The LSM algorithm works based on the L2-norm 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 gray-value differences (SSD) between the two templates is minimized [13,18,19]. Assume a discrete two dimensional function F(x, y) represents intensity data of a subset of an image taken over a certain area at a time t₁. G(x′, y′) represents intensity data of a subset of another image taken over the same area at some other time t₂. In principle, if the two subsets (templates) match, the two functions will be equal. However, due to the likely presence of noises in both images, random noise (e) is added (Equation (1)). In an ideal situation e or its expectation equals zero. In the displacement measurement of Earth surface mass movements, the two images are often taken with long temporal baselines. Consequently, systematic radiometric changes can exist due to various factors such as changes in surface condition, illumination, noise, imaging condition and geometric distortions. It is often assumed that the systematic radiometric differences can be accounted for by gain (λ) and offset (η) parameters as in Equation (2).

A moving slope, here represented by an image template, may also change geometrically. A geometric transformation model, characterized by parameters p₁ to p_n, relates the pixels of the reference template to those of the search template for each dimension, as shown for the affine model in Equations (3) and (4). The relationship between the two intensity data will consequently incorporate the parameters of the model (Equation (5)). This relationship can be linearized using Taylor-series expansion and rewritten in matrix form as in Equation (6). Here, v is the residual, A are the differential quotations computed from the gray-value gradients, l is the difference between the gray-values of the matching templates, and dp is an element of the adjustments (dp₁,…, dp_n, dλ, dη) to the initial parameter values that assume the same size, shape and orientation. The solution for dp is then computed using Equation (7). The process is iterated until the computed dp converges close to zero. LSM is an unbiased minimum variance estimator with the variance computed as in Equation (8) where RC is the total number of observations (pixels in the template) and n is the number of model parameters. Further details of the LSM algorithm are given in [18,20,21].

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, i.e., strain rates, can be computed at sub-template level, even if computed after the matching. Additionally, the spatial transformation parameters (p_i) are measures of the geometric deformation of the masses between the acquisition times of the two images. The specific meanings of each of the spatial transformation parameter in mass movements are presented in Table 1 together with a sketch of the geometric changes represented. The spatial transformation parameters can be exploited to automatically compute strain rates simultaneous with the matching as outlined below.

Figure 1 reduces the reality of Earth surface mass movements to the square template depicted, which has unit sizes in both dimensions (denoted as l_xo and l_yo respectively). The unit lengths can also be thought of as diameters (2r) of a circle as shown in the figure. After deformation, the template moves to a new location, attains new sizes (l_xi and l_yi) and gets rotated and/or sheared by angles γ₁ and γ₂ to a new shape. The circle now becomes an ellipse with maximum extension and compression shown orthogonal to each other (dashed axes of the ellipse). The orientation of the ellipse is not necessarily along the movement direction due to shearing and rotation. Notice that the central pixel itself is also deformed and displaced from the center. The Euclidian distance between B and B′ (parallel to the longitudinal direction) is the horizontal displacement magnitude while ω is the angular displacement direction measured clockwise from the north. The change in size can be quantified using extension ratios (S_x and S_y) as in Equations (9) and (10). S_x and S_y are the same as the scaling factors p₂ and p₆ of Table 1. The corresponding normal strains (ε_x and ε_y) and their relations with the scaling factors are given in Equations (11) and (12), adapted from [22,23]. Scaling factors greater than 1 (i.e., ε_x and ε_y values greater than 0) imply extension, where as scaling factors less than 1 (i.e., ε_x and ε_y values less than 0) imply compression.

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 (p₃ and p₅) of the transformation models. Shear results in both shearing and rotation. Thus, shear strain (γ_xy) and rotation angle (φ_xy) are derived from the shearing angles as given in Equations (13) and (14) respectively [24]. Strain is better expressed in terms of strain rate (ɛ̇_i) which is strain per unit time (Equation (15)). The strain rates in this study are expressed in the time unit that is covered by the dataset in order to avoid extrapolating to time periods not supported by data.

