These authors contributed equally to this work.

Ground-penetrating radar (GPR) is widely used for determining mountain glacier thickness. However, this method provides thickness data only along the acquisition lines, and therefore interpolation has to be made between them. Depending on the interpolation strategy, calculated ice volumes can differ and can lack an accurate error estimation. Furthermore, glacial basal topography is often characterized by complex geomorphological features, which can be hard to reproduce using classical interpolation methods, especially when the field data are sparse or when the morphological features are too complex. This study investigates the applicability of multiple-point statistics (MPS) simulations to interpolate glacier bedrock topography using GPR measurements. In 2018, a dense GPR data set was acquired on the Tsanfleuron Glacier (Switzerland). These data were used as the source for a bedrock interpolation. The results obtained with the direct-sampling MPS method are compared against those obtained with kriging and sequential Gaussian simulations (SGSs) on both a synthetic data set – with known reference volume and bedrock topography – and the real data underlying the Tsanfleuron Glacier. Using the MPS modeled bedrock, the ice volume for the Scex Rouge and Tsanfleuron glaciers is estimated to be 113.9

It is widely accepted that global climatic changes impact future precipitation rates and temperatures. In Switzerland, these changes will inevitably induce new stresses on alpine environments and on glacier mass balance

Ice volume estimation relies on two components: (1) the surface topography of the glacier and (2) the underlying bedrock topography. The first one is easily measurable using lidar

One classical interpolation strategy used for basal estimation is the ordinary kriging method

Other empirical methods using volume–area (

In the last decades, new geostatistical methods have arisen with the aim to improve the realism of the simulation using another form of information than the one expressed by two-point statistics and variogram interpretation. Multiple-point statistics (MPS) simulation algorithms use a training image (TI) to infer the spatial statistics of the model and generate random fields reproducing these spatial statistics. The TI represents a conceptual knowledge of the variable that is aimed to be simulated. It can be created by experts or can be extracted from analog data sets. Unlike other geostatistical approaches, MPS does not require the definition of an analytical two-point statistics model to represent the spatial variability but instead infers it implicitly from the TI provided by the user

Aerial image and digital elevation model captured from drone images of the Tsanfleuron Glacier, Switzerland.

The aims of this paper are both methodological and applied. Regarding the methodological aspect, this work aims to demonstrate the use of the MPS method to combine information provided by GPR data points and a digital elevation model (DEM) to simulate a realistic glacial basal topography. The benefits of using a MPS approach are highlighted by comparing its results with those obtained with more classical geostatistical methods. Using synthetic test cases, the different methods are compared by calculating for each one an ice volume and a roughness estimation, which are then compared against the true synthetic values. A set of scores are computed to compare the methods. Through this comparison process different parameter sets are also tested for each method. This methodological aspect helps to select the most suitable parameter sets on a synthetic case where the target topography is known. This highlights the advantages of the multiple-point simulation approach for estimating glacier volume and its associated basal geomorphology. On the applied side, the objectives are to present new field data and new estimations of the volume of the Tsanfleuron Glacier, roughly 10 years after the last detailed published estimation

The Tsanfleuron Glacier and the Scex Rouge Glacier (Fig.

The core MPS technique used in this study is the direct-sampling algorithm

In this paper, the exposed basal surface of the melted glacier zones is employed as a training image for the simulation of the covered glacier basal topography. The justification for this is that the lithology and general topographical slope below the glacier and in the exposed area are similar, and therefore the geomorphological features should also be similar. This idea is validated by the analysis of the area where the glacier has retreated in the last dozen years, which exposed geomorphological structures similar to the older part. The GPR-inferred depths are then used as conditioning points as well as the topographical data around the glacier.

