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Displacement and deformation are fundamental measures of Earth surface mass movements such as glacier flow, rockglacier creep and rockslides. Groundbased methods of monitoring such mass movements can be costly, time consuming and limited in spatial and temporal coverage. Remote sensing techniques, here matching of repeat optical images, are increasingly used to obtain displacement and deformation fields. Strain rates are usually computed in a postprocessing step based on the gradients of the measured velocity field. This study explores the potential of automatically and directly computing velocity, rotation and strain rates on Earth surface mass movements simultaneously from the matching positions and the parameters of the geometric transformation models using the least squares matching (LSM) approach. The procedures are exemplified using bitemporal high resolution satellite and aerial images of glacier flow, rockglacier creep and land sliding. The results show that LSM matches the images and computes longitudinal strain rates, transverse strain rates and shear strain rates reliably with mean absolute deviations in the order of 10^{−4} (one level of significance below the measured values) as evaluated on stable grounds. The LSM also improves the accuracy of displacement estimation of the pixelprecision normalized crosscorrelation by over 90% under ideal (simulated) circumstances and by about 25% for real multitemporal images of mass movements.
Remote sensing is highly suited for slope monitoring in inaccessible areas such as high mountains and cold regions where mass movement processes such as glacier flow, permafrost creep and rock sliding are common. Repeat optical image matching is used to compute displacements on slope movements within the temporal baseline of the images’ acquisitions [
The NCC, and other similarity measures, is subject to a number of shortcomings: (1) The NCC is only reliable in cases where the template is not significantly deformed, but only shifted in position. (2) The precision with which the matching position is located is limited to the pixel size unless subpixel precision procedures [
Matching of orthorectified and coregistered bi/multitemporal images of the Earth surface can be thought of as looking to the changing object from a fixed camera multiple times with small (or absent) spatial baselines and dominant temporal baselines. The spatial transformation models are in this case used to model the distortion of the mass, not the perspective one as done in multiangular parallax matching [
There is limited application of this powerful image matching approach in mass movement analysis [
The LSM algorithm works based on the L2norm theorem to determine the best matching position by adjusting the geometry and radiometry of the matching templates so that the sum of squares of the grayvalue differences (SSD) between the two templates is minimized [
A moving slope, here represented by an image template, may also change geometrically. A geometric transformation model, characterized by parameters
LSM, as a precise matching algorithm, is expected to produce precise velocity for each pixel of the template. This gives the advantage that velocity gradients,
The tangent of the shearing angle or the angle in radians (as the angles are often very small) for each direction is the same as the shearing factors (
The horizontal length of the ground surface plane changes when there is compression or extension. As ice is incompressible, and thus also to a large extent masses that are supersaturated with ice, horizontal compression is usually balanced by vertical extension or vice versa [
In this study, the radiometric parameters are not used further than estimation of the matching position although it can potentially be used in estimation of surface albedo changes.
In this study, the computations were conducted on three real mass movement image pairs and two pairs of simulated deformed images. The first bitemporal set of images (
The second image pair is a section from panchromatic aerial images over the Nigardsbreen glacier in Southern Norway (approx. 61.68°N, 7.20°E). The images were acquired on 19 and 29 August 2001 within the EU Glaciorisk project (
The third bitemporal image set (
A simulated image pair was created by analytically deforming a section of the Nigardsbreen aerial photographic image by using an affine geometric model. Gaussian noise of mean zero and variance (σ^{2}_{n}) 0.01 was added. Another simulated image pair was created by using the same deformation model but adding a higher level of noise, σ^{2}_{n} of 0.1. The two simulated deformation image pairs are used for the evaluation of the algorithm as the actual displacements and transformation parameter values are precisely known for these images, in contrast to the real mass movement images.
