ATMOSPHERIC PHASE SCREEN IN GROUND-BASED RADAR: STATISTICS AND COMPENSATION

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1 ATMOSPHERIC PHASE SCREEN IN GROUND-BASED RADAR: STATISTICS AND COMPENSATION A. Monti Guarnieri 1, L. Iannini 1, and D. Giudici 2 1 Dipartimento di Elettrica e Informazione, Politecnico di Milano, Via Ponzio 34/5, Milano, Italy 2 Aresys s.r.l., Via Bistolfi 49, Milano, Italy ABSTRACT The paper deals with the Atmospheric Phase Screen issue in Ground Based SAR interferometry. The ku-band RADAR used, the IBIS by IDS, is capable to continuously image scenes of a few square kilometers with a repeat interval that ranges from a small fraction of a second (in unfocused mode), up to a few minutes (in focused SAR mode). Such features allowed an assessment of the time-varying statistics of the APS, leading to some interesting quantitative considerations. With the aim of estimating and removing the APS from the data, an alternative approach with respect to the currently well-known techniques has been investigated: a compensation strategy based on the available on-site meteo information has indeed been developed. The results of the technique are critically discussed in the case of a real campaign dataset. Key words: Ground-Based Radar; Atmospheric phase screen; Ku-band interferometry. 1. INTRODUCTION Ground-Based SAR Interferometry ([1], [2], [3], [4]) is a remote sensing technique which finds its perfect application field in the monitoring of small scale areas, such as buildings or single hillsides, because of the fine spatial resolution provided (inferior to a meter) together with a very small revisiting times and fitting illumination angles for these specific site geometries. In radar interferometry accuracy is strictly connected to the concept of coherence, i.e., the degree of correlation showed by the phase of the target with respect to itself at two different times. The most relevant decorrelation sources for targets that exhibit a stable behaviour are: target displacement, changes in the atmospheric configuration and thermal noise. In this paper we ll deal with the atmospheric phase screen (APS), which certainly represents the major responsable for the severe phase fluctuations in the received signal. Our analysis will make use of the data gathered by the IBIS-L tool installed in Bolzano (Italy) during a time interval of 7 days from 15th July to 22th July Parameter Value Wavelength, λ 1.7 cm Bandwidth, B 300 MHz Synthetic Aperture length 2 m Range interval 0-4 km Azimuth aperture ±30 Azimuth sampling step 5 mm Table 1. Technical specifications of the IBIS-L instrument The instrument (whose specifications are reported in Table 1) consists in a GB-SAR device manufactured by IDS Ingegneria Dei Sistemi S.p.A. of Pisa (Italy), with a front-end operating at Ku frequencies covering a bandwidth of 300 MHz by means of the CW-SF (Continuous Wave Stepped Frequencies) technique. The geographic settings consist of an almost flat inhabited terrain in the foreground (first 600 m circa) just underneath a mountainside which extends up to a range of about 1200 m. In Fig.1 the picture of the instrument view together with the amplitude map of the targets in the scene is shown. In addition, for our investigation, we have at disposal a set of meteo measures collected every 8 minutes by a single meteo station installed nearby the radar. These data represent a very important asset, since our work will focus on studying the informativeness of the meteo information with regards to APS removal, i.e. we ll explore the compensation performance achievable by exploiting pressure, temperature and humidity measures. In section 2 a mathematic formalization of the received signal is provided, with particular concern to the modelization of the APS superimposed to the real displacement. We ll then proceed by reporting and commenting in section 3 some statistics on APS extracted from both radar acquisitions and meteo data. Section 4 is finally dedicated to the proposed compensation approach, which will be hereafter called Model-based approach since it relies on the physical modelization of the delay. The results achieved by the model feeded with the raw meteo parameters lead us to explore the possibility of a further model refinement in order to obtain a better decorrelation from meteo data: a meteo calibration solution based on humidity tuning has therefore been developed. The calibration scheme together with its results are in the end illustrated and discussed. Proc. Fringe 2009 Workshop, Frascati, Italy, 30 November 4 December 2009 (ESA SP-677, March 2010)

