# Novel Techniques for Void Filling in Glacier Elevation Change Data Sets

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## Abstract

**:**

^{2}of glacier area in southeast Alaska, USA, covered by two void-free DEMs as the study site to test different inpainting methods. Different artificially voided setups were generated using manually defined voids and a correlation mask based on stereoscopic processing of Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) acquisition. Three “novel” (Telea, Navier–Stokes and shearlet) as well as three “classical” (bilinear interpolation, local and global hypsometric methods) void filling approaches for glacier elevation data sets were implemented and evaluated. The hypsometric approaches showed, in general, the worst performance, leading to high average and local offsets. Telea and Navier–Stokes void filling showed an overall stable and reasonable quality. The best results are obtained for shearlet and bilinear void filling, if certain criteria are met. Considering also computational costs and feasibility, we recommend using the bilinear void filling method in glacier volume change analyses. Moreover, we propose and validate a formula to estimate the uncertainties caused by void filling in glacier volume change computations. The formula is transferable to other study sites, where no ground truth data on the void areas exist, and leads to higher accuracy of the error estimates on void-filled areas. In the spirit of reproducible research, we publish a software repository with the implementation of the novel void filling algorithms and the code reproducing the statistical analysis of the data, along with the data sets themselves.

## 1. Introduction

## 2. Study Site and Data Set

^{2}consisting of over 700 individual glacier catchments according to the Randolph Glacier Inventory (RGI) 6.0 [19]. The ice-covered areas range from sea level to more than 4000 m a.s.l. Various glacier types such as surge-type glaciers and large and small valley glaciers as well as retreating and advancing tidewater glaciers can be found within the analyzed area.

## 3. Methods

#### 3.1. Void-Free Volume Change and Artificial Voids

^{2}glacier area) with a relatively smooth ice topography and mainly small elevation changes (called the “Juneau” setup in the following). The other covers mainly the RGI60-01.20686 glacier at the center of the study region, which expands from close to sea level up to ~2200 m a.s.l. over a ~142 km

^{2}ice area (called the “Center” setup in the following). At both setups, artificial voids were generated to simulate large data gaps. The strip-shaped voids (called “Strip” voids in the following) simulate gaps between satellite acquisitions. Cloud cover, low image contrast or sensor saturation can lead to large voids in the accumulation areas, when using optical satellite data to generate elevation information. The large circular voids (called “Circle” voids in the following) simulate these issues. Additionally, the complete absence of a glacier section is simulated at the Center setup. Therefore, the lower most section at the ablation area of the glacier was cut off (called the “Terminus” void in the following).

#### 3.2. Void Filling

#### 3.2.1. Telea Approach

#### 3.2.2. Navier–Stokes Approach

#### 3.2.3. Shearlet Approach

**,**that agrees with the measured data, $PI$ (the non-void pixels). Once $\widehat{c}$ is found, an estimate for the true image may be easily obtained by taking the inverse shearlet transform $\widehat{I}=\Phi \widehat{c}$

**.**

#### 3.2.4. Classical Void Filling

#### 3.3. Error Metrics and Comparison Methodology

#### 3.3.1. Large Voids—Juneau and Center Setups

#### 3.3.2. Large Region—Correlation Setup

#### 3.3.3. Impact of Void Filling on Different Scales

## 4. Results and Discussion

#### 4.1. Void-Free Volume Changes

^{2}, a void-free volume change of −52.9 ± 11.9 km

^{3}(corresponding to an average elevation change of −8.52 ± 1.92 m) was found for the period 2000–2012/13 (based on the data sets provided by R. McNabb; [9]). The hypsometric distribution of the elevation changes binned in 50 m intervals is plotted in Figure 3. Average surface lowering of up to ~35 m is obtained for the lower-elevation intervals. Above ~1300 m a.s.l., a positive surface elevation change is found. The contribution of glacier sections above 2000 m a.s.l. to the total volume change is minimal, due to the small glacier area.

#### 4.2. Evaluation of Void-Filled Data Sets

#### 4.2.1. Large Voids—Juneau and Center Setup

**Center setup:**Telea and Navier–Stokes achieve reasonable ${\overline{dh}}^{r}$ and $AA{E}^{f}$ values for all voids. The shearlet inpainting led to considerably larger errors (in terms of ${\overline{dh}}^{r}$) at the Circle and Terminus voids. The hypsometric approach shows the lowest $AA{E}^{f}$ and a small ${\overline{dh}}^{r}$ at the Circle and Terminus voids. The visual inspection of its reconstruction results (Figure S1) also suggests the conclusion that the hypsometric approach leads to the best reconstruction (among all methods) in these two regions. However, at the Strip void, the hypsometric approach shows much higher ${\overline{dh}}^{r}$ and $AA{E}^{f}$ values than the competing methods.

**Juneau setup:**At the Circle void, the Navier–Stokes approach shows the smallest ${\overline{dh}}^{r}$ value. Telea and shearlet (nscales = 7) inpainting methods also show a good performance. The lowest $AA{E}^{f}$ is found for the shearlet and bilinear void filling methods. At the Strip void, Telea and Navier–Stokes approaches lead to the smallest ${\overline{dh}}^{r}$ and $AA{E}^{f}$ values. The hypsometric approach shows the worst performance for both voids and metrics.

