In recent years a vast amount of glacier surface velocity data from satellite imagery has emerged based on correlation between repeat images. Thereby, much emphasis has been put on the fast processing of large data volumes and products with complete spatial coverage. The metadata of such measurements are often highly simplified when the measurement precision is lumped into a single number for the whole dataset, although the error budget of image matching is in reality neither isotropic nor constant over the whole velocity field. The spread of the correlation peak of individual image offset measurements is dependent on the image structure and the non-uniform flow of the ice and is used here to extract a proxy for measurement uncertainty. A quantification of estimation error or dispersion for each individual velocity measurement can be important for the inversion of, for instance, rheology, ice thickness and/or bedrock friction. Errors in the velocity data can propagate into derived results in a complex and exaggerating way, making the outcomes very sensitive to velocity noise and outliers. Here, we present a computationally fast method to estimate the matching precision of individual displacement measurements from repeat imaging data, focusing on satellite data. The approach is based upon Gaussian fitting directly on the correlation peak and is formulated as a linear least-squares estimation, making its implementation into current pipelines straightforward. The methodology is demonstrated for Sermeq Kujalleq (Jakobshavn Isbræ), Greenland, a glacier with regions of strong shear flow and with clearly oriented crevasses, and Malaspina Glacier, Alaska. Directionality within an image seems to be the dominant factor influencing the correlation dispersion. In our cases these are crevasses and moraine bands, while a relation to differential flow, such as shear, is less pronounced on the correlation spread.

The increased global availability of satellites images has created unprecedented archives of velocity products over glaciers, ice caps

In our opinion the assumption of constant variance (homoscedasticity) does not hold, as displacement extraction is based upon pattern matching of small subsets of imagery, where the image content influences the displacement precision.
Pattern matching is based upon similarity metrics between the matched image across its extent.
Such an image subset can have texture with a strong directionality, such as crevasses, or the texture in an image subset is distorted due to skewed flow, such as shear

The issue of homoscedasticity can also be approached from the perspective of optical flow.
Pattern matching and optical flow can be seen as interchangeable techniques, as they are mathematically similar

In this contribution, we demonstrate a fast estimation approach for dispersion characteristics for individual displacement estimates from image matching. These dispersion characteristics are then used to explore the connection between the correlation spread and the processes of shear flow and crevasse orientation. This gives a better understanding of the image regions where displacement estimates need to be interpreted with caution. Furthermore, our method enables better quantification of error propagation into the remote sensing and derived model products, which can improve inferences about, for instance, strain rates, glacier depth, bed roughness and rheology.

The backbone of velocity extraction from imaging satellites is image correlation (a.k.a. pattern matching and feature tracking).
For a general overview of an image-matching pipeline, see Appendix

Apart from the displacement information, other metrics can also be extracted from the correlation surface.
For an extensive assessment of such metrics, see

However, upon close inspection the width and form of the highest peak in a correlation surface changes and depends greatly on the image structure, for example, surfaces with a preferred orientation, such as crevasses (Fig.

Examples of cross-correlation results with anisotropy, due to the image content or the underlying surface flow. For both panels, the lower-left panel is the image template to be correlated with the upper-left search area. The resulting correlation surface is displayed in the lower right, and a zoom around the correlation peak is in the upper right. Both examples

A second process influencing the spread of correlation scores is when significant shear occurs within the template (Fig.

Formulating the precision of a match can be done by looking directly at the variation in intensities within an image

We perceive the close surroundings of the correlation function as a probability density function.
This is a standard perception in the field of fluid mechanics

Here, we draw up a linear formulation to describe the variance of the correlation peak, which also considers its orientation.
At a certain location in this search space (

This gives the possibility for directly estimating the unknowns (in

This estimation procedure is an extension of

The dispersion matrix (

For the dependencies between two-dimensional displacements, as presented here, interpretation of the elements within the dispersion matrix might not be intuitive.
For example, an equal variance can still produce an orientation dependency, as can be seen for example in Fig.