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 super-saturated with ice, horizontal compression is usually balanced by vertical extension or vice versa [25], making the total sum of the horizontal and vertical strain rates equal to zero. Compressive/extending motion is well discussed for glaciers [24,26,27] and rockglaciers [25,28,29]. In all cases, longitudinal (i.e., along the flow direction) and transverse (i.e., across flow direction) extension and compression are mentioned to lead to such processes as the formation of crevasses, calving and surface micro-topography. Shearing may lead to cracking of the masses and eventually collapses.

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 bi-temporal set of images (Figure 2) was taken over a rockglacier in the Muragl valley of the Upper Engadine area in the Swiss Alps (approx. 9.92°E, 46.50°N). The orthoimages used in the present study were based on aerial images taken on 7 September 1981 and 23 August 1994 with 13 years of temporal baseline and 0.2 m of spatial resolution. The Muragl rockglacier has been under investigation for decades using technologies such as photogrammetry, geodesy and geophysics to understand its mechanics [30,31].

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 (Figure 3). The images were orthorectified using photogrammetric stereo pairs and automatic DEM extraction of the two dates. The ground resolution of the orthoimages is 0.3 m. Surface changes within the very short time period of 10 days were very small, besides glacier flow. Additional details on the images and on their glaciological analysis can be found in [32].

The third bi-temporal image set (Figure 4) is from panchromatic images taken by the high spatial resolution QuickBird satellite over the La Clapière landslide (approx. 44.25°N, 6.94°E,) in the French Alps near the town of Saint-Etienne-de-Tinée. The first image was taken on 6 September 2003 with a mean in-track view angle of −0.5° and a mean cross-track off-nadir view angle of 9°; whereas, the second image was taken on 27 September 2010 with a mean in-track view angle of −0.4° and a mean cross-track off-nadir view angle of 4.2°. The ground pixel size of the images is 0.6 m. The images were orthorectified in PCI-Geomatica using the SRTM DEM and GCPs collected from stable areas based on the French 3D web service (www.geoportail.fr). As the orthorectification based on this base-map could not result in accurate co-registration, subsequent co-registration using polynomial transformation model was performed using precise GCPs collected from outside the landslide area. Finally, a mean co-registration error of about 1.2 m (2 pixels) is recorded from the horizontal shifts outside the landslide area. The La Clapière landslide has been intensively investigated over the years using different techniques [3,33–35].

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 (σ²_n) 0.01 was added. Another simulated image pair was created by using the same deformation model but adding a higher level of noise, σ²_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.

The precision of the LSM algorithm in matching and computation of displacement and strains are evaluated using error propagation principles [18,38]. The precision of the estimated parameters can be expressed by the covariance matrix (Equation (20)). However for the precision of the matching, Gruen [18] suggests the use of the standard deviation of the shift parameters alone (K_p1 and K_p4) as they determine how precise the matching position is (Equation (21)). Precision measures based on the covariance matrix are however known to be optimistic due to the high data redundancy [16,18].

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 [39], assuming that accurate matching leads to accurate reconstruction. The SNR is computed as the ratio between the correlation coefficient (ρ) and 1 − ρ. The relative gain of the SNR is used as a measure of relative performance.

The horizontal surface velocity statistics of the three mass movements investigated is presented in Table 2 to give an overview of the mean, standard deviation (variation of the magnitudes) and the maximum values of the horizontal surface velocities of the mass movements. The velocities of the rockglacier and the landslide are in the range of (tens of) centimeters per year while that of the glacier is in the range of centimeters per day or meters per year (ma⁻¹). In the table, we use velocity units that are relevant for the time period covered by the image pairs. For example, we use md⁻¹ as opposed to ma⁻¹ for the velocity of the Nigardsbreen as the data cover only 10 days from late summer. The data on maximum velocity, especially those of the landslide, needs to be read with caution. Experience shows that in image matching, ambiguous (noise-based) matches often lead to large displacement estimates. Even when they are filtered manually and systematically, some of such false matches would still remain. Mismatches often occur in fast moving areas where surface destruction is likely. Although such outliers may not have considerable influence on the mean, they may inflate the maximum values.

