Análisis de las variables cuantitativas

2.1.2 InSAR measurement errors

2.1.2.1 InSAR error sources

The error term ( ) of Equation (2.3) is composed of contributions from variable error sources (Berardino et al., 2002) that are usually present in most InSAR applications. It can be extended as

(2.10)

where represents orbital errors, the contribution from atmosphere path delays, and

describes the source from ionopheric anomaly (Meyer et al., 2006; Meyer, 2010; Rosen et al., 2010; Heki, 2011). The inaccuracy of an external DEM used in the repeat-pass InSAR processing also results in phase errors ( ). is partly due to non-efficient data processing, which is not easy to quantify and can be grouped into the random noise term ( ).

Atmospheric phase screen (APS) is one of the major error sources in the conventional InSAR measurements (e.g. Hanssen et al., 1999; Hanssen, 2001; Li et al., 2006), which can lead to an order of ~0.1 m in deformation products (Zebker et al., 1997). In small deformation analysis, the magnitude of APS can be even greater than targeted signals. Utilizing external atmospheric water vapour datasets (e.g. Li, 2004; Li et al., 2006; Li et al., 2009c; Walters et al., 2013; Jolivet et al., 2014), e.g. GPS, MERIS, MODIS and meteorological data, its effects on SAR interferograms can be partly reduced. In cases without external datasets, time-series analysis can provide an alternative way to separate deformation signals from APS (e.g. Lanari et al., 2007; Li et al., 2009d; Reeves et al., 2011; Cetin et al., 2012).

Regular and long-wavelength fringes in unwrapped interferograms are usually induced by orbit-related errors. Recent studies have also pointed out that timing errors and oscillator clock drifts over time may also contribute to orbit-like fringes (Bekaert et al., 2013; Marinkovic and Larsen, 2013; Teng et al., 2014). A best-fitting polynomial of order one or higher can be sufficient to reduce its effects in an individual interferogram. A spectrum domain method (Shirzaei and Walter, 2011) is also feasible to estimate its distribution. For the applications in interseismic creep rate estimations, a network approach (Biggs et al., 2007) with multiple interferograms is proposed to split orbital errors into master and slave components. This method can avoid effects of long-wavelength tectonic signals and APS on the orbital errors estimates to an extent. In this method, the rank of the designed matrix is insufficient. SVD is suggested to solve such an underdetermined linear problem (Biggs et al., 2007). An improved network approach with a two-step strategy is developed in this thesis. Details can be found in Chapter 3.

2.1.2.2 Spatial characterization of APS

APS is a common issue when applying InSAR techniques since radar signals travel through the troposphere. All methods for correcting for APS in an individual interferogram remains challenging

22 (Knospe and Jonsson, 2010). The confidence intervals and uncertainties in model parameters induced by InSAR errors must be provided in geodetic applications (e.g. Menke, 1989; Lohman and Simons, 2005). The geostatistic method is an effective way to characterize the dispersion of the observations using the structure function (or variogram), variance and covariance (Hanssen, 2001).

Without loss of generality, the anisotropic covariance can be expanded from the Hanssen's (2001) isotropic model by combining a scalar distance and azimuth between any two points as

where is always non-negative, is the observation at position in an interferogram.

The variogram originates from geostatistics, whilst the structure function is widely used in the turbulence literature (e.g. Antonia and Smalley, 2001; Emardson et al., 2003). Both can be defined as a variance of the difference of two points separated by a distance and azimuth angle ,

(2.12)

and

(2.13)

where is the variance of the interferogram. Since elements in covariance are always positive, the variances from the azimuths of and with the same distance should be equivalent.

and can be computed by either a full variogram where all point pairs between any two points are computed, or a sample variogram where only a certain number of random pixel pairs are chosen for computation. A full variogram should always be given priority to unless the computational burden in spatial domain is an issue. In this study, a FFT method (Marcotte, 1996) is suggested to compute full interferogram variograms in the frequency domain, which can provide a very fast computation. As shown in Figure 2.7, the original interferogram contains 649 by 663 pixels. All of possible point pairs reach to 1.9*10¹¹. It took only 3 seconds to produce the full variogram (Figure 2.7 (b)) using the FFT method. Note that null value in the interferograms should be interpolated before the calculation.

23 Figure 2.7 Example of an interferogram, a full variogram and structure function with distance and azimuth . The Interferogram used in (a) was generated from ascending ASAR track 026 covering the Damxung area in the southern Tibet.

24 Figure 2.7 shows the two-dimensional full variogram (Figure 2.7(b)) of the interferogram (Figure 2.7 (a)) and one-dimensional structure function along different azimuth angles (Figure 2.7 (c)). The structure functions over 20 km are relatively variable. In practice, APS spatial distribution is usually simplified to be an isotropic problem as used in most of previous cases (e.g. Hanssen, 2001;

Parsons et al., 2006). An isotropic theoretical function can be modelled by an exponential function over distance as

(2.14) where is the parameter that is theoretically equal to the standard variation of the data errors, is another controlling parameter. Because of the presence of white noise, is not zero at each pixel, which is usually estimated using a small window. In the case of Figure 2.7, the best-fit model of the structure function shows the variance ( ) of 44.92 mm², and model parameters for of 25.93 and of 0.065. The data of coseismic interferogram is larger than the parameter , which is similar with the phenominon in the 2011 Iceland earthquake (Sudhaus and Jonsson, 2009), suggesting that white noise in the InSAR measurements is common.

Note that data in the far-field without no detectable deformation, are used to estimate the variogram.

If insufficient far field pixels are available in the case of large shallow earthquakes, the residuals after removing the modelled displacements with a best-fit slip model can be used instead (Elliott et al., 2010).