Similarly, the orientation of the ellipse (

Surface strain rates are used in this study to assess the relation between the correlation ridge and ice deformation.
Such strain rates can be extracted from a velocity field; however remote sensing results contain holes and patches without estimates, since similarity could not be established.
Hence a robust estimation framework is given in Appendix

To assess the impact of directionality in the input images on our approach to compute and use the dispersion of individual correlations, we need to quantify the directional characteristics of glacier images.
In particular crevasse fields have strong directional properties, which can be composed of cracks with several predominant orientations.
In order to extract the local crevasse characteristic for each matching template, a Radon transform is used, as described in earlier work

Color-coded speed and streamlines of Sermeq Kujalleq (Jakobshavn Isbræ) between 20 and 30 July 2020 based on Sentinel-2A imagery. The upper inset

Here we present results from two sites, namely Sermeq Kujalleq (Jakobshavn Isbræ), Greenland, and Malaspina Glacier system, Alaska.

We demonstrate and assess our method to estimate the uncertainty in displacement matching using a small subset of two orthorectified Sentinel-2A scenes over Sermeq Kujalleq, a large and fast outlet glacier of the Greenland ice sheet.
High-pass filtered imagery (following

The velocity magnitude between the two images (20 and 30 July 2020), derived streamlines, and the resulting along- and across-flow variance estimates are shown in Fig.

Descriptions of directionality for the case study of Sermeq Kujalleq.

The dominant crevasse orientation (Fig.

Figure

The strength of asymmetry of the correlation peak

Results from the surroundings of Malaspina Glacier and Agassiz Glacier in the St. Elias Mountains are presented here as well.
The region exhibits more supraglacial features than Sermeq Kujalleq, which is an outlet of the Greenland ice sheet with predominantly clean ice.
For example, a large collection of moraines, ogives, foliations, meltwater channels and more diverse orientations of flow is present on both Malaspina Glacier and Agassiz Glacier, as can be seen in Fig.

Here we use two subsets of Sentinel-2 scenes from 21 August and the 15 September 2019, a 25 d difference, and from the same orbit. Processing parameters are similar to the Sermeq Kujalleq study: a high-pass-filtered band 4 image was matched, with a template window of 200 m wide, and velocities are estimated every 100 m, with a search window of 800 m. No co-registration over stable ground was done, so the velocities should be seen as displacement (being real surface displacements or artificially created due to sensor/processing biases).

Image displacement

The estimated displacements over the study region (Fig.

Estimated surface shear, derived from the estimated velocity (Fig.

The pattern of elongation (Fig.

The dominance of the feature orientation (Fig.

Orientation descriptors over the Malaspina case study, estimated through Radon transform (Fig.

In general the main orientation of the crevasses at Sermeq Kujalleq (Fig.

Probability density plots of results for Sermeq Kujalleq of correlation peak versus crevasse orientation (Figs.

Correlation descriptors over the Malaspina case study showing the absolute correlation value for each match

In earlier work the handing of dispersion has been estimated through sampling statistics (standard deviation and mean absolute difference), where displacement estimates are compared against in situ measurements or stable terrain.
The use of stable terrain for dispersion estimation has drawbacks, apart from assuming constant variance of the whole scene as mentioned earlier.
Specifically, image matching in the frequency domain is hampered by peak locking that favors integer displacements

Thus the method presented here can be a direction to formulate measurement precision, without biases introduced by sample statistics and peak locking.
Another advantage of our method is the possibility for using statistical testing

We postulate that the correlation coefficient is a proxy for the confidence of a match and are therefore less suited to function as a descriptor of precision.
The maximum correlation coefficient and the signal-to-noise proxy are dissimilar proxies.
For example, the narrow and crevassed outlet of Malaspina Glacier has low correlation scores (Fig.

A second commonly used proxy for precision is the signal-to-noise ratio.
Here we postulate that this proxy might describe the uniqueness of a match.
Very high signal-to-noise values (Fig.

Estimated surface divergence, derived from the estimated velocity (Fig.

In this study we propose to use a Gaussian formulation to describe the matching precision.
If the maximum correlation or signal-to-noise ratio would be a good proxy for precision, then one can expect a correlation or some form of agreement between the major axis (Fig.

Probability scatterplots between different matching descriptors for the Malaspina Glacier system.

The implementation done here for our correlation-dispersion-based method is a simple least-squares adjustment, and no robust re-weighting is applied.
This can result in negative variances or rank deficiency, corresponding to the white data voids in Fig.