Considering older images as reference and newer images as search images, the images are first matched using the pixelprecision NCC algorithm to estimate the initial model parameters for the least squares adjustment. Initially, the matching templates are assumed to have the same geometry (except position). Therefore, for each dimension a unit geometric and radiometric scaling factor is used keeping the other parameters at zero. The least squares iteration starts by using these initial values. Template sizes of 51 by 51 pixels are used for the matching. Smaller template sizes were not used to suppress noises and avoid ambiguity. As LSM is applied in a later step, the presence of displacement gradients in such large templates is not of concern, or even desired as we aim at deformation measurement in addition to displacement. In fact, as the images are of high resolution, the templates are not large in ground size,
LSM is implemented using the affine transformation model following the procedures explained in Section 2.1. The gradients are computed from the matching template, not from the reference template. Intensity values at subpixel positions are interpolated using the cubic convolution [
After the parameters have converged, the horizontal surface displacement of each pixel is computed separately as Euclidian distance between the pixels’ positions in the reference and the matching templates. Velocity is then computed as the displacement divided by the temporal baseline of the image pair. The movement direction is computed as the arctangent of the ratio of the displacements of the easting and northing directions in angles from the north. For later comparison, the displacements for each mass movement are also registered for the pixelprecision NCC. The mean and standard deviation of the velocities are computed. The standard deviation is computed as the square root of the sum of square differences between individual values and the mean divided by the total number of observations.
The transformation parameters are computed for the
The precision of the LSM algorithm in matching and computation of displacement and strains are evaluated using error propagation principles [
Accuracy (validity) assessment of the measurements such as displacement and strain rates is conducted on stable grounds and simulated deformation images. The stable grounds are expected to have zero values for the displacement, strain rates and rotation. Simulated deformed images have known displacement and deformation parameter values. The mean absolute deviation (MAD) between the actual and the computed values is used for the evaluation. Stable grounds are, however, not a complete representation for the mass movements as one source of error, which is the deformation, does strictly speaking not exist on the stable ground. Thus, to compare the performance of the algorithm over the moving masses to that of the NCC, the SNR of reconstructing the reference image from the search image is computed as detailed in [
The horizontal surface velocity statistics of the three mass movements investigated is presented in
The velocity magnitude map of the Muragl rockglacier computed using the LSM is presented in
The velocity magnitude map of the La Clapière landslide is presented in
As stated above, the SNR of reconstructing the reference image of
The deformation parameters investigated in this study are the longitudinal strain rate (
The
The horizontal compression and extension are compensated for by the vertical extension and compression respectively as the rockglacier can roughly be assumed to be incompressible due to its substantial ice content. This means the negative sum of the horizontal strain rates presented in
The
Summary statistics of the computed deformation parameters for the Nigardsbreen glacier (maps not shown here) are given in
The computed
When it comes to the surface shear strain rate (
Error propagation using the covariance matrix of the geometric parameters of the affine transformation model (Kp_{i}) shows high precision (
The MAD of the strain rates on the stable grounds are in the order of 10^{−4} for the glacier and rock glacier and 10^{−3} for the landslide (
Although no formal quantitative comparison is conducted, the velocities obtained for the mass movements using the LSM are in agreement with those obtained in other studies using similar and other methods. For example, similar velocities are registered for the same section of the Nigardsbreen glacier during the same period [
Realistic values of longitudinal, transverse and shear strain rates together with rotation rate are also obtained. The technique computes strain rates at higher resolution than the conventional technique of computing them from velocity gradients after the matching. When computing strain rates of a template from the velocity gradients, two neighboring templates are used for each orthogonal dimension. Therefore, the computed negative total sum of strain rate is in a way averaged over neighboring templates. Additionally, such strain rates are simply measures of velocity changes between the central pixels of the neighboring templates especially when the NCC is used for the matching. Thus it can appear smoothed even before filtering. Changes in the size and shape of the masses are not directly computed as is done when using the LSM algorithm as presented here. As a result microscale deformations are detected with LSM.
