Phase error

The geophysical results presented in this thesis are derived from measurements o f the phase-difference Ogf (e=l,2; f=2,3) between two SAR images, and so the error, AOgf, in this measurement is the hmiting factor in their quality.

Equations (2.8) and (2.13) may be rewritten as :

Pf =

e x p ( - i ( 2 * ( r ^ +

+ 4 . ) + AO,^ ) ] e x p ( î ( < P , + A0,^^ ) j

(4.1) SO that equations (2.9) and (2.16) become :

0 ^ = 2k[A T^^+ A,) + A ^ , ,

(4.2) where :

(4 3) The terms and A w i l l here be referred to as, respectively, the phase "scatter" error and phase "travel" error. The origins of these two types o f phase error will now be discussed separately.

4.1.1 Phase errors due to changes in the target's scattering properties

Equations (2.8) and (2.13) made the fundamental assumption that the scattered phase 0^ detected at Sf was identical to that detected at Si. Backscatter is the sum o f returns from many scatterers randomly positioned within a resolution cell. W hen the resolution cell is m uch larger than a wavelength, the phases o f the individual scatterers are uniformly random. Therefore only an exact repeat of the viewing scenario will allow our assumption to hold. As an interferometric system views its t a r ç t from two different positions, and at

Zebker and Villasenor (1992) and Hagberg et al. (1995) have discussed the sources o f this error in detail. There is an underlying value due to the signal-to-noise ratio of the system, which includes artefacts of the raw-to-SLC processing procedure. In addition there is a baseline-dependent error introduced because the target is viewed from two slightly different angles, resulting in different surface and volume scattering effects. Further errors are introduced if the images are less than perfectly registered. Physical processes acting on the scattering surface, such as wind, precipitation, and surface melting, can cause temporal changes in the distribution of scatterers, and are often the largest source of scatter error.

Such effects reduce correlation betw een im ages, and so by m easuring the correlation, using equation (3.6), we can quantify the scatter error. The relationship between and depends on the probability density function (PDF) of 0^. This has been derived by Joughin et al.(1994) and numerically integrated by Joughin (1995) to obtain Figure 4.1, which shows the relationship between correlation and scatter error for different averaging parameters. The plot agrees well with similar results obtained using Monte Carlo simulations (Li and Goldstein, 1990; Zebker et al., 1994).

2.10 1.75 "3 1.40 1.05 0.70 0.35 0.00 1.0 0.6 0.2 0.4 C o r re la tio n 0.0

Figure 4.1 After Joughin (1995), showing the relationship between phase error and correlation, where n^ is the number o f looks. For the data used in this research the equivalent o f 30 looks have been taken.

These results show that increasing the amount of averaging, or "number o f looks", can significantly reduce phase error. The concept of taking looks is explained in Barber (1985). The interferograms produced in this work are averaged over 4 pixels in range and

2 0 pixels in azimuth, but as adjacent pixels are not completely independent, this is only equivalent to taking 30 looks (Joughin, 1995). Consequently, we can use the %^=30 curve from Figure 4.1 to quantify the scatter error in our measurements of interferometric phase.

Rather than repeating the numerical integration of the backscatter PDF (Joughin, 1995) a th ir-i polynomial was fitted to Figure 4.1's «^=30 curve, and used to

calculate from the correlation images C^/J) shown in Figure 3.5. An example of is shown in Figure 4.2. This image shows peaks in scatter error relating to dark streaks in the corresponding correlation image. The origin of these effects will now be discussed. 2 0 km b) P hase error AOs (rad) 0.701 0 .3 5 0.0 i 1 Itlil i . _{I mil :} 4 0

t

P hase error AOg

0.35 0.35 2.10 (rad)

/

6 0 8 0 (k m )

t

B

Figure 4.2 a) Phase scatter error f o r the im age p a ir acquired 13th/16th F ebruary 1992. The corresponding correlation image can be seen in Figure 3.5a. The error w as calculated using the 30-look curve fro m p lo t 4.1. The image is streaked by local a n o m a lies a p p roxim ately p a ra llel to the x axis, b) P rofile a cro ss the A B line, show ing that the streaks correspond to sm all localised peaks in p h ase error (m arked by arrows). The origin o f these anom alies is discussed in the text.

