significantly differ in winter and summer at mid-latitudes, Alshawaf et al. (2015b) recommend to deduce Π from surface temperature measurements, if available. The IWV estimates of MERIS are converted to a wet tropospheric delay according to Bekaert et al. (2015b):
φtropo,wet= −4π λ
Q
cosθ ·IWV . (6.8)
Neglecting seasonal variations ofΠ=Q−1, balloon sounding data can be used to estimate an appropriate value of Π. The freely available sounding data provided by the Department of Atmospheric Science of the University of Wyoming (UWDAS, 2015) is downloaded at a site located close to the URG, i. e. at Idar-Oberstein, Rheinland-Pfalz. From data between 2003 and 2009,Qresults in 6.41 at the chosen site.
MERIS data is available for registered users at ESA (2015) with a spatial resolution of approx-imately 1 km ×1 km. The advantage of using MERIS data for the correction of tropospheric phase delays present in Envisat data is the fact that the measurements have been performed at the same time. However, MERIS is an optical sensor and therefore strongly depending on the daily cloud cover of both master and slave images. Hence, a master scene with as less cloud coverage as possible, but still minimising decorrelation, see Sect. 6.1.1, is selected for the analysis of atmospheric phase delays from the stack of 43 usable SAR scenes in Envisat track 294S. The scene acquired on 2007-08-06 is almost cloud-free enabling the calculation of phase delays at 99.9 % of the PS locations. At 27 scenes of the images in the analysed stack, the percentage of PS locations with available IWV values from MERIS is below 30 %. Only for 12 scenes including the master, more than 70 % of the PS pixels could be used for the calculation of φtropo,wet from MERIS IWV estimates. The atmospheric phase screen (APS) of interferograms follows from subtraction ofφtropo,wetof the slave acquisition date fromφtropo,wet of the master date. Images of APS for all interferograms of the stack are shown in Fig. C.1(a) in Appendix C. The results indicate that the interferometric delays may contain large-scale effects as well as variations with a spatial scale of several km. Note that the analysis of phase delays under more or less cloud-free conditions may bias the assumptions on the magnitude of the delays as much larger variations are possible in case of extreme weather events along with thunderclouds.
6.3.2 Phase correction using weather model data
Following Jolivet et al. (2011), the ERA-Interim global meteorological model can be used to calculate phase delays at the acquisition dates. ERA-Interim is a global atmospheric reanaly-sis starting in 1979 provided by the European Centre for Medium-Range Weather Forecasts (ECMWF, 2015). ERA-Interim data is available free of charge and accessible via automated download scripts from the ECMWF server. Atmospheric and surface parameters are given on different vertical levels, but with a rather sparse spatial resolution of about 80 km (horizontal) and a temporal resolution of 6 h. The phase delay along the InSAR LOS results from the temperature T, air pressure pand partial pressure of water vapour e, deduced from relative humidity, at different height levels. The atmospheric parameters are integrated along the ver-tical profiles at the grid nodes of the weather model data between surface height H0 up to the maximum height of the troposphereHmax(Bekaert et al., 2015b):
φtropo = −4π λ
1 106cosθ
Z Hmax
H=H0
k1p
T +
k2−Rd· k1 Rw
e
T +k3 e T2
dH . (6.9)
6.3 Atmospheric influences and corrections 133 Note that the dry component of the tropospheric delay is included in Eq. (6.9), in contrast to the phase delay shown in Eq. (6.8). The phase delays are calculated for each acquisition time and interpolated at the spatial sampling of the PS pixels.
The APS for the interferograms in the analysed image stack in Envisat track 294S follows from subtraction ofφtropo between slave and master acquisition dates and is shown in Fig. C.1(b).
It becomes obvious that the weak resolution of the ECMWF weather model is able to account only for the long-wavelength atmospheric variation. The high temporal and spatial variations of water vapour with spatial wavelengths of several km are not resolvable from the ERA-Interim data. Weather models with higher temporal and spatial resolution are needed in order to model the atmospheric phase delays present in InSAR data, such as atmospheric models calculated with the Weather Research and Forecasting Modelling System (WRF, see Alshawaf, 2013, p. 28 ff.). However, the phase delays are in a similar order of magnitude compared to the MERIS estimates, and some visible correlation to MERIS is found, particularly at dates revealing strong long-wavelength atmospheric signals, such as on 2005-08-01 or 2006-07-17.