**Summary:**The findings of the case study lead to the following conclusions.

#### 4.2.2. Large Region—Correlation Setup

#### 4.2.3. Impact of Void Filling on Different Scales

^{2}and a coverage $p$ of 80% (number of void pixels $N=172,200$). Applying Equation (20) with ${f}_{V}=2$ leads to an ${\delta}_{dV}^{V}$ of 0.262 km

^{3}. Using Equation (21) with ${\delta}_{dh}^{V}$ obtained from Equation (19) for ${\delta}_{dV}^{V}$, we obtain 0.138, 0.138 and 0.229 km

^{3}for shearlet (nscales = 5), bilinear and local hypsometric void filling methods, respectively. The contribution by ${\delta}_{dh}$ to ${\delta}_{dV}^{V}$ (0.131 km

^{3}) dominates. Thus, the additional uncertainty caused by the void filling is negligible for the shearlet and bilinear approaches. For the local hypsometric approach, a contribution of ~37% to ${\delta}_{dV}^{V}$ by the void filling is found. All ${\delta}_{dV}^{V}$ estimates based on Equation (21) are smaller (53–87%) than the estimation based on Equation (20), leading to overall lower uncertainties in glacier mass balance computations.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Data and Code Availability

## References

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**Figure 1.**Overview of study area in southeast Alaska, USA. Glacier outlines are based on the Randolph Glacier Inventory 6.0. Elevation change is derived from SRTM (2000) and IfSAR (2012/13) data. Black areas indicate artificial voids on glacier surfaces using an ASTER correlation threshold of 50%. Background: ASTER GDEMV2 hillshade. Light blue dashed rectangle indicates the extent of the Juneau setup and pink dashed rectangle indicates the extent of the Center setup (see Figure 2).

**Figure 2.**Original elevation change fields of subsets of the study region (Figure 1) used for analysis of the void filling approaches on large data voids (purple polygons). Results of the different void filling approaches are provided in Figure 4, Figures S1 and S2.

**Figure 3.**Hypsometric distribution of glacier area (gray bars) and surface elevation changes $d{h}^{O}$ (blue dots: mean $d{h}^{O}$, error bars: median absolute deviation of $d{h}^{O}$) of the complete void-free data set (Correlation setup), grouped in 50 m altitude bins.

**Figure 4.**Void filling results of the different approaches (one per approach) at the Center (left column) and Juneau (right column) setups. See Figures S1 and S2 for more results.

**Figure 5.**Mean relative offset ${\overline{dh}}^{r}$ and absolute average error $AA{E}^{f}$ of the individual voids for the Center and Juneau setups. * values out of bounds.

**Figure 6.**Mean $\overline{dh}$ (${\overline{dh}}^{r}$: relative offset to original data) and standard deviation ${\sigma}_{dh}$ of sampled offset values for the Correlation setup including simultaneous confidence intervals.

**Figure 7.**Scatter plot of $\left(\overline{d{h}_{gl}}-\overline{dh}\right)$ versus ${\sigma}_{dh}/\sqrt{N}$ for the three tested void filling approaches. Black vertical lines indicate correlation area approximation in pixels (${d}_{cor}^{2}$). Gray dashed lines indicate the bounds from Equations (16) and (18) approximated by $2{d}_{cor}{\sigma}_{dh}/\sqrt{N}$ for $N>{d}_{cor}^{2}$ and by $2{\sigma}_{dh}$ for $N<{d}_{cor}^{2}$. ${f}^{\mathrm{int}}$: fraction of points within the bounds.

Approach | Parametrization | Abbreviation | |
---|---|---|---|

Novel Void Filling Approaches | |||

Telea | search radius | 2 | Telea-02 |

5 | Telea-05 | ||

8 | Telea-08 | ||

10 | Telea-10 | ||

15 | Telea-15 | ||

20 | Telea-20 | ||

Navier–Stokes | search radius | 2 | NS-02 |

5 | NS-05 | ||

8 | NS-08 | ||

10 | NS-10 | ||

15 | NS-15 | ||

20 | NS-20 | ||

Shearlet | nscales | 5 | SL-5 |

6 | SL-6 | ||

7 | SL-7 | ||

Classical Void Filling Approaches | |||

Bilinear | CL-BL | ||

Hypsometric | local | CL-LO | |

global | CL-GL |

**Table 2.**Correspondence between the variables of an incompressible fluid flow problem and the Navier–Stokes inpainting problem.

Fluid Dynamics | Inpainting |
---|---|

Stream function $\mathsf{\Psi}$ | Image intensity $I$ |

Fluid velocity $v={\nabla}_{p}\mathsf{\Psi}$ | $\mathrm{Isophote}\text{}\mathrm{direction}\text{}{\nabla}_{p}I$ |

Vorticity $\omega =\mathsf{\Delta}\psi $ | Image smoothness $\mathsf{\Delta}I$ |

Viscosity $\upsilon $ | Anisotropic diffusion $\upsilon $ |

**Table 3.**Mean $\overline{dh}\text{}$ and standard deviation ${\sigma}_{dh}$ of sampled offset values for the Correlation setup including lower, ${\overline{dh}}^{L}$, and upper, ${\overline{dh}}^{U}$, bounds of simultaneous 95% confidence intervals. ${d}_{cor}$: