Measuring thermal expansion using X-band persistent scatterer interferometry

This paper is focused on the estimation of the thermal expansion of buildings and infrastructures using X-band Persistent Scatterer Interferometry (PSI) observations. For this purpose an extended PSI model is used, which allows separating the thermal expansion from the total observed deformation thus generating a new PSI product: the map of the thermal expansion parameter, named thermal map. The core of the paper is devoted to the exploitation of the information contained in the thermal maps: three examples are discussed in detail, which concern a viaduct, a set of industrial buildings and two skyscrapers. The thermal maps can be used to derive the thermal expansion coefficient of the observed objects and information on their static structure. In addition, the paper illustrates the distortions in the PSI deformation products that occur if the thermal expansion is not explicitly modelled. Finally, an inter-comparison exercise is described, where the thermal expansion coefficients estimated by PSI are compared with those derived by a Ku-band ground-based SAR campaign.

One of the main advantages of PSI is its sensitivity to small deformations. Several validation experiments have shown this property for C-band PSI. Ferretti et al. (2007) show submillimeter accuracy with an experiment based on dihedral reflectors. In the PSIC4 project, the inter-comparison of deformation velocities estimated by different teams yielded standard deviations that range from 0.6 to 1.86 mm/yr (Raucoules et al., 2009). Another intercomparison experiment, from the Terrafirma Validation project, gave standard deviations of the velocities that range from 0.45 to 0.66 mm/yr (Crosetto et al., 2007).
With the advent of X-band data, PSI has experienced a further improvement of its sensitivity to deformations, e.g. see Crosetto et al. (2010). The sensitivity is so high that the interferometric phases (i.e. the main PSI observations) often contain a non negligible component related to thermal expansion, i.e. PSI senses the displacements that are caused by temperature differences in the imaged area between SAR acquisitions. It is worth noting that this has also been reported in some C-band PSI studies, e.g. Ferretti et al. (2005), Perissin and Rocca (2006) and Crosetto et al. (2008). However, these results are often related to the deformation time series of single Persistent Scatterers (PSs), while with X-band PSI the thermal expansion is evident over large sets of PSs. This makes possible the analysis and interpretation of the thermal expansion signal of single objects like buildings, bridges, etc. (Monserrat et al., 2011;Fornaro et al., 2013).
The X-band PSI capability of sensing the thermal expansion, which is illustrated in the following sections, represents a powerful argument to promote this satellite-based deformation monitoring technique. However, it has an important disadvantage: the total observed deformation contains both the thermal expansion and the deformation signal of interest. Note that in deformation analysis what usually matters is the total observed deformation without thermal expansion effects. For this reason, in order to derive proper deformation estimates, the two components have to be separated. Note that the same problem occurs in many other highprecision deformation monitoring techniques, e.g. see Dong et al. (2002). This paper discusses two main issues related to thermal expansion: how to separate the thermal expansion from the total deformation and how to exploit the information related to the thermal expansion. The first issue is addressed in the following section, which describes an extended model for X-band PSI.
This model generates a new PSI product: the so-called thermal maps. This is followed by a section which provides a discussion and the interpretation of three examples of thermal maps: a viaduct, a set of industrial buildings and two skyscrapers. Then an inter-comparison exercise is described, where the thermal expansion coefficients estimated by PSI are compared with those derived by Ku-band Ground-Based SAR (GBSAR) data. Finally, the conclusions of this paper are summarized.

An extended model for X-band PSI
In this section we briefly discuss how the thermal expansion component can be properly modelled and estimated starting from X-band PSI interferometric phases, , see Monserrat et al. (2011). This estimation requires an extension of the classical two-parameter PSI model (deformation velocity, v, and residual topographic error, RTE), by including a thermal expansion parameter, which is related to the average temperature differences between the acquisition of the images. The extended model is: where: k e  is the differential interferometric wrapped phase, where k is the interferogram number, e the edge that connects two PSs and  indicates the difference of the phases of the two PSs. The precision achievable in the estimation of this parameter is discussed in Monserrat et al. (2011) and Fornaro et al. (2013). The former reference reports a standard deviation Th  of 0.04 mm/•C, which was estimated over homogeneous portions of the runways of an airport.
Alternatively to the above model, other approaches have been proposed in the literature, which make use of a seasonal sinusoidal deformation model (Gernhardt et al., 2010;Duro et al., 2010). However, the above model is more accurate because it explicitly makes use of the temperature, thus avoiding the hypothesis of a sinusoidal temporal evolution. ], with k varying from 1 to the N th available interferogram.