Since the exact topography below the glacier is unknown, to analyze and benchmark the performances of different interpolation methods, a numerical experiment was designed in which references can be compared to the simulation outputs. For that purpose, the exposed part of the bedrock is also used (besides being used as TI) to compare the performances of the MPS, kriging, and sequential Gaussian simulation (SGS) approaches. In total 20 zones are extracted from the exposed DEM and sampled to create fake GPR data sets. Using only the sampled data set, the topography in the test zones are interpolated using MPS, SGS, and kriging and compared to the reference topography. The true volumes of the synthetic tests are defined as being the space between the simulated topography and a flat surface representing the top of the ice sheet. The altitude of this ice sheet is defined as being 4 m above the maximum altitude of the simulation. The absolute volumes of the simulations are then compared to this true volume, using the same ice sheet altitude. Moreover, an estimation of hydrological flow accumulation is calculated on the simulated bedrock and on the reference. The flow accumulation map outlines the link between structures of the topography and the connectivity of the cells. In addition, different parameter sets are tested for each method through this experiment. Finally, different scores are used to compare the methods. This numerical experiment helps to understand and visualize the impact of each method on the simulated bedrock shape and volume estimation distribution.

Lastly, the Tsanfleuron and Scex Rouge bedrock's topography is interpolated using the previously tested methods and parameter sets. The simulated topography is also compared to recently uncovered bedrock, using a simple estimation of flow accumulation. A brief overview of the glacier volume distributions and their past evolution is finally carried out using the calculated bedrock surfaces and a different DEM.

In summer 2018, a dense GPR acquisition on the Tsanfleuron Glacier was performed (Fig.

In addition, during the summer in 2019 several UAV flights were undertaken above the Tsanfleuron Glacier. We conducted five flights with a Sense Fly EBEE UAV equipped with a 20 megapixel RGB camera. The objective was to have a resolution of at least 10

Acquisition lines from August 2018. The cumulative length is about 18 km. Hill shade from the DEM derived from drone images. The straight lines visible on the hill shade are ski lifts and transit tracks for snow groomers. A GPR line is displayed, with the basal reflector outlined.

The MPS algorithm used in this paper is DeeSse

The three main parameters of the method are the maximum number of neighbors (

In the case of the Tsanfleuron Glacier, the acquired DEM of the exposed bedrock is used as the TI. The conditioning data are the GPR lines and the altitude of the bedrock (from the DEM) surrounding the glacier.
To obtain the best parameters for this data set, a series of experiments with different parameter sets were conducted. The methodology to conduct these tests is detailed in Sect.

For all the simulations, the multi-scale mode using Gaussian pyramids and relative distance options is activated to get the best reproduction of the patterns. The first feature improves the simulation of patterns at different scales and produces more consistent simulation outputs

Kriging and sequential Gaussian simulations (SGSs) are standard geostatistical techniques that are well described in many textbooks

At the scale of the study site, the exposed bedrock topography presents a general slope toward the east and is therefore non-stationary. To account for this general slope, we decided to first remove the trend present in the DEM and in the hard (GPR) data before conducting the variogram analysis. To do so, a polynomial surface is fitted and removed from the data. By removing the polynomial surface from the data, only the deviation from this surface, which corresponds to the deviation from the general slope of the glacier, is simulated. At the end of the process, the trend is then added to the interpolated values to obtain the final basal topography. It is important to note that since the Scex Rouge and the Tsanfleuron glaciers have two different orientations, two different trends were interpolated and applied to each glacier, the transition being set at Tsanfleuron pass. The polynomial interpolated trends in the form of

Variogram analysis.

In order to benchmark the different geostatistical algorithms and parameter sets, a systematic testing approach is applied (Fig.

For the MPS approach, nine sets of parameters are considered. They are given in Table

SGS and ordinary kriging are applied using the same variogram model presented in Sect.

Once the simulations are performed, quality indicators are computed from the predicted and actual topography.

The approach used for the systematic tests. Test zones are extracted from the exposed glacier bedrock. GPR lines are also extracted from these zones and used to constrain the geostatistical simulations. It is then possible to compare the different sets with the actual topography. Flow accumulations maps are calculated from the simulated topography and from the reference extracted DEM for comparison.