2 topic widely explored by the scientific community since the middle of the previous century (see [5], [6], [7]). Refractivity model is computed as the sum of a dry (or hydrostatic) component N hyd, linearly related to the partial pressure of dry gases P d, and a wet component N wet, proportional to the partial pressure of water vapour e. In the present work the empirical model proposed in [6] is taken as reference. The refractivity equation is: N = N (P,T,H) = N dry +N wet = P ) d (71.7+ T e T T (4) Figure 1. On the left: picture of the Bolzano scenario from the radar point of view. On the right: average amplitude map of the scene in the radar coordinates plane. 2. TARGET PHASE MODEL We can write the received signal associated to the generic p th target in the i th image as: y p (i) = a(i) (b p exp(jϕ p (i))+ω C (p,i))+ω T (i) (1) where a is the overall gain of the i th image, b is the specific amplitude of the target, ω C (p,i) and ω T (p,i) account respectively for the clutter noise and the thermal noise: both can be assumed circular complex gaussian (ccg) processes independently distributed with zero mean and non stationary variance for the clutter, while stationary variance for the thermal noise. In the endϕ p (i) is the unwrapped phase of the target, which is composed by its own backscatter phaseϕ 0,p, by the propagation delay term proportional to the LOS distance of the scatterer ρ p and by the additional one-way delay d atm,p (i) introduced by the non-idealities of the troposphere. ϕ p (i) = ϕ 0,p + 4π λ (ρ p +d atm,p (i)) (2) with λ refering to the central wavelength of the transmitted band. The approximated formulation of the equivalent atmospheric delay expressed in meters for almost straight paths is: d atm,p (i) = 10 6 L p N( r(l),i)dl (3) where N is the refractivity index of the atmosphere. It s related to the index of refraction n through the equation N = 10 6 (n 1) and it s a function of the space r and the time of i th image acquisition. The above expression states that the delay for the p th target results from the integration of the refractivity function along the ray pathl p which links the radar to the target. The refractivity dependency at radio frequencies from the three meteo parameters P,T and H, which are respectively pressure, temperature, and relative humidity, represents a research where the pressures P, P d,and e, linked by the relationship P d = P e, are expressed in mbar, the temperature T is expressed in K and the relative humidity in percentual points (%). The water vapour pressure e can be computed using the Magnus-Teten formula [8]: e(t,h) = H 100 e sat(t) = H ( 7.5 T T+237.3) (5) with e sat standing for the saturation water vapour pressure (mbar), and T measured in C. Though the dry component is the larger of the two, the wet refractivity is the most problematic one, as it presents the greatest variations between close acquisitions. We should indeed remember that in radar interferometry we are not concerned in estimating the distance of the target but only its displacement. Thus we re interested in evaluating: φ p (i 1,i 2 ) = y p (i 2 ) y p (i 1 ) = 4π λ (d atm,p(i 2 ) d atm,p (i 1 ))+w φ,p (i 1,i 2 ) (6) where w φ,p (i 1,i 2 ) is the phase noise coming from the combination of ω C (p,i 1 ) and ω T (p,i 1 ) with ω C (p,i 2 ) and ω T (p,i 2 ), which behaves, in case of a high SNR, as an indinpendentely distributed gaussian process with zero mean and non stationary variance σ 2 φ,p (i 1,i 2 ) << π. In this last case scenario we could also ignore the contribute of noise coming to the relationship: 3. APS STATISTICS φ p = 4π λ d atm,p (7) Our previous campaigns allowed us to quantitatively assess the entity of APS disturbances on the acquired data. The variogram certainly offers the most convenient statistical tool for a comprehensive characterization of atmospheric behaviour: both short-time statistics (Fig.2a), up to about ten minutes, and longer time statistics (Fig.2b) are estimated. The relevance of statistics evaluated on a short time basis relies on the fact that short intervals phase fluctuations can have significant role in the SAR

