrapped interferogram $\Delta \phi _{tot}$ we can remove the tropospheric effect from interferogram.

Name	Variable	Name	Variable
divergence	d	specific_cloud_liquid_water_content	clwc
fraction_of_cloud_cover	cc	specific_humidity	q
geopotential	z	specific_rain_water_content	crwc
ozone_mass_mixing_ratio	o3	specific_snow_water_content	cswc
potential_vorticity	pv	temperature	t
relative_humidity	r	u_component_of_wind	u
specific_cloud_ice_water_content	ciwc	v_component_of_wind	v
vertical_velocity	w	vorticity	vo

Table 1. Variables of ERA5 data

Conclusion

Tropospheric effects can limit the accuracy of InSAR measurements and cause misunderstanding in the interpretation of the geophysical processes. For instance, an error of 0.10–0.14 m for land surface deformation monitoring and possibly 80–290 m for topography mapping may be introduced by a 20% relative humidity change . Thus, properly characterizing this noise remains a challenge and reduces the ability of researchers to take full advantage of the available InSAR time series.

1- Different methods correct for different components of the troposphere and no method is exclusive the best in reducing tropospheric delays. In other words, the different tropospheric correction techniques all have their own limitations, and are not always sensitive to the same component of the tropospheric delay. For example, GPS-derived corrections performed comparably well when and where data was available and elevation-phase dependence is a more useful metric for areas of high relief and less effective in areas of low relief. Thus, rather than opting for a single method, jointly invert for tropospheric properties such different technique constrains each other, this will require relative weighting, quality measure of methods

2- Global measurements of tectonic/volcanic deformation need global atmospheric corrections. Although, ECMWF based on atmospheric reanalysis data provides a global data, low spatial resolution of it regarded as drawback. In order to optimally utilize InSAR atmospheric correction methods and to avoid potential uncertainty caused, it is urgent to develop a generic assessment method without the use of ‘ground truth’ to evaluate the atmospheric correction performance.

3- Compared with GPS data, InSAR is a promising technique to retrieve the water vapor product with global coverage and higher resolution. On the other hand, the InSAR technique can also provide potential enrichment of datasets used for research using numerical weather models, especially in very localized areas for turbulent atmosphere research. Even now, the latest suite of numerical weather prediction models has difficulty in accurately predicting meso- and micro-scale atmospheric dynamics, due mainly to computational discretization and observational scarcity. It is believed that atmospheric delays from InSAR can provide a promising observation dataset for weather data assimilation