In this study, the correlation computation is done in the spatial domain.
When transformed back to the spatial domain, frequency domain methods produce sharp peaks in the correlation surface in the form of a two-dimensional Dirichlet function, as they prescribe consistent rigid displacement at integer resolution

Finally to demonstrate its application domain, we introduce a generalized least-squares framework to use our dispersion estimation (see Appendix

Quantifying the measurement precision of individual displacement estimates from matching repeat spaceborne images has received little attention in recent years despite the increasing efforts to produce large displacement datasets from an increasing number of suitable data. Here, we introduce a simple procedure to estimate the correlation dispersion of such displacement measurements (either optical or SAR), through characterizing the shape of the correlation surface. We demonstrate this technique for Sermeq Kujalleq, a fast-flowing and heavily crevassed outlet of the Greenland ice sheet and the Malaspina Glacier system. Dispersion results are compared to shear strain rates and crevasse orientation. These results indicate that crevasses are the dominant driver for asymmetry in the correlation surface. We suggest this simple procedure to estimate uncertainty in individual image matches can be useful in processing pipelines for large-volume image displacement measurements, so error-propagation can be applied on a large scale and will improve inversion of other geophysical properties. In all, we hope this demonstrates the rich information present in satellite imagery and its processing chain and might make it easier to extract a more detailed physical signal from such noisy remote sensing products.

In order to clarify where in a displacement processing scheme the dispersion estimation can be implemented, a schematic of an image-matching pipeline is drawn in Fig.

Schematic of the main procedure to generate a displacement field from a pair of remote sensing images.

Given the extent of the imagery, a mask is generated indicating what is ocean, land and glacier.

A regular grid is generated, where for each location the land cover is recorded.

For each post of the grid, a subset of the satellite imagery is used. A kernel is moved over a base image, and at every location a similarity score is estimated. This generates a correlation surface. The highest value is taken as the correct displacement. The neighboring correlation values of this peak can be used for sub-pixel localization, but the same values can also be used for the dispersion calculation following the method presented in this study.

The displacements over stable ground are used to correct offsets due to misalignment of the satellite platform.

The co-registration parameters are subtracted from the displacement vectors, resulting in a grid of velocities and its precision.

A two-dimensional normal distribution, with a dependency (

Knowing

With the resulting parameters (

Example of ellipses with different dispersion parameters. Illustration adopted from

Flow descriptors like strain rates can also give an insight into the geometric bedrock configuration or properties related to subglacial sliding. Strain rates can be formulated in relation to the local flow direction, giving longitudinal, transverse or shear flow, respectively. These properties are computed from velocity estimates over a close neighborhood of surrounding pixels. As strain rates are derivatives of velocities, they are particularly sensitive to the propagation of noise and errors in the input velocities. Applying thresholds and filters to the strain rates based on variations or low quality of the input velocities can lead to voids in the resulting strain rate field. Here, a methodology is introduced that is somewhat resistant to such cases caused by velocity errors or missing data.

The methodology presented here is based upon the redundancy of a kernel, since it is typically formulated as a smoothed differentiation.
The steps are schematically illustrated in Fig.

Schematic of a computation of a convolution, in this case the first derivative in the vertical and horizontal direction.

In the example shown in Fig.

Nevertheless, improvement is only made on a local level in a direct neighborhood covered by the kernel, so when large parts are affected with regions of missing values or when the outlier detection is false, spurious fluctuations can still propagate into the final product.

In this appendix, additional illustrations are shown for the Malaspina Glacier to ease interpretation of the results and to highlight the information present in dispersion peak.

Here we also show some sub-pixel displacement plots, as the integer (Fig.

Sentinel-2 scene over Malaspina Glacier (center) and Agassiz Glacier (left) with annotations in red to enhance interpretation.

Rainbow-color-coded remainder of the modulus of displacement, for the horizontal and vertical direction (

A simple MATLAB and Python implementation for the dispersion estimation is included in the submission. The implementation for the Radon transform can be found at

In this study we use optical data from the Sentinel-2 satellites. Since these satellites are part of the Copernicus satellite system, which is the European Commission's earth observation program, all data are freely available. Hence, current acquisitions can be retrieved from

The supplement related to this article is available online at:

BA conceived the study; AK and BW contributed with comments and suggestions to the work.

At least one of the (co-)authors is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to thank two anonymous reviewers for their input, which helped to improve the paper, and Kang Yang for handling the review process as the editor.

This research has been supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Dutch Research Council, NWO; grant no. ALWGO.2018.044; and 016.Vidi.171.063) and the European Space Agency (grant no. 4000125560/18/I-NS).

This paper was edited by Kang Yang and reviewed by two anonymous referees.