3 995 Pressure [mbar] (a) Temperature [ C] Figure 2. The variograms refer to a 1200 m far motionless target for both the shorter time statistics in (a) and longer time statistics in (b). The red and blue curves in the panels have been extracted from the radar datasets collected during a urban (red) and a mountain (blue) campaigns, while the green curve in (b) has been simulated using the meteo information provided by a meteo station installed inbolzano. image focusing process. Since the the device takes about 5 minutes to complete the scan operation along the rail, a change in atmospheric conditions would produce a defocusing effect whose extent grows with range. The longertime statistics provide instead an interesting quantitative evidence of the fake displacement which affects the interferograms with a time baseline in the order of a few days: the figure hints that for a target at 1200 m range the mean squared value for the delay difference in a 10 days interval would approximatively amount to 10 cm 2, which is a considerable value. The red and blue curves showed in Fig.2a and Fig.2b were extracted from a motionless, as well as very reflective, target positioned at a 1200 m range, therefore we were sure that all the registered phase fluctuations were belonging to APS. The meteo information collected in Bolzano were also exploited in order to assess the delay statistics: the result is the green variogram in Fig.2b. It s easy to notice that mean squared values of atmospheric delay differences are indeed bigger for the urban scenarios: the explanation by all means resides in the higher temperature and humidity values registered in lower elevation. 4. MODEL-BASED COMPENSATION The APS compensation in GB-SAR interferometry is currently performed by means of the Permanent Scatterers technique (introduced by Ferretti et al. in [9]) as suggested in [10] and [11], or by exploiting the a-priori knowledge on the nature of the scenario, thus extrapolating the APS from those safe motionless targets (Ground Control Points) as carried out in [2]. Both approach has proven to be powerful, though each one presents its drawbacks: the PS technique can mistake the real motion for the APS if their behaviour with respect to range is similar, while the GCP technique has to rely on the existence of some known steady targets, which cannot be easily found in every scenario. In this paper a removal procedure based on the supplementary information of pressure, temperature and humidity data has been in- Relative Humidity [%] (b) (c) Figure 3. Bolzano meteo parameters: pressure (a), temperature (b) and humidity (c) Refractivity [ppm] Figure 4. Total refractivity in Bolzano vestigated. When compared to the aforementioned techniques, such a compensation approach has the advantage of being free from any of the conjectures upon the targets displacement; however other kind of assumptions, concerning the spatial configuration of the atmosphere, has to be done. The results is a loss in the model accuracy, which has therefore to be assessed. Since in our test campaign (Bolzano) we had at disposal a single meteo sample captured nearby the radar, we choosed to rely on a very coarse yet reasonable (particularly for short distances) approximation: i.e. refractivity has been assumed uniform with respect to space. A more complex atmopheric model, such as the ones considering vertical profiles for pressure, temperature and humidty, didn t seem justified for the modest range and altitude variations in the scenario. The generic delay expression in (3) is semplified in the linear relationship: d atm,p (i) = 10 6 N (0,i) ρ p (8) The temperature, pressure and humidity data measured by the meteo station are shown in Fig.3a, 3b and 3c. They exhibit a critical behaviour, since both temperature and humidity reach very high values with wide fluctuations, especially in the last three days. The associated refractivity curve is then plotted in Fig.4. The analysis will be performed on a limited group of targets chosen as the most coherent targets according to the PS technique. The candidates amount to 1025 and their position is shown in the right panel of Fig.6.