Parameter sets used for the synthetic tests.

We expect that the geostatistical methods that are used to interpolate the basal surface perform differently depending on which derived quantities from the simulated bedrock we compare them to. We compare the fidelity of the different DEMs by evaluating different performance metrics. To illustrate this idea, three quality indicators were designed, based on different uses of the basal interpolated bed. They are defined related to the estimation of the topography itself, the estimation of the overall ice volume between the topography and the top of an arbitrary ice cover, and finally the estimation of flow accumulation on the basal surface.

A further aspect is that we wish to evaluate both the average match between the forecast and the reference and the predicted uncertainty range. This is why we consider not only the absolute bias but also the continuous ranked probability score (CRPS)

While the bias quantifies the mismatch between the expected value and the true value, CRPS compares a single true value

In the following subsections, we define the scores that were applied to the synthetic test cases. Note that to form final scores, each presented score is averaged over all 20 synthetic test cases.

For all the test cases, we define the ice volume by fixing (arbitrarily) the altitude

The volume of ice

The mean value of

Following the same logic, the CRPS score of the volume prediction is computed by first summing the elevation values in the domain

To test if the altitude is properly estimated in all locations of the domain, we compute the absolute value of the bias at each location: it is defined as the absolute difference between the mean DEM (ensemble average over all the simulations) and the reference. To form a single score, this map is averaged over all points in the domain. The mean absolute bias (MAB) for the DEM estimations is therefore defined as follows:

The units of MAB(DEM) and AB(volume) are identical. This allows us to directly compare ice volume errors with DEM errors. Furthermore, MAB(DEM) provides an upper bound for AB(volume).

To compute the CRPS score, let us consider that at each point with indices

Flow accumulation is considered here because DEMs are often used to make predictions that are highly affected by geomorphological structures or roughness. Reliably predicting the geomorphological patterns of a DEM is critical, for example, to estimate the velocity at which the glacier may move or to simulate how meltwater can be channelized at the base of the glacier. Flow accumulation maps are thus used in this study because they can be computed rapidly and easily. More importantly, they illustrate the concept of complex geomorphological and roughness description. A flow accumulation map represents the number of cells in each cell that are located upstream and are used to estimate flow direction and catchment delineation

To quantify these differences, the probability density function (PDF) of the flow accumulations for each individual simulation is compared with the PDF of the reference. A standard indicator to compare two PDFs

Supposing that the flow accumulation map is given by

Let us call

The last step consists of applying the MPS and SGS simulation methods as well as kriging to the actual data set below the Tsanfleuron and Scex Rouge glaciers. The conditioning data are identical for the three methods: the GPR data below the glacier and the DEM around the glacier to ensure the continuity between the border of the simulated area and the exposed altitude of the glacier.

For the MPS simulations, we use the parameters and setup described in detail in Sect.

The training data set uses the complete exposed part of the glacier's bedrock as a primary variable and its computed gradient as a secondary variable. The DEM is also used directly as hard conditioning data. In total, 120 MPS simulations (40 simulations per parameter set) are generated.

For the sequential Gaussian simulations and the kriging estimate, the variogram that is employed is the one introduced earlier in Sect.

The ice volumes are then computed between all the simulated basal surfaces and the ice topography measured in August (end of summer) 2011 and 2019. For 2011, we use the Alti3D DEM from the Swiss Federal Office of Topography and our DEM for 2019.

For 2019, we use the DEM that was acquired in this work (see Sect.

A 3D view of topographic interpolation from the synthetic test case. The color corresponds to the simulated elevation. Panel

Figure

Figure

Regarding the cross sections, Fig.

Regarding the volume, Fig.