6.3.3 Spatial phase filtering using the power-law method
The power-law method developed by Bekaert et al. (2015a) applies a spatial phase filter to the interferometric phase after best possible reduction of other phase terms. The proposed model describes the variation of phase delays with heightHusing a heightHmaxat which the tropospheric delays reduced to a value close to zero and a decay coefficientκ:
φtropo =K∆φ(Hmax−H)κ . (6.10)
Hmax andκ are estimated from balloon sounding data of a nearby site available at UWDAS (2015). Again site Idar-Oberstein is used resulting in Hmax = 13.5 km and κ = 2.0. The coef-ficientK∆φ relates the tropospheric phase to topography and varies for different acquisitions.
According to Bekaert et al. (2015a), it can be estimated from filtering of phase values in dif-ferent spatial bands selected in a way that the contribution from other signals is reduced. For the investigation of interferograms in Envisat track 294S, the spatial filter bands are selected based on empirical studies of correlation to the APS calculated from MERIS data. A high correlation is found for spatial filter bands between 2 and 18 km. Fig. C.2(a) displays the re-sulting APS estimated using the power-law technique. The general distribution of positive and negative phase values is mainly in accordance with the results derived from spatio-temporal filtering in Fig. C.2(b), see Sect. 6.3.4. The mean correlation between APS resulting from the power-law method and from the spatio-temporal filtering of StaMPS is 0.65, with 80 % of the interferograms revealing a correlation larger than 0.5. However, the power-law method seems to overestimate the magnitude of atmospheric delays for most of the interferograms. In ad-dition, the resulting APS largely depends on the chosen filter bands for the analysed SAR data in the URG area. A proper validation with MERIS or weather model data is therefore advisable if the method shall be applied.
6.3.4 Spatio-temporal filtering of the phase
As noted in Sect. 6.2.3, a spatio-temporal filtering method can be applied to the residual interferometric phaseφres of each interferogram in order to separate the atmospheric signal.
In a first step, a temporal low-pass filter f of lengthTf iltis applied to the phase signal of alln interferograms using, e. g. a Gaussian exponential window function:
φlow-pass,i =
∑
n i0=1φres(i)·f(Ti,i0,Tf ilt) =
∑
n i0=1φres(i)·exp(−0.5Ti,2i0
T2f ilt) , (6.11) where Ti,i0 represents the temporal differences between one slave acquisition iand all other slave acquisitionsi0 within the stack. The values of f are scaled, so that the sum of all weights in f equals one, resulting in a filter kernel of the form:
f(T1,i0,Tf ilt) = 0.093 0.089 0.083 0.081 0.079 . . . 0.023 0.004
, (6.12)
exemplarily calculated for the first interferogram of Envisat track 294N withTf ilt= 730 days.
All phase values of every interferogram are then multiplied with the corresponding filter ker-nel. The filtered low-pass phase values result from summation of the weighted phase values.
In a second step,φlow−passis subtracted fromφres. The resulting phase values represent the de-sired atmospheric signal, which is subsequently smoothed in space. A spatial low-pass filter is applied in a similar way as for the temporal filtering using a given spatial filter wavelength Sf ilt. The filter lengths of the temporal and spatial filtering are previously unknown. Gong et al. (2015) propose to estimate temporal filter parameters from a cross-check with numeri-cal weather prediction models. Within this work, a validation of the remaining deformation signal in areas with known displacement from levelling analyses is performed, such as in the area close to Landau, see Fig. 6.10. In addition, the resulting atmospheric signal is validated with the APS calculated from MERIS data.
(a) (b) (c)
(d) (e) (f)
−5 −2.5 0 +2.5 +5
Figure 6.10:LOS displacement rates [mm/a] at PS of Envisat track 294N applying the spatio-temporal filtering with different parameters. The reference area, located in Karlsuhe, is marked by a black triangle and represents the mean of 235 PS within a 500 m radius. The spatial extent of figures (a)–(c) is the same as in Fig. 6.7, 6.8 and 6.9; (d)–(f) show a zoom on the area close to Landau with an extent of 8.030◦–8.230◦E, 49.150◦–49.270◦N, marked by a black frame in (a). The filter parameters are Tf ilt = 365 days, Sf ilt = 100 m for (a) and (d), Tf ilt = 365 days,Sf ilt = 2, 000 m for (b) and (e), Tf ilt =730 days, Sf ilt =2, 000 m for (c) and (f).
