Three data sets were used in the case studies. The levelling data set usually comprises the last 2 to 4 years. The levelling measures the level of two markers installed in the facades of the buildings at low elevation, relative to a network of fixed points in the inner city. Figure 1 gives an example of a levelling time series. The accuracy of the levelling measurements is in the order of 1 mm. The precision, expressed as the a-priori standard deviation, is around 0.5 mm (Amsterdam, personal communication).

Two pre-processed InSAR data sets were supplied by SkyGeo: a general data set covering the entire city (2009–2018), and a locally optimized InSAR data set (2014–2019). The locally optimised data set gives a larger number of data points and better reliability than the general Amsterdam InSAR data set. The accuracy of the InSAR measurements has not be assessed by direct comparison with reference measurements e.g. from permanent GNSS stations. Every InSAR processing uses 2000 to 3000 of the most stable data points in a radius of 5 km around the area of interest to determine reference time series. The precision, expressed as the a-priori standard deviation, is approximately 3 mm (Marinkovic et al., 2007).

For all InSAR datasets the deformation in the direction of sight of the satellite (line-of-sight deformation) is converted to the vertical deformation using the incidence angle of 31.1∘. This procedure assumes that the buildings do not move in the horizontal direction because their movement results from subsidence processes acting over a larger area. The analysis only uses InSAR data points with an elevation at least 2.5 m above street level to make sure that the points represent the buildings.

The data set includes all points within 2 m outside the footprint of the buildings, because the location precision of the InSAR points is 1 to 2 m (SkyGeo, 2020). Figure 2 gives an example of an InSAR time series.

The procedure to determine the measurement reliability has a statistical basis. The final result of the procedure is the probability that the deformation rate exceeds the threshold of 2 mm yr-1. The probability is calculated from the test statistic T=(rate-2)/srate assuming a Student t distribution with n-2 degrees of freedom. Here, rate is the deformation rate (positive when downward), srate is its standard deviation and n is the number of observations in the time series.

If this probability is larger than 5 %, the building is tagged for further investigation of its foundation. It is not enough to determine the average deformation rate, since it does not consider the scatter of the measurements. For this reason, the reliability of its outcome is unknown.

The main steps in the procedure are (Fig. 3):

Visualisation in GIS to assess the coverage of the data points.

A direct comparison between levelling and satellite measurements is meaningless for several reasons. Firstly, the location of the data points on the buildings is different. The levelling markers are located low in the facade of the buildings. The InSAR data points are located on the rooftops. This implies that the deformation observed in the levelling is determined foremost by the behaviour of the foundation. The deformation observed through the satellite is also determined by the deformation of the building itself.

Secondly, the levelling measures vertical deformation, the satellite data points are assumed to move only in the vertical direction. If in reality the data points also move horizontally this will affect the interpreted vertical deformation.

Thirdly, both types of measurements are sensitive in a different way to external changes such as seasonal effects. The frequency of the levelling is too low and the scatter too high to detect seasonal effects. Some InSAR time series display seasonal effects with a deformation amplitude up to several millimetres. Both types of measurements are sensitive to deformation of their reference points. Many levelling time series of neighbouring datapoints exhibit simultaneous random movements up and down in the order of several millimetres. This indicates a movement of the reference point. Also, the InSAR reference points show a periodic movement with an amplitude of 0.5 mm.

The locally optimised InSAR data set contains a significant number of time series with a non-linear trend, with the deformation rate increasing or decreasing in time. The current procedure can only handle linear trends. For this reason, the data set has been cut in two, and the analysis has been performed on the 2014–2019 data only. This operation produced a data set consisting of the most recent and relevant observations.

Figure 5 compares the deformation rates with probability larger than 5 % that the deformation rate exceeds 2 mm yr-1, for the buildings in the case study presented in Sect. 3.

The deformation rate in the levelling time series usually is smaller than in the satellite time series. The reason is not clear but could for example be due to settlement of the reference points.

The locally optimised InSAR data set shows the largest number of buildings with data points with probability larger than 5 % of exceeding the threshold deformation rate. Most of these data points are located in the middle or back of buildings, although data points in the front facades facing the canals indicate a probability less than 5 % of exceeding the threshold. At the moment is not clear how this observation should be interpreted in terms of the condition of the foundation of the building.

The general Amsterdam InSAR data set hardly has buildings with probability larger than 5 % of the deformation rate exceeding 2 mm yr-1. This may be caused by a large number of non-linear time series that were eliminated in step 4 of the procedure.

Both levelling and InSAR measurements indicate that the probability of a deformation rate exceeding 2 mm yr-1 is larger than 5 % for buildings 10, 11 and 12. Also, the levelling indicates that the probability of a deformation rate exceeding 2 mm yr-1 is larger than 5 % for buildings 15, 19 and 42, whereas the InSAR measurements do not. The levelling time series for these buildings may be less reliable because of jumps up to 1.5 mm between subsequent measurements and the limited (4 to 6) number of measurements in the time series. The jumps do not correlate with a seasonal pattern.

Because of the large number of measurements in a time series the conclusions about the deformation rate are much more robust for the InSAR measurements in comparison with the levelling. The standard deviation of the deformation rate of a levelling time series in the case study typically is in the order of 0.5 mm. This implies that a data point may have a 5 % probability that the deformation exceeds 2 mm yr-1, although its average deformation rate is below 1 mm yr-1. This is the case for buildings 10, 11, 12 19 and 42 in Fig. 5. For the same case study, the standard deviation of the deformation rate of an InSAR time series typically is in the order of 0.05 mm. For this reason, InSAR measurements provide a much more reliable estimate of the deformation rate.

Table 1 summarises the coverage of the different data sets for all three case studies. The traditional levelling is capable of providing a reliable conclusion for only 67 % of the buildings. The locally optimized InSAR data set has a 100 % coverage of the buildings, meaning that every building has one or more reliable data points.

Locally optimized PS-InSAR satellite measurements are most suitable for assessing the deformation rate of buildings in the Amsterdam inner-city. The case study presented shows that deformation rates of the satellite measurements are significantly higher than those of the levelling on the same buildings. The other two case studies (not shown in this paper) show a general agreement of the deformation rates of the levelling and the satellite measurements.

The locally optimised InSAR data set with measurements between 2014 and 2019 has reliable data points on 100 % of the buildings in all case studies. The large number of measurements in an InSAR time series makes the standard deviation of the deformation rate much lower than in a typical levelling time series, in spite of the larger scatter of individual InSAR measurements.

Future application of the procedure should include a comparison of the deformation rates derived from satellite measurements and the foundation condition of the buildings. This will improve the interpretation of the satellite measurements and reduce the number of false positives and false negatives. The improvement may include the interpretation of data points with high probability of the deformation rate exceeding 2 mm yr-1 that are not located near the facade of the buildings.

Also, the procedure may be extended to include analysis of non-linear trends such as second order trends and seasonal effects.