Synthetic test results. Panel

The quality scores presented in Table

Another important indicator to take into account is the mean Jensen–Shannon divergence of the flow accumulation scores. It reflects how similar distributions of flow accumulation are compared to the reference. The scores of the MPS sets are 10 times better than the score of the SGS and 4 times better than kriging. Figure

Finally, the systematic tests showed that the best parameter sets were numbers 6, 3, and 8 for MPS and the 24-neighbor parameter set for SGS. These parameters were then used for the practical application of Tsanfleuron Glacier.

Quality indicators averaged over all realizations for different simulation methods (MPS, SGS, kriging) and for different parameter sets. The rows are sorted by best (lowest) CRPS score of ice volume estimation. CRPS score of ice volume estimation, mean absolute error of the ice volume, CRPS score of DEM averaged over all points, mean absolute error of DEM averaged over all points, and Jensen–Shannon divergence of flow accumulation distributions averaged over all simulations.

Probability distributions of flow accumulation values for example realization 11. The probability distribution of flow accumulation for the true DEM is compared to an example MPS simulation, kriging map, and SGS simulation. The corresponding Jensen–Shannon divergences with respect to the true distribution are 0.015 (MPS), 0.228 (kriging), and 0.107 (SGS).

Figure

Table

Volumes of ice computed with the 2019 and 2011 DEMs for the Tsanfleuron and Scex Rouge glaciers.

So far, the results obtained by the three methods are very close, because the differences in the spatial distribution of the basal elevation values are compensated for when we integrate them over the whole glacier area to compute the volume or mean altitude.

However, the flow accumulation results (Fig.

Simulated basal topography for the MPS method (in blue) and for the SGS method (in brown). The kriging-estimated topography is displayed alongside the SGS simulation in green.

Flow accumulations calculated from the three methods and compared against the one computed in the exposed part of the bedrock used as reference and the probability density of accumulation values. The SGS simulation underestimates the length of the connected path, while kriging overestimates it. The visual patterns and the flow accumulation distribution obtained with MPS are the closest to the exposed bedrock reference.

When comparing the interpolated basal topographies with the three methods, our results show that the MPS approach provides the simulations that display geomorphological features that are the closest to the data set. But to obtain those results, the DeeSse algorithm needs to be properly parameterized, and adequate secondary variables have to be used. During this project, we have tested various options. Using the topography gradient as a secondary variable proved to be a simple and efficient solution, but further improvements could certainly be made. One possibility that we considered but did not implement would be to use as secondary variables two hillshade projections, 45

The results from the previous section show that the three geostatistical methods are able to provide consistent and comparable estimates of the mean basal topography and the overall ice volume. This similarity is expected for the SGS and kriging because the volume calculation is a linear function of the basal topography, and in this situation kriging and SGS simulations will provide the same mean value

Ice thickness calculated from the three methods using the 2019 surface DEM. The SGS and the MPS basal models used are the averaged model over all the realizations.

Only the SGS and MPS methods are able to estimate the uncertainties on the total volume. As explained before, even if kriging can, at any point, provide the variance of the altitude, it cannot be used directly to infer uncertainties on the volume. Covariances between any pair of points would need to be considered and integrated over the whole domain. The volume uncertainties estimated with the SGS and multi-Gaussian model are in general reliable

For the Scex Rouge and Tsanfleuron glaciers, these three methods allow us to obtain an estimation of 149.8

Another indirect comparison with existing data was made using year-to-year mass balance

To finalize this discussion, we would like to recall that several operators did the picking of the depth of the reflectors independently on our GPR data and that the order in which the data were presented to the operator was randomized. We also compared the result of the picking from the different operators and removed the parts of the data that were not consistent. Even if the surface and volume estimations may still suffer from some remaining errors, we tried to apply these strict procedures to avoid bias, and therefore, we expect that the data set and the estimated volumes are as reliable as possible.

As expressed earlier, the present analysis did not consider the uncertainty regarding the velocity used to convert the two-way travel time data from the GPR to depth. We used a uniform value that corresponds to the wave velocity in cold and non-wet ice generally used. However, Tsanfleuron Glacier is described as a polythermal glacier

To rapidly estimate the volume of ice, our results show that the kriging method provides a value that is reasonable. However, kriging cannot be used to directly obtain the uncertainty on the volume.