The example shows that besides the atmospheric signal also parts of the deformation sig-nal may be filtered out by the spatio-temporal filtering. The default parameter settings of
6.3 Atmospheric influences and corrections 135 Tf ilt=365 days andSf ilt =100 m in StaMPS are not suitable for a separation of atmospheric effects from the anthropogenic deformation signal close to the city of Landau. A larger min-imum wavelength prevents the Landau deformation from being filtered out, see Fig. 6.10(e), while a larger time window reduces the spatially correlated noise present in the whole ana-lysed area, see Fig. 6.10(c). A time window of two years along with a minimum wavelength of 2,000 m turns out to provide a good solution for the filtering of spatially correlated nuisance terms without deteriorating the deformation signal, which is validated by a comparison with the results from levelling. Furthermore, a higher correlation of the estimated atmospheric signal with APS resulting from MERIS data, see Sect. 6.3.1, is achieved with the adapted filter parametersTf ilt =730 days andSf ilt= 2, 000 m. Note that some of the PS shown in Fig. 6.10 reveal a different displacement behaviour compared to their neighbourhood and are assumed to be noisy. Therefore, an additional filtering for high quality PS is performed, described in Sect. 6.5.2.
In addition to the filtering approach of StaMPS, a slightly different atmospheric filter method provided within the Delft PS-InSAR software (DePSI, van Leijen, 2014, p. 29 ff.) is applied to the image stack of Envisat track 294S. The temporal filtering of DePSI is similar to the StaMPS method shown in Eq. (6.11) and is performed with the same time window. In contrast to StaMPS, Kriging is used for spatial interpolation of atmospheric signals in DePSI. In general, the atmospheric filtering is performed at an earlier stage of the processing within DePSI and only applied to a subset of high quality PS pixels, the so-called first-order network. For the comparison of tropospheric correction methods within this work, the filter approach of DePSI is implemented within the StaMPS processing chain and compared to the other correction methods in Sect. 6.3.5.
6.3.5 Comparison of results
The APS estimated from the five different correction methods as described above is shown for one interferogram of Envisat track 294 in Fig. 6.11. APS for all interferograms of the stack are provided in Fig. C.1 and C.2. The atmospheric signal resulting from MERIS, shown in Fig. 6.11(a), varies within approximately three phase cycles over the 100 km×100 km scene.
From the MERIS estimate, the high spatial variability of the phase delays present on differ-ent spatial scales becomes obvious, which is hardly reconstructible by the other correction methods. As already discussed, only large-scale tropospheric effects are resolvable using the ERA-Interim weather model for the calculation of APS. The power-law technique is work-ing fine for the interferogram shown in Fig. 6.11, but significantly deviates for others, see Fig. C.2(a). The spatio-temporal filtering approaches of StaMPS and DePSI yield similar re-sults and a high correlation to the MERIS phase delays. Only some small-scale effects cannot be resolved, which is partly due to the spatial wavelength used with the filtering, discussed in Sect. 6.3.4.
In addition to the visual interpretation of different correction methods, a numerical compar-ison is performed using the MERIS estimates as validation for the other methods. Tab. 6.5 presents 2D correlation coefficients between APS of MERIS and the other methods for all interferograms with a MERIS pixel coverage of more than 70 %. Highest correlations are achieved for the spatio-temporal filtering of StaMPS and DePSI. A generally lower correlation level is observable for some interferograms, e. g. for 2005-09-05 or 2008-09-29, which may as well be attributed to inaccuracies of the MERIS water vapour estimation or theΠ-factor used for transformation of water vapour into phase delay. For the reduction of atmospheric effects in the analysed URG data stacks the spatio-temporal filtering is used as it is applicable to all analysed scenes and shows the highest correlation with MERIS water vapour estimates.
(a) (b)
−2π −π 0 π 2π
(c) (d) (e)
Figure 6.11:APS [rad] of a slave of Envisat track 294S acquired on 2006-07-17 w.r.t. the master acquired on 2007-08-06, calculated using different methods and data sources. The spatial extent of the figures is 6.709◦–8.400◦E, 47.237◦–48.196◦N. The correction data and methods are (a) MERIS data, (b) ERA-Interim data, (c) Power-law method, (d) Spatio-temporal filtering of StaMPS, (e) Spatio-temporal filtering of DePSI.
Table 6.5:2D correlation coefficients of APS estimates using ERA-Interim data, Power-law method and spatio-temporal filtering of StaMPS and DePSI w.r.t. APS deduced from MERIS data
Slave date ERA-Interim Power-law StaMPS DePSI
2006-06-27 0.55 0.52 0.77 0.77
2005-08-01 0.46 0.44 0.60 0.59
2005-09-05 −0.03 0.25 0.49 0.48
2006-06-12 0.33 0.65 0.73 0.73
2006-07-17 0.60 0.73 0.89 0.89
2007-04-23 0.35 0.80 0.80 0.80
2007-09-10 0.50 0.62 0.70 0.70
2007-10-15 0.17 0.56 0.76 0.77
2008-05-12 −0.04 0.50 0.76 0.76
2008-08-25 0.45 0.65 0.77 0.76
2008-09-29 0.34 0.34 0.40 0.40
Mean 0.34 0.55 0.70 0.70