Topographic mapping processor

The complete implementation of the topographic mapping processor in absence of snow is outlined in this section. The major elements of the processor can be seen in the block diagram in Figure 6.2, where dashed boxes indicate the novel steps. First of all, the frequency domain backscatter is acquired simultaneously at each of the two receiving antennas of the sensor Rx1 and Rx2, where n indicates the number of successive acquisitions over time. The raw data is then focused in a rectangular grid with any of the algorithms presented in Section 2.3. In order to minimize the impact on the final averaged interferogram of different atmospheric conditions in each of the N images acquired at different instants of time, a calibration of the radar images’ phase is applied. This calibration simply shifts pixel by pixel the phase φ#,n of the whole image to match the

phase value φ#,0 at a reference point: #, #, ( #,0 #,n)

j

n n

I =I e ϕ −ϕ . # = {1,2} depending on the Rx channel being processed. A reference point is selected such that it presents the highest absolute value of zero-baseline temporal coherence accumulated over all the time series analyzed. It is important to note that for the calibration step of the image phase a reference point not covered by snow is necessary to be visible in the radar FOV.

Once the phase calibration has been applied, the interferometric coherence between the two images I1,n and I2,n is obtained according to (2.21) with a spatial averaging of 3×3 pixels (4×1 resolution cells approximately). Taking advantage of the high acquisition rate of the GB-SAR sensors (typically on the order of several images per hour) a time series of

N coherence images can be averaged to reduce the impact of the phase instability due to

the atmospheric effects and the radar phase noise introduced during the aperture synthesis.

Fig. 6.2. Block diagram of the interferometric processor for DTM generation, where the dashed boxes indicate the novel steps of the processor.

It is interesting to note that the simultaneous acquisition in the LISA instrument of both reception channels separated by a vertical baseline cancels out possible atmospheric effects introduced by a temporal delay in the acquisitions (as is the case of repeat-pass interferometers). Nevertheless, the temporal interferogram averaging proposed in this technique re-introduces the problem of temporal decorrelation of the scene. This problem is addressed through the phase calibration step before mentioned, in which the whole image phase is shifted to match the phase value at a reference point.

While space-borne interferograms may suffer from spectral shift, as demonstrated in Section 2.4.4 GB-SAR interferograms typically present very high common frequency content. Actually, the interferograms used in this study have a spectral shift of 200 kHz in the worst case over 60 MHz bandwidth (i.e., a common frequency content higher than 99%). This and the fact that a frequency windowing function (e.g., a four terms Blackman Harris window) is needed to suppress side-lobes in the imagery eliminate the need for the common band pre-filtering.

Phase unwrapping, instead, is necessary since fringes are present in the interferometric phase as shown in Figure 6.3-a. A weighted least-squares algorithm solved by the preconditioned conjugate gradient (PCG) method is used [33], with the weight function given by the averaged absolute value of the coherence. The unwrapped phase of the example in Figure 6.3-a can be observed in Figure 6.3-b.

Fig. 6.3. Wrapped and unwrapped interferometric phase, scale in radians.

Next after the phase unwrapping, a calibration of the differential signal path Δr is

performed in order to minimize the effects of an inaccurate measurement of the baseline components Bh and Bv in the sensor. Considering that the unwrapping procedure

reconstructs the absolute phase up to a constant, knowledge of the precise terrain height at least in one ground control point (pREF) is necessary. The calibration consists first in

calculating the theoretical differential path ΔrREF corresponding to the reference point

pREF = (xr,yr,hr): 2 2 REF 2 1 1 1 1 2 2 2 1 2 cos , ( ) . r r r r r r r B r B r r x y H h τ Δ = − = + − − = + + − (6.18)

Then, the differential path at each pixel of the whole image is shifted in order to match ΔrREF in the reference point (pREF):

CAL ( REF) REF.

r r r p r

Δ = Δ − Δ + Δ (6.19)

At this stage the phase to height conversion can be done by any of the three methods reported in (6.6), (6.11) and (6.17) of the preceding section. To complete the process a final calibration of the height with the ground control point pREF is necessary in order to

determine the value of the constant H. As mentioned, the unwrapping procedure

reconstructs the phase up to an unknown constant, so with the knowledge of pREF it is

possible to provide the absolute DTM heights h.

6.3.2. Results without snow

The processor has been assessed with some 20 days of GB-SAR C-Band interferometric imagery acquired in August 2005 in the Sion Valley test site. A high resolution DTM of the study area was available from the Swiss Federal Office of Topography, produced from an aerial survey with a state-of-the-art laser scanner [38]. The

ground-truth DTM pixel size was 0.84 m×0.84 m, while the height information had ±0.5 m accuracy.