The velocity magnitude map of the Muragl rockglacier computed using the LSM is presented in Figure 5(a) with the velocity vectors overlaid. The tongue-like shape of the rockglacier also manifests in its velocity distribution creating a similar pattern. The mean velocity of the rockglacier is computed to be 0.18 ma⁻¹, with the maximum reaching up to 0.50 ma⁻¹. The area with highest speeds in the middle of the rockglacier causes considerable velocity gradients. Upstream of this area velocity increases significantly while velocity decreases downstream along the creep direction. The smooth transitions in the velocity field instead of stepwise ones are ascribed to the capability of the LSM to detect and estimate intra-template displacement variation unlike the constant displacement assumed for each template by the NCC-based matching, for instance. The displacement voids in the figure indicate templates that failed to converge during the LSM iteration.

Figure 5(b) presents the spatial variation of velocity on the Nigardsbreen glacier computed using the LSM with the velocity vectors overlaid. The glacier velocity is high in the center decreasing towards the boundaries. The velocity vectors show a coherent field of flow directions.

The velocity magnitude map of the La Clapière landslide is presented in Figure 5(c) together with the velocity vectors. The upper part of the landslide moves faster than the lower. The recently detached part on the upper left part is also moving with high velocity. The reliability of matches in this area is, however, questionable as it is close to the head scarp of the landslide where significant surface changes are likely to lead to mismatching. The velocity vectors show that the major part of the landslide moves towards the south-west with its lower part turning slightly to the south-east direction.

Table 3 presents the precision of the shift parameters of the three mass movements averaged over all templates. The highest precision is recorded for the Muragl rockglacier image while the lowest is recorded for the La Clapière landslide. The mean precision ranges between about 1/20th of a pixel (Muragl) to 1/6th of a pixel (La Clapière). These values show the power of LSM in locating the matching positions under different radiometric and geometric distortions. After consultation of their histograms, templates with precision of the shift parameters over 0.2 were removed from the analyses. The accuracy (validity) is however dependent on the precision of the precision of the orthorectification and co-registration of the images, deformation of the templates, and noise. Stable grounds and simulated deformation images are used to evaluate the validity of the LSM and the NCC algorithms.

Table 4 presents the mean absolute deviation (MAD) between the measured displacement and the expected displacement for the simulated images and the stable grounds of real mass movements. For the simulated image with low noise variance, about 90% reduction in MAD by the LSM compared to that of the pixel-precision NCC is obtained. However, when σ²_n is raised to 0.1, the reduction in the MAD by the LSM from that of the NCC is only 52%. When the noise level (σ²_n) increased ten-fold, the error in the estimated displacement increases five-fold for the LSM and only 10% for the NCC. We do not investigate further at this point how far this trend is a general one for LSM applied to noisy images. However, the finding supports the widely known general robustness of the NCC algorithm with regard to noise. The reduction in the MAD of displacement error on the stable grounds of the real mass movements is also presented. As the table shows, the MAD is the lowest for the glacier (due to precise orthorectification and co-registration of the images) and the highest for the landslide. The percentage of the error (MAD) of the NCC reduced by the use of LSM is the highest in the Muragl rockglacier case followed by the glacier and the landslide respectively.

As stated above, the SNR of reconstructing the reference image of t₁ from the deformed image of t₂ is also computed over the mass movements. For the analytically deformed image pairs, the relative gain in the SNR of the LSM in relation to that of the NCC is close to 100% (double) in the case where the noise standard deviation is 0.01 and 47% where it is 0.1. For the real mass movement images, around 25% gain in SNR of matching is obtained by using the LSM compared to that of the NCC. The LSM algorithm again makes the best improvement on the Muragl rockglacier. The expectation is that this improvement in the SNR of the image reconstruction translates to improvements in the precision of the estimated displacements (and thus velocities). For the analytically deformed images, the gain in the SNR is comparable to the reduction in MAD of the estimated velocities. The improvements in the SNR for the real mass movements are also comparable to the reductions in the MAD of the velocities on stable ground except for the La Clapière landslide. The general trend of agreement between the SNR gain and the reduction in MAD of displacement shows some important points. (1) The LSM performs clearly better than the NCC. (2) The SNR gain of image reconstruction is directly related to the precision of the displacement estimation. (3) Increased noise level reduces the performance of the LSM, which strongly indicates that the LSM is more sensitive to noise than NCC.