Application of the extended model
This section provides the discussion and interpretation of three different applications of the extended model. They concern different types of structures that can be found in an urban and peri-urban environment: a viaduct, a set of industrial buildings and two skyscrapers.

Thermal map over a viaduct
This section describes the first example of thermal map, which concerns a viaduct, see Fig. 1.
This structure shows a periodic pattern, which is clearly visible in single interferograms ( Fig.   1a). Using 27 StripMap TerraSAR-X images and 51 interferograms, which cover the period December 2007 to November 2009, the extended model was used to analyse this structure. Fig.   1b shows the deformation velocity map, which indicates that the structure is stable. Fig. 1c shows the corresponding thermal map, while Fig. 1d shows an enlargement of this map. These two maps clearly display a periodic pattern, with 96 m, that corresponds to the locations of the thermal joints of the viaducts, which are highlighted by white squares in Fig. 1d. Between two subsequent joints, the thermal map has values that range approximately from -0.35 to +0.3 mm/ºC. This makes a total difference of 0.65 mm/ºC, which is observed along the satellite Line-Of-Sight (LOS) direction. Assuming that this is caused by horizontal thermal expansion along the main bridge direction and for a length of 96 m, this corresponds to an estimated coefficient of linear thermal expansion of 11.7·10 −6 /ºC, which is typical for reinforced concrete, e.g. see Merritt et al. (1976). The interpretation of the result from Fig. 1c and 1d involves three basic properties of the viaduct: the material and hence the thermal expansion coefficient, the geometry and the static structure. Fig. 2 shows a scheme of the viaduct, where the geometry was derived from Google Earth and the static structure was inferred directly from the thermal expansion map: it includes a pinned joint and three slider joints. The white arrows indicate the vertical and horizontal components of the parameter Th in correspondence of the four supports, while the black arrows show the sum of the above components projected in the LOS direction. The above components were computed from the static scheme and using the coefficient of linear thermal expansion of 11.7·10 −6 /ºC. The static scheme was validated by an in situ inspection of the viaduct.

Thermal map over industrial buildings
The second example concerns a set of industrial buildings located in Barcelona (Spain). Fig. 3a shows an orthoimage of the area at hand. In this case the PSI analysis was performed using 13 StripMap This result differs substantially from the velocity map estimated using the extended PSI model, see Fig. 3d. The difference reflects the effect of thermal expansion, which is not modelled in Fig. 3b and, as a consequence, generates distortions in the velocity map, thus creating a kind of "virtual displacement rates". By contrast, when the thermal expansion component is properly modelled it generates a much more accurate deformation velocity map and, in addition, the thermal map is obtained, see Fig. 3c. This example is useful to illustrate the importance of properly modelling thermal expansion: if this is not performed, the PSI end users should be warned on the risk of obtaining deformation products that potentially contain spurious patterns due to unmodelled thermal expansion effects. Note that this risk increases if the PSI products are derived from PSI datasets that cover short periods e.g. of several months. It is worth noting that the velocity map shown in Fig. 3d still shows a residual thermal pattern: this is due to the limited PSI dataset that does not allow the velocity to be perfectly separated from the thermal parameter. The thermal map shown in Fig. 3c displays values that range approximately between -0.6 and +0.5 mm/ºC. These values reflect the displacements of the roofs of the analysed buildings. Note that the pattern that is visible in the thermal map is directly related to the LOS direction, which is indicated in Fig. 3a.