Since the objective is to obtain a DTM with the best possible accuracy, results are measured against the standard deviation of the differences between the ground-truth DTM and those obtained with the GB-SAR in its topographic mode (one transmitting and two receiving antennas).

First of all, the standard deviation in meters of the DTM differences without the Δr

calibration step is presented in Table 6.1. Results are presented for n = {1, 4, 8, 16, 32, 64

and 128} averaged interferograms using the formulation presented in Sections 6.2.1, 6.2.2 and 6.2.3. It is worth noting that at an acquisition rate of 12 minutes per image, n=128

corresponds approximately to the data acquired in a complete day, from 0h to 24h for example.

Table 6.1. Standard deviation of the DTM differences without Δr calibration

n 1 4 8 16 32 64 128

A 11.7 11.1 11.2 11.3 11.1 11.0 11.0

B 6.62 5.89 5.74 5.68 5.37 5.28 5.21

C 6.58 5.86 5.71 5.64 5.33 5.24 5.17

Note also that in Tables 6.1 and 6.2 the rows A, B and C stand respectively for the formulations presented in Sections 6.2.1, 6.2.2 and 6.2.3. Hereinafter these formulations will be referred directly as A (Taylor and r2 approximation), B (Taylor approximation, exact r2) and C (No Taylor approximation, exact r2).

Examining Table 6.1, the following conclusions can be drawn:

a) The standard deviation is clearly reduced by increasing n. A 20% decrease is

achieved when comparing n=1 to n=128 in B and C formulations.

b) The DTM accuracy obtained with the formulation C is higher than that with formulation B. Likewise, formulation B gives more accurate results than those obtained with formulation A.

c) The standard deviation values are on the order of 5-6 m, as reported in [32], [35] and [49] with similar sensor parameters, but different instruments and test areas. Table 6.2, instead, shows the same results, but incorporating to the processor the Δr

calibration step. The accuracy improvement is clearly observed in the column n=128. The

standard deviation is reduced by 37% when compared to the results without Δr calibration

(from 5.17 m to 3.28 m). In addition, it is observed that the formulation used has a minor impact on the DTM accuracy in this case.

Table 6.2. Standard deviation of the DTM differences after Δr calibration

n 1 4 8 16 32 64 128

A 5.68 5.03 4.53 4.33 4.06 3.98 3.84

B 5.33 4.64 4.09 3.86 3.55 3.41 3.28

Concerning the number of coherence images averaged, it has been observed that after n=128 the standard deviation tends to saturate or even slightly increase in the order of

centimetres. For example, with Δr calibration using formulation B or C the following

standard deviations are obtained: n=192 → 3.31 m; n=256 → 3.39 m; n=384 → 3.41 m.

The saturation may indicate that the remaining height uncertainty comes from the radar instrument itself and does not depend on the measurement conditions. n=128 corresponds

to a time window of about 1 day considering the acquisition rate of the LISA instrument reported in Table 4.1 (12 minutes per image approximately). Nevertheless, with the available data it is difficult to extract conclusions on whether the accuracy saturation is achieved after a given number of averaged samples or after a given temporal window regardless of the number of interferograms used for averaging. Different atmospheric conditions during the aperture synthesis itself could also be the cause of the slight increase in the DTM differences’ standard deviation after n=128.

Similar experiments have been conducted to observe the behavior of the DTM differences as a function of the phase calibration step, but no relevant impact on this processor step has been observed. The reason for this result is the homogenous atmospheric conditions observed during the field experiments, with temperature gradients equal or less than 8 °C over a 24h period. Nevertheless, the phase calibration step has been kept in the processor summarized in the block diagram of Figure 6.2 for the sake of completeness.

Figure 6.4 shows the DTM differences in a map and histogram for the best case achieved with the presented processor: Δr calibration, n=128 and C formulation.

Fig. 6.4. DTM differences after Δr calibration for n=128 and using the C formulation. a) Masked map of the whole radar FOV; b) Corresponding histogram.

Figure 6.5 instead shows the DTM absolute heights map and the normals map of the Sion Valley test site obtained with the LISA instrument. Normals are defined as the angle subtended by the vector (x,y,z) = (0,0,1) and the vector normal to each facet of the DTM. Normals can be seen as a qualitative indicator of the interferometric DTM, since they show evident random values in the areas where interferograms are noisy.

Note that the results presented in Figures 6.4-a and 6.5 are masked out to discard noisy pixels (white background in the figures). The mask is computed with a double

criterion: backscattered intensity level and zero-baseline temporal coherence. Only pixels within 50 dB dynamic range of backscattered signal over a whole day and with zero- baseline (only one reception channel used) coherence values greater than 0.85 are selected.

Fig. 6.5. a) DTM heights and b) DTM normals of the test site retrieved with the novel processor: n=128, Δr calibration and C formulation.