Error propagation using the covariance matrix of the geometric parameters of the affine transformation model (Kp_i) shows high precision (Table 8). The parameters are related directly to the strains as shown in the bracket. The mean absolute value of the estimated parameters is given in the parenthesis for overview of the relative magnitude of the precisions. The uncertainty is therefore less than 10% of the estimated parameter values, i.e., at least one order of magnitude lower. The scaling parameters are relatively more precise than the shearing parameters.

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 (Table 9). This means the values estimated for each deformation parameter in reality lies within the estimated value ± the MAD. The error values for rotation are somewhat higher in all cases. This might be an indication of a slight rotation of one of the images against the other. The error magnitudes are however below (at least one order of magnitude lower than) the estimated values of the deformation (rotation and strain rates) except that of the La Clapière landslide for which the computed strain rates are not significantly outside the error margin. These MAD values can even be optimistic as they are computed on non-deformed area, i.e., stable ground.

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 [32]. The average and maximum velocities of the Muragl rockglacier are similar to what is reported in [30] which is validated using different approaches, including ground measurements. The surface velocity data in [30] agrees well with that of borehole data [40]. The limited surface change during the 13 years period contributes to the success of optical image matching on this rockglacier, and rockglaciers in general. The spatial pattern of the La Clapière landslide velocity variation is in agreement with that computed by other studies, particularly [3]. The velocity magnitudes show a slight slowdown from earlier records (e.g., [33,41]). The decreased magnitudes are in agreement with the general observation of the slowdown of the landslide since the year 2000 (http://gravitaire.oca.eu/spip.php?rubrique15). Errors in the orthorectification and the presence of vegetation on the surface leading to radiometric noises imply the presence of blunders in the image matching.

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 micro-scale 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 [8], agree with those of the present study. The negative total of the horizontal strain rates of the present study (Figure 7) agrees with Figure 3 of [8]. Notice that the symbols in the present study and [8] are inverses of each other and scaled differently. Areas of vertical compression (horizontal extension) in our study correspond with negative elevation changes presented in Figure 2 of [8], indicating dynamic loss of mass from the vertically compressed areas. For the Nigardsbreen glacier, the negative sum of the horizontal strain rates shows that the glacier is dominantly extended vertically, which is in agreement with the thickening of the glacier as reported in [32], and with the general ice compression in glacier ablation areas. High shear strain rate is registered at the margins of flow, especially where the moving glacier borders with stable ground. The deformation maps of the La Clapière landslide require cautious interpretation as more sources of error can influence the reliability compared to the other two mass movement types as discussed in the following section. Specifically, propagated orthorectification error and surface changes such as vegetation cover contribute to major blunders in the strain rate data. Computation of strain rates on its stable ground shows that the computed strain rates are not outside the error margin, i.e., not statistically significant.

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 sub-pixel 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 bi-temporal 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 [20,21,42].

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 co-registration) or the ground itself (deformation and temporal surface changes). Both major error sources can technically be grouped into three, i.e., geometric errors, radiometric errors and propagated sensor or processing errors. The possible sources that cause geometric error are: (1) the formation of crevasses or micro-topography, (2) the boundary effect where the velocity gradient between the moving body and the stable ground is so large that it creates outlying deformation parameter values, (3) vegetation cover can also create geometric change that is not actually ground deformation in addition to its contribution to intensity noise. Signal- (in fact the SNR-) related causes include the presence of shadows, surface changes, illumination differences, presence of dirt, vegetation cover, etc., that can lead to false and outlying convergence parameter values in the least squares iteration. Propagated errors can be attributed to the sensor or image preprocessing, such as orthorectification and co-registration. If the images are not perfectly orthorectified and co-registered, the geometric adjustment includes the mass deformation, sensor projective distortion and change of geometry between the two images [17,43].

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 high-resolution 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. Pixel-based constraining such as data snooping or template-based constraining such as raising convergence precision can be used [18,21]. This would affect only the highly noisy (and maybe the highly deformed) templates, leaving the well-converging templates unaffected. An initial test shows that increasing the demanded precision of the parameter adjustment during the LSM iteration discards more templates, resulting in more data voids. However, the strong spatial variability is not substantially smoothed, implying that it stems from real strain rate variations.