Correlation streaks

Streaks of low correlation, forming bands that are approximately parallel to the look direction are prominent in several of Figure 3.5's correlation images. Joughin et al. (1996a) and Jezek and Rignot (1994) have seen similar structures in Greenland images. Jezek and Rignot demonstrated that the streaks were related to high frequency variations of up to ± 2 pixels in the azimuth fine registration offsets, though their cause is unclear.

The intensity of the feature is independent of baseline length. As the bands extend indiscriminately across areas of ice cap, bedrock and fast-ice, they appear not be linked to any surface geophysical process. Figure 4.3 overlays correlation images of overlapping ascending and descending tracks that are almost temporally coincident. This composite clearly shows that the streaks must be an artefact of the data rather than being related to any physical phenomenon, as their orientation is solely related to look-direction.

The DPAF raw-to-SLC processing procedure is not a source of the streak effects, as they have appeared in the products of other processors. However, to date they have

only been reported in data recorded at the K iruna G round Station. It is therefore currently th ought that their m ost likely cause is a p roblem w ith the low b it rate tape reco rd er dow nlinks at K iruna (I. Joughin, personal com m unication, 1997).

correlation

lo o k d irectio n

"^Austfonnia

2 0 km

Figure 4.3 O verlapping ascending/descending correlation im ages f o r locations A and

C in F igure 3.2. The ascending observations were acquired on F eb 13th/16th '92, a nd the descending observations w ere a cq uired on F eb 14th/17th '92. N ote the change in orientation o f the dark correlation streaks in the descending image. A ll descending pairs view ed to date that have produced correlation streaks have resulted in the sam e orientation. A ll ascending p a irs have correspondingly show n the sam e streak orientation as the A ustfonna im age show n here. The evidence is therefore co n c lu sive th a t co rrela tio n strea ks c o u ld n o t be re la te d to any g e o p h y sic a l phenomenon, and m ust instead be an artefact o f the imaging process.

4.1.2 Phase error due to travel-tim e anom alies

Phase travel errors do not cause decorrelation within interferom etric pairs, and so cannot be m easured directly. T here are two m ain causes; clock tim ing anom alies and atm ospheric attenuation.

a) Clock tim ing errors

The slant range term r^, in equation 4.1, is calculated from the associated tw o-w ay travel tim e, as m easured by the SA R internal clock. Errors in this m easurem ent w ill therefore contribute to

Such errors can be large, as shown by M assonnet and Vadon (1995) who attributed phase ramps of up to 6 fringes (37 radians) across a 100km SAR frame to a clock drift that was within E R S -l's design tolerances. However, as these errors also tend to have a linear y

dependence they may be partially com pensated by the baseline constraint procedure (Appendix 2).

bl Atm ospheric.

Variations in the velocity of the radar signal as it travels along its path will also contribute to Spatially and tem porally varying atm ospheric anom alies can cause the electromagnetic wave to be introducing significant phase errors (e.g. Tarayre and Massonnet, 1996).

H anssen and Feijt (1997) m odelled the effects o f tropospheric variations in temperature, pressure and relative humidity, and showed that interferometric phase errors can be significant with respect to the wavelength of the ERS radar signal. For instance, they showed that at a temperature of 0°C, a 20% horizontal change in relative humidity would produce a half cycle phase error in an interferogram. Such changes are common in the vicinity of cumulus clouds. Horizontal pressure changes consistent with meteorological frontal zones were predicted to cause phase errors of a similar magnitude. Tropospheric anomalies can therefore result in phase errors on both sub-kilometre and multi-kilometre spatial scales.

Horizontal and temporal variations in electron density in the ionosphere can also produce significant phase travel errors. The two main sorts of irregularity that are thought to be o f importance are travelling ionospheric disturbances (TID) and F-irregularities (Tarayre and Massonnet, 1996). TIDs are internal waves that create sinusoidal fluctuations of the electron density over wavelengths of between 30 and 300km, and with amplitudes of the order o f 1% of the total electron content. Tarayre and M assonnet (1996) performed simulations which showed that a typical TID creates a 1.2 radian phase error in a standard ionosphere.

The same workers showed that F-irregularities have produced phase errors of up to 19 radians. These errors are very distinctive, being aligned on the magnetic field in a cigar shape. Close to the pole the length-to-width ratio is approximately 5, and typical widths vary between several metres to several kilometres.

In addition, geomagnetic storms, caused mainly by solar wind events, can cause dramatic changes in the ionosphere, particularly in the poles, and can produce phase errors of the order of a factor of two higher than those discussed above (Callahan, 1984). These would not be expected to display any regularity of form or occurrence.