The spatial patterns of the strain rates and elevation changes of the Muragl rockglacier, previously computed from velocity gradients by [
The results of the study show that the LSM computes horizontal displacements in Earth surface mass movements with significantly higher precision (level of detail of measurement) and accuracy (truthiness of the estimated values) compared to the NCC. The mean precision of the LSM algorithm in locating the matching position is found to be between 0.06 and 0.15 pixel; whereas, the matching precision of NCC without subpixel extensions is generally ±0.5 pixel. In addition to the precision of matching, the accuracy of the computed displacements is also higher when computed using LSM than using NCC as evaluated on test images and stable grounds of the bitemporal mass movement images. The better performance of the LSM is in agreement with theoretical claims and earlier findings in photogrammetry on image pairs of shorter temporal baselines [
When computing deformation and displacement of mass movements from repeat images using a precise algorithm such as LSM, the sources of error are basically related to either the image (noise, orthorectification and coregistration) or the ground itself (deformation and temporal surface changes). Both major error sources can technically be grouped into three,
Due to the high resolution of strain computation, the maps of the strain rates look noisy when visually observed, suggesting the application of noise filters. However, in the case of the strain rate maps, it might well be that highresolution deformations actually are somewhat noisy due to real local deformation of the masses, such as from crevasse formation, or due to the error sources mentioned above. Filtering would lead to smoothing of the map. In so doing it affects both the realistic values and the blunders. The use of larger template sizes also leads to a more smoothed strain rate map. Recall that the criteria for the right template size in the NCC is the presence of adequate SNR and constant displacement within the template. In the LSM, constant displacement is not anymore a criteria but rather a constant displacement gradient, at least for the affine model. Thus, for very large templates, as the parameters of the transformation model are forced to be constant within the template, the computed strain rates visually look like as if they are filtered. Such smoothed or filtered strain rate map may be sufficient or even wanted for some geoscientific applications such as numerical models. However, the detailed variability may be needed for other current and future applications, and provide new insights into the mechanics of mass movements.
A better approach towards removing noises than filtering or the use of much larger template sizes would therefore be further restricting the least squares iteration process. Pixelbased constraining such as data snooping or templatebased constraining such as raising convergence precision can be used [
This study explored the possibility of automatically and simultaneously computing displacement, strain rates and rotation of Earth surface mass movements from repeat high resolution satellite and aerial optical images. The performance of least squares matching (LSM) with an affine geometric transformation model is evaluated in relation to that of the most widely used algorithm for such purposes,
The results of the study clarified that the LSM estimates the displacement of Earth surface masses with better precision and accuracy than the conventional NCC. Around 25% improvement in the SNR gain of image matching over that of the NCC is registered in real images reaching up to double in the case of the analytically deformed images. The improvements in the SNR gain lead to comparable improvements in the accuracy of the estimated displacement. Up to 35% reduction in the MAD of displacement by the LSM from that of the NCC is observed on the stable grounds of the mass movement images reaching up to 90% in the case of the simulated deformation. The exact magnitude obviously varies depending on the application scenario. The improvements are dependent on the level of noise in the images.
The study has also demonstrated the capability of the LSM in deriving surface strain rates and rotation rate simultaneously with the image matching process. This has the potential of replacing earlier approaches based on postprocessing from displacement gradients. Additionally, the spatial density of deformation parameters measured, and thus the unprecedented level of detail of deformation fields obtained, might allow for new insights into the mechanics of the masses observed. The strain rate data obtained through such processes are found to be realistic when compared with data from different sources and when logically evaluated. The spatial transformation parameters from which the strain rates are derived are computed with precisions better than 10% of the measured values in all cases. However, evaluation of the accuracy (validity) of the rotation and strain rates on the stable grounds of the mass movement images shows that the accuracy is dependent on the mass movement type,
The capability of deriving surface strains from images through the LSM algorithm advances the application of image matching in mass movement analysis. Once strain is reliably computed from repeat images automatically through image matching, the stress exerted on the masses can also be computed using the stressstrain relationship for the specific type of mass under investigation. This is very important in early warning related to slope instabilities and failures, and in understanding terrain kinematics, and potentially dynamics, in high mountain areas. Nevertheless, validation of the computed strain under different conditions and possibility of extending to the computation of stress requires further research.
The algorithm is computationally expensive as it involves iteration for each template. The ever improving computer processor speeds coupled with smarter computational approaches can deal with this limitation. Initial tests indicate that the algorithm is very sensitive to noises in the images. More work is thus needed to define the sensitivity and applicability range of the LSM approach for repeat images of lower resolution, of more strongly deforming masses, and with longer temporal baselines.
The orthoimages of the Muragl rockglacier are based on aerial photos acquired by the Swisstopo/flight service (BA057212, BA081844). The Nigardsbreen aerial images are courtesy of EU Glaciorisk. This study is funded by the Research Council of Norway (NFR) through the CORRIA project (no. 185906/V30) and contributes to the NFR International Centre for Geohazards (SFFICG 146035/420), the ESA Glaciers_cci project (4000101778/10/IAM), and the Nordic excellence center SVALI. The QuickBird images are courtesy of DigitalGlobe. Special thanks go to the anonymous referees, and to EditorinChief, Prasad Thenkabail for the comments that substantially improved the paper. We would like to thank Kimberly Casey for proofreading the manuscript.
