OR/19/043 Methodology

Novellino, A, Terrington, R, Christodoulou, V, Smith, H and Bateson, L. 2019. Ground Motion and Stratum Thickness Comparison in Tower Hamlets, London. British Geological Survey Internal Report, OR/19/043.

The newly defined AMG layer has been used as the model cap to adjust the thicknesses of the Lithoframe 50 model within the Tower Hamlets Borough where, on the other hand, values have been calculated considering the DTM as the model cap (Figure 5).

While superficial units (ALV – Figure 5a, RTDU – Figure 5b, LASI – Figure 5c, KPGR – Figure 5d, TPGR – Figure 5e, HAGR Figure 5f) tend to be very limited in spatial extent and of limited thickness (≤5 m), deeper units (LC Figure – 5g, LMBE – Figure 5h and TAB – Figure 5i) tend to cover a larger area and be thicker on average (≥10 m).

As the calculation of the thickness was completed using GSI3D, the thicknesses across this fault were calculated accurately.

Figure 5    Thickness map in the AoI after removal of the AMG thickness (see Figure 2) for ALV (a), RTDU (b), LASI (c), KPGR (d), TPGR (e), HAGR (f), LC (g), LMBE (h) and TAB (i). Contains Ordnance Data © Crown Copyright and database rights 2019. Ordnance Survey Licence no. 100021290.

The AMG and the underlying units have then been individually compared in space and time with ground motion information derived from InSAR, in order to disentangle trends and patterns of the terrain displacement that can be connected to the underground geology. InSAR is a remote sensing technique allowing the measurement of ground deformation from the phase difference between SAR images acquired over the same area at different times by ground, air or space platforms (Rosen et al., 2000). This approach has proved to be particularly suitable solution for long term monitoring (>20 years) at a relative low cost and high precision especially in urban areas like London where intense exploitation of aquifers occurs and vertical movements might be significant (Bateson et al., 2009; Aldiss et al., 2014; Bonì et al., 2018 ). The study has been performed combining data from 105 ascending and 111 descending SAR images acquired by Sentinel-1A/B satellites of the European Space Agency (ESA) between May 2015 to January 2018 with a nominal revisit cycle of 6/12 days. The small spatial and temporal separation (baseline) between satellite orbits provided by Sentinel-1A/B reduces decorrelation noise effects (Zebker and Villasenor, 1992) affecting the interferograms and to maximize the number of reliable measurement points and is the reason why only this dataset has been considered among the ESA SAR data available in the last 25 years for the area.

Processing was carried out using the GAMMA software by CGG’s NPA Satellite Mapping group within the framework of GIRP project (www.business.esa.int/projects/girp). Displacements are provided for 110 synthetic dates along the vertical direction resulting from the combination of the ascending and descending results for a total of 762 606 Measuring Points (MPs) minimally affected by temporal and geometric decorrelation over the Greater London Area (1,596 km² extended; Figure 6). Given an average standard deviation associated with the vertical velocity of ~1.5 mm/yr, a conservative velocity threshold of ±5 mm/yr has been applied to consider a MP unstable.

In the investigated area, the coordinates of each MP can be expressed in term of X, Y, Z with the latter usually corresponding to radar reflection from building roofs. During integral analysis of the building deformation state, however, buildings usually can be considered as a rigid body without considering elastic deformation because they are made of reinforced concrete and a rigid motion model is sufficient to describe the deformation state of the whole structure (Yang et al., 2016). Under this assumption, the displacement information from the roof can be assimilated to the displacement information from the bottom of the building that could be either the ground surface or the sub-surface where its foundations lie.

Figure 6    Vertical velocity measured by Sentinel-1 data during the period 2015 to 2018 across Greater London and the AOI (in red). Positive values indicate uplift and negative values indicate subsidence. InSAR data © CGG NPA Satellite Mapping 2018. Contains modified Copernicus Sentinel data 2014–2018. Contains Ordnance Data © Crown Copyright and database rights 2019. Ordnance Survey Licence no. 100021290.

The thickness of the AMG and the superficial and bedrock units has been analysed in connection to the 23 245 InSAR MPs within the AoI by considering two different semi-automatic ways of sorting MPs into groups that, in turns, could be correlated to the local geological or anthropogenic conditions:

1. MP average displacement rates.
2. Cluster analysis of the time series of the MP displacements.

Average displacement rates can represent an important clue especially when identifying linear patterns of motion like compaction of alluvial deposits but misleading when looking for shared patterns of deformation or when detecting non-linear or cyclic patterns of motion that have historically occurred in the Tower Hamlets area.

Clustering is one of the most common methods for unsupervised learning, where each data entry is given a cluster without any prior knowledge or input from the user.

The shape-based distance (SBD) clustering method was applied in the Tower Hamlets dataset as implemented in the k-shape algorithm (Paparrizos et al., 2015) from the R ‘dtwclust’ package (Sarda-Espinosa R., 2017). The Dynamic time warping Barycenter Averaging (DBA) method was used for the cluster’s centroid computation. With this process, the time series sequences are grouped into clusters where they have the most similar distance. Given x, y are two z-normalized time series subsequences and with m being their length, the shape-based distance (SBD) is calculated by the following equation:

(1)     

Where, CC_w is the cross-correlation in the position w, for each shift of x over time series y when x is slided by shifts 𝑠𝑠 ∈ [−m, m] and is computed by the formula:

(2)     

Where R_s if s=w-m, is computed in turn, as:

(3)     

The goal of the process is to find the position w at which the cross-correlation maximizes equation (1). The SBD takes values between 0 and 2, where a value of 0 indicates perfect similarity between two time series x and y. The above process is performed k times, with k being the number of clusters. The clustering algorithm requires a single user parameter, k, to cluster,the n time series observations (the satellite acquisitions in this case). However, because we do not know the number of clusters that exist in the dataset, an objective measure has to be used to find this optimal number 𝑘𝑘 < 𝑛𝑛. A variety of cluster validity indices are used in the literature and in this work the elbow method was applied to decide the optimal number of clusters (Kodinariya et al., 2013). To find the ‘elbow’, the variance of the distribution represented by total sum of square (TSS) distances for each cluster has to be computed. In this case, for a set of clusters k, the TSS is computed by the function:

(4)     

This calculation is performed for each different configuration of k-clusters and the TSS is plotted against the number of clusters. The algorithm is run several times because it returns a non- deterministic result with each experiment and as the number of clusters increases the TSS decreases and it will eventually become 0 if k becomes equal to the number of the time series sequences in the dataset.

The objective of this process is to find, either visually or by calculating the curvature, the number of clusters that resemble an ‘elbow’ in the graph. The elbow point shows the highest drop in the TSS and means that the algorithm has achieved a well grouped clustering.