We argue that any scientific estimation should always be accompanied by an uncertainty estimation when possible. Hence, it is therefore preferable to directly use the SGS or MPS approach to get not only the volume but also the corresponding uncertainty. The two methods provide comparable results. The SGS method requires a variogram model from the experimental variogram of the data. By contrast, the MPS method simply requires providing an exhaustive data set that represents the type of spatial variability that is analog to the patterns that are expected below the glacier. In a previous study

Finally, if the estimated topographies of the bedrock below the glacier have to be used to estimate a quantity that derives non-linearly from the topography, the MPS method should be used. Indeed, we have shown that MPS provides a much better reproduction of the geomorphology of the simulated basal surfaces: the results are much closer to the reference than the other two techniques. This result confirms the observations made by

This study presents an example of the benefits of using advanced geostatistical methods for basal topography interpolation and compares three methods: kriging, sequential Gaussian simulations, and multiple-point statistics.

The three methods are able to provide consistent and similar volume estimations. Kriging and SGS require the analysis of the statistics of the data, adjust a spatial trend, and identify a variogram model. Once this is done, these two methods provide local estimation of uncertainty. But to get the uncertainty on the volume, one needs to use the SGS method.

The MPS technique is the most versatile: it provides a comprehensive estimation of the volume, as well as local uncertainties comparable with the other methods. But in addition, it is able to produce realistic basal structures, even in areas where the data are scarce or the structure complex. Compared to MPS, SGS and kriging tend to produce interpolated surfaces that are respectively too smooth or too noisy. Therefore, they can lead to biased predictions when they are used to derive quantities that depend strongly on the detailed geomorphological structures of the basal topography as illustrated with the flow accumulation calculations done in this paper. The same types of errors are expected if these topographies are used to simulate the glacier movement or the basal flow of melted water and the channeling of this flux. In these situations, the detailed structures may be even more important than the global trend, and the MPS approach is recommended.
The main limitation of the MPS approach is that it requires a TI that is representative of the glacier basal structures. Finding the proper analog data is therefore an important part of the approach and may be difficult for large glaciers with little information about the underlying geology. In these situations, a possibility could be to test various training images using

Based on the results obtained in this paper and those published by

Finally, the ice volumes calculated for the Scex Rouge and Tsanfleuron glaciers with the three methods are in accordance with the mass balance calculation and are linked with robust error estimation. Our results indicate that there has been a significant mass loss at this glacier and that these methods enable higher-accuracy ice loss estimates and could enable improvements in glacier retreat projections. Such improvements could be important for global awareness, political decisions, and preparing the mountains' infrastructure for its future evolution.

A simulation obtained with each method, the mean simulations of SGS and MPS, and the DEM and the conditioning point sets are available at

AN coordinated and conducted a part of the fieldwork. AN processed the GPR and drone data. AN, VD, JS, and PR designed and tested the different geostatistical procedures. AN and VD prepared the data. PJ participated in the data acquisition and designed and implemented the quality indicators. AN, VD, and PJ wrote the paper. PR initiated and supervised the work, conducted the field acquisition, and was involved in the writing and editing of the paper.

The contact author has declared that neither they nor their co-authors have any competing interests.

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

The authors would like to thank Marie Vallat and Cyprien Louis, two students who participated in the GPR data collection and initial geostatistical analysis. The authors are also thankful to James Irving at the University of Lausanne, who provided the GPR equipment and support. They would like to thanks the Prarochet Hut and the “Glacier 3000” company for the logistic support. Finally, they want to thank the editor Adam Booth, the anonymous reviewer, Clemens Schannwell, and Emma MacKie for their numerous comments that helped improve the quality of the paper.

This paper was edited by Adam Booth and reviewed by Clemens Schannwell, Emma MacKie, and one anonymous referee.