Deformation of a template and transformation of its parameters from eastingnorthing to longitudinaltransverse axis.
A section of the aerial photographic image (400 m × 600 m) over part of the Muragl rockglacier taken on 7 September 1981, North to the top.
A section of the aerial photographic image (1,233 m × 1,233 m) over part of the Nigardsbreen taken on 19 August 2001, North to the top.
A section of the QuickBird panchromatic satellite image (1,600 m × 2,430 m) over the La Clapière landslide taken on 06 September 2003, North to the top.
Velocity (ma^{−1}) maps and vectors of (
The longitudinal
The negative sum of the horizontal strain rates (assumed to be equal to the vertical strain rate for an incompressible medium) for the Muragl rockglacier.
The negative sum of the horizontal strain rates (assumed to be equal to the vertical strain rate) of the Nigardsbreen glacier.
The longitudinal
Parameters of the spatial transformation models related to mass movement processes.
p_{1} and p_{4}  Translation 

Quantifies shift, creep, slip, and slide. Change in position disregarding shape and size. 
p_{2} and p_{6}  Scaling 

Measures change in length due to compressive/extending motion. Measures normal strain (elongation). 
p_{3} and p_{5}  Shearing (and/or rotation) 

The slippage of orthogonal masses in relation to one another (and/or rigid rotation of the mass). Measures shear strain and/or rotation angle. 
Overview statistics for horizontal surface velocity for the three mass movements.
Mean  0.18  0.57  0.4 
Standard deviation  0.09  0.32  0.32 
Maximum  0.5  1.1  2.75 
Mean precision of the shift parameters for the three bitemporal mass movement image pairs.
Muragl rockglacier creep  0.07  0.06 
Nigardsbreen glacier flow  0.11  0.12 
La Clapière landslide  0.13  0.15 
The MAD of the errors of displacement on the simulated deformation images and the stable grounds of the real mass movement images.
Simulated (σ^{2}_{n} = 0.01)  0.38  0.04  90 
Simulated (σ^{2}_{n} = 0.1)  0.42  0.2  52 
Muragl rockglacier creep  2.25  1.4  37 
Nigardsbreen glacier flow  0.47  0.34  27 
La Clapière landslide  2.5  2.01  19 
Summary statistics of the computed deformation parameters of the Muragl rockglacier.
Longitudinal strain rate (ma^{−1})  0.00012  0.0018  −0.0077  0.0085 
Transverse strain rate (a^{−1})  0.00039  0.0016  −0.0061  0.0077 
Shear strain rate (a^{−1})  0.0015  0.0016  0  0.01 
Rotation rate (degrees a^{−1})  0.068  0.066  0  0.44 
Summary statistics of the computed deformation parameters of the Nigardsbreen glacier section.
Longitudinal strain rate (d^{−1})  −0.0003  0.0024  −0.01  0.012 
Transverse strain rate (d^{−1})  −0.00067  0.002  −0.01  0.007 
Shear strain rate (d^{−1})  0.0017  0.002  0  0.019 
Rotation rate (degrees d^{−1})  0.072  0.076  0  0.54 
Summary statistics of the computed deformation parameters of the La Clapière landslide.
Longitudinal strain rate (a^{−1})  −0.0009  0.005  −0.028  0.017 
Transverse strain rate (a^{−1})  −0.0007  0.005  −0.01  0.017 
Shear strain rate (a^{−1})  0.005  0.005  0  0.03 
Rotation rate (degrees a^{−1})  0.15  0.13  0  0.7 
Precision of the geometric (shape and size) parameters of the spatial transformation model for the three mass movement types.
Muragl rockglacier creep  0.002 (0.997)  0.002 (0.021)  0.002 (0.022)  0.002 (0.998) 
Nigardsbreen glacier flow  0.003 (0.997)  0.003 (0.013)  0.003 (0.021)  0.003 (0.999) 
La Clapière landslide  0.003 (1.001)  0.003 (0.038)  0.003 (0.044)  0.003 (0.098) 
The standard deviation of the errors of the computed rotation and strain rates as computed on the stable grounds of the three mass movement images.
0.0001  0.0005  0.006  
0.0001  0.0006  0.002  
0.0002  0.001  0.005  
0.008  0.05  0.1 