4 Figure 5. Results of APS compensation over limited image intervals of 100 samples (13 hours): (a) Coherence of the phase data before compensation, (b) Coherence after compensation, (c) Differences in coherence matrices Figure 6. Residual analysis after the compensation process: the residual displacement is plotted in left part of the image for two targets at different ranges, chosen as representatives among the selected PSs shown in the right panel Compensation with raw meteo data The compensation is performed subtracting from the acquired data the APS model computed through equations (8) and (4) and its performance is be evaluated measuring the targets coherence before and after the compensation, either over the entire sequence at disposal or into a limited time interval. The latter approach exploits time windows of 100 images (a single day duration in terms of number of images is about 180 samples), and the result of the processing is the coherence matrix C where the first index accounts for the scatterer number and the second refers to the position of the window along the sequence. Let s call N w the length of the image window and θ p the phase signal associated to the p-th target. The elements of C matrix are then computed as: C(p,i) = 1 N w i+n w/2 n=i N w/2+1 exp(jθ p (n)) (9) where the phase θ p can represent either the received signal phase φ p or the residual after the atmospheric model subtraction, i.e. ǫ p = φ p ˆϕ p,atm. The results in Fig.5 register an overall positive behaviour, though negative spots can also be found. The panels convey that the first part of the sequence is the one benefiting the most from the compensation. Besides, a significant trend in the residual s behaviour can be found: the coherence tend to be higher (with some exceptions, such as the 16 June trend) in the first hours following the midnight. The most critical situations indeed occur at noon when temperature is very high: in these cases a bad humidity evaluation (which is more likely to happen then inaccuracies in temperature measurement) leads to the greatest errors in the phase estimation. Regarding the coherence achieved on the whole 7-days sequence (Fig.10), the APS removal process produces a visible improvement for almost every target (with the exception of the farthest ones). The unwrapped residual together with the unwrapped phase data Figure 7. Diagram for model tuning. Humidity calibration is performed by means of a linear transformation; the new humidity is substituted to measured humidity in the refractivity model described by eq. (4). The two best candidates ˆα and ˆβ are elected by maximizing the overall coherence for the target group Ω. In our analyasis γ has been set to 300m. φ p and phase model ˆϕ p,atm are displayed in Fig.6 for two scatterers located at respectively 327 and 570 meters in range. By means of these two examples we can see how the residual is still affected by oscillations whose amplitude rises with range, thus leading to a lower coherence for increasing target distance Humidity calibration Basically, in the residuals plotted in Fig.6, two components can be identified: the first is almost modelizable as a noise, though its sources have for the most a deterministic nature (such as small errors in the refractivity model estimation or clutter noise), while the second is a more regular component characterized by periodic fluctuations. The latter is the contribute which captures our interest, since it seems to bear the effects of a systematic error in the delay model, presumably produced

5 Figure 8. Results of the humidity calibration process on 100-images windows coherences: (a) Coherence of residuals before calibration, (b) Coherence of residuals after calibration, (c) Improvements brought by calibration by the underestimation of the oscillations amplitude regarding one or more of the atmospheric data (pressure, temperature and humidity) gathered by the meteo station. Indeed, by simply taking a look at both the residual and humidity curves, a close relationship between these two data appears evident. We can therefore state that the proposed delay model fails to completely remove from the initial phases the dependence from humidity. It s important to make clear that this phenomenon is hardly to be attributed to the delay formulas themselves, but rather the explanation must be searched in meteo measures offsets and/or inadequate spatial meteo sampling (since we have a single sample) having to deal with atmospheric inhomogeneities. In order to further improve the residual s coherence, a rather harsh but yet effective empirical approach whose scheme is presented in Fig.7. It is based on a first order tuning upon humidity: the measured humidity H is simply switched with the calibrated humidityh cal inside the model equations (4) and (8). The best parameters ˆα and ˆβ are chosen by the algorithm as the values able to register the maximum increase in average coherence for the group of targetsω. In order to make sense, calibration has to be performed only on stable targets near to the meteo station, so our choise was to select all the PSs up to the range of 300 m. With respect to this last step, a proper observation has to pointed out: by introducing the calibration procedure, the APS estimation loses the previous independency from the radar data and consequently it has to be now considered a hybrid approach. The assessment methodology exploited in the previous section is now applied to these new compensated data. Fig.8 refers to the coherence computed over short time windows (100 samples): the panel immediatly confirms, when compared to the ones in Fig.5, the presence of significant improvements for almost every target. The same positive results can be found by looking at the coherences evaluated over all the available image sequence (green curve in Fig.10), Figure 9. Residual example for a 570m range PS after the compensation with calibration: the improvements in APS estimation are evident, though critical zones (pointed out by the blue markers) still exist around noon. In the central hours of the day temperature and humidity register indeed the widest fluctuations, and most importantly temperature reaches the highest values. In these conditions even small drifts in humidity lead to big errors. or by directly observing the compensated phases of the targets (Fig.9), which in the most straighforward way convey the fact that the residuals have effectively been flatted. 5. CONCLUSIONS The brief but significant statistical analysis on the APS phenomenon confirmed the importance of a proper compensation process in order to reduce the heavy fluctuations affecting the data. The approach proposed in this paper, based on the meteo parameters (pressure, temperature and humidity) collected nearby the GB-SAR device, Coherence W/out Compensation Compensation w/out calibration Compensation with calibration Range [m] Figure 10. Coherence evaluated over the whole sequence (7 days) before compensation (blue) and after APS removal either without humidity calibration (magenta) or with calibration (green). The procedure without calibration marks great improvements for nearer targets, while the introduction of calibration benefits the farther scatterers.

6 offers the capability of building a delay model without having to rely on the acquired phase data, and thus being able to avoid the ambiguities intrinsically connected to the current techniques. The model has been tested on a 7-days real campaign dataset succeeding, on near range targets, in reducing the APS to fluctuations in the order of 2 mm (starting from centimeters). The subsequent model tuning by means of humidity calibration further lowered the residual displacement to less than 2 mm even for farher targets, up to approximatively m. Still, the technique has to face a few problems: the model loses accuracy in presence of turbolent atmospheric delay (typically around noon) and compensation at far ranges is still critical. However, solutions to these issues (by a better exploitation of meteo local statistics and/or by using more meteo measures) seem possible and are currently under investigation. ACKNOWLEDGMENTS The authors would like to thank prof. Paolo Mazzanti of the Universitá di Roma Sapienza and the IMG s.r.l. company for providing us with both the IBIS-L and meteo data collected in the Bolzano campaign. [7] CCIR Recommendations and reports of the ccir Technical report, International Radio Consultative Committee, International Telecommunication Union (ITU), Geneva, XVIth Plenary Assembly Dubrovnik [8] F. W. Murray. On the computation of saturation vapor pressure. Journal of Applied Meteorology, 6: , [9] A. Ferretti, C. Prat, and F. Rocca. Permanent scatterers in sar interferometry. IEEE Trans. Geosci. Remote Sens., vol. 39, no. 11:8 20, Jan [10] L. Noferini, M. Pieraccini, D. Mecatti, G. Luzi, C. Atzeni, A. Tamburini, and M. Broccolato. Permanent scatterers analysis for atmospheric correction in ground-based sar interferometry. IEEE Trans. Geosci. Remote Sens., vol. 43, no. 7: , Jul [11] L. Pipia, X. Fbregas, A. Aguasca, and C. Lopez- Martinez. Atmospheric artifact compensation in ground-based dinsar applications. IEEE Geosci. and Remote Sens. Letters, vol. 5, no. 3:88 92, Jan REFERENCES [1] D. Tarchi, H. Rudolf, M. Pieraccini, and C. Atzeni. Remote monitoring of buildings using a groundbased sar: Application to cultural heritage survey. Int. J. Remote Sens., vol. 21, no. 18: , [2] D. Leva, G. Nico, D. Tarchi, J. Fortuny-Guash, and A. J. Sieber. Temporal analysis of a landslide by means of a ground-based sar interferometer. IEEE Trans. Geosci. Remote Sens., vol. 41, no. 4:745752, Apr [3] G. Luzi, M. Pieraccini, D. Mecatti, L. Noferini, G. Guidi, F. Moia, and C. Atzeni. Ground-based radar interferometry for landslides monitoring: atmospheric and instrumental decorrelation sources on experimental data. IEEE Trans. Geosci. Remote Sens., vol. 42, no. 11: , Nov [4] Z. Zheng-Shu, W.-M. Boerner, and M. Sato. Development of a ground-based polarimetric broadband sar system for noninvasive ground-truth validation in vegetation monitoring. IEEE Trans. Geosci. Remote Sens., vol. 42, no. 9: , Sept [5] L. Essen and K. D. Froome. The refractive indices and dielectric constants of air and its principal constituents at 24 ghz. Proc. of the Physical Society (London), Section B, 64: , [6] H. J. Liebe. Modeling attenuation and phase of radio waves in air at frequencies below 100 ghz. Radio Sci., vol. 16, no. 6: , 1981.