Thermal map over two skyscrapers
The results discussed in this section concern two skyscrapers observed with two datasets: 28 Let's discuss the results of SB. Fig. 4a  are strongly correlated and this results in a velocity map that is strongly distorted by the thermal expansion. By contrast, in the TSX dataset, which almost doubles the number of SAR images, these two vectors are much less correlated, preventing the corruption of the velocity map by the thermal expansion, which, however, remains as an unmodelled term. This has a direct impact on the temporal model coherence, which is the goodness-of-fit parameter used to measure the quality of the modelling (Ferretti et al., 2001). In the case of CSM this parameter is quite high, and hence the PSs are accepted, while in the case of TSX it is low and the PSs are rejected (the used threshold was 0.6). Fig. 4c and 4d show the deformation velocity maps estimated with the extended model and both indicate stability. Finally, Fig. 4e and 4f show the thermal maps, with values that roughly range from 0 to +1 mm/ºC.
It is worth noting that the behaviour of SA slightly differs from that of SB: the thermal expansion parameter Th is sensibly lower. In the estimation based on the TSX dataset, which is the more reliable, the Th values approximately range between 0 mm/ºC and 0.55 mm/ºC, while in the neighbour skyscraper, which has exactly the same height, these values reach 1 mm/ºC. This difference is probably due to the different static structure of the two skyscrapers: for SB the main structure is, at least partially, external and hence directly exposed to the seasonal temperature changes. In contrast, for SA these changes can be more moderate due to the fact that its main static structure is internal, i.e. not directly exposed to the seasonal temperature changes.
The reduced magnitude of Th values of SA is reflected in Fig. 4b, where the velocity map is correctly estimated even using the two-parameter model. A similar result is shown in Fig. 4a, even though it is noisier. Finally, it is worth noting that the CSM solution (both velocity and thermal maps) for SA is slightly different from the homologous TSX solution. For instance, the CSM thermal values are slightly overestimated with respect to the TSX ones. This is caused by the limited size of the CSM dataset, which does not allow us to correctly separate the deformation velocity from the thermal expansion parameter.

Inter-comparison based on ground-based SAR data
This section describes an experiment where the thermal expansion coefficients estimated from TSX X-band PSI were compared with those derived by Ku-band GBSAR data. The experiment was focused on the skyscraper SB described in the previous section: 139 GBSAR images were acquired over a 12-h monitoring using an IBIS-L GBSAR manufactured by IDS Spa (www.idscorporation.com). At the same time, thermal camera measurements were performed using the Thermo Tracer TH9100 WR manufactured by NEC, to monitor the temperature of the SB façades.
The LOS displacement time series were derived from the 139 GBSAR images. The maximum observed LOS displacement is about 13 mm as shown in Fig. 5. It is worth noting that the above displacements were computed by neglecting the atmospheric effects. In fact, we found that the atmospheric effects, in terms of displacement, were below 0.2 mm by using a refractive model (e.g. see Iannini and Guarnieri, 2011) and the measured air humidity, pressure and temperature values.
The analysis was then focused on the data that concern one of the corners of SB. For a given point of this corner, the GBSAR Th value in the GBSAR LOS was computed as the ratio between its displacement magnitude (estimated from the GBSAR time series) and the temperature  The TSX PSI estimates, PSI Th , were taken from the data shown in Fig. 4e using an average temperature of the city at the time of acquisition of the SAR images.

Conclusions
In this paper the capability of X-band PSI to estimate the thermal expansion of buildings and infrastructures has been discussed. Two main issues related to this topic have been addressed.  X-band PSI has a high sensitivity to subtle displacements, including those caused by thermal expansion.
 The thermal maps provide a new type of information, which is related to a physical property of the observed objects (the thermal expansion coefficient) and to their static structure. This has been shown in the example of the viaduct, where a structural static scheme that includes a pinned joint and three slider joints was derived.
 The thermal maps can highlight different thermal expansion behaviours that are related to the type of static structure of the building or infrastructure at hand. This has been shown in the analysis of the two skyscrapers: even though they have the same height, their thermal expansion behaviour is sensibly different.
 If the thermal expansion is not explicitly modelled, the PSI deformation products can be affected by strong distortions, which can be particularly severe if limited PSI datasets in relatively short periods are analysed.
 A weak point of the proposed procedure is given by the problems of parameter estimability, which depend on the number of available SAR images (the larger, the better) and the correlation (the lower, the better) between the vectors [ k ], with k varying from 1 to the N th interferogram. Some of these problems have been illustrated in the case study of the skyscrapers.
Finally, an inter-comparison exercise has been described in the last part of the paper. The thermal expansion coefficients estimated with PSI have been compared with those derived by a Ku-band GBSAR campaign. The good agreement shown by the analysed data confirms the soundness of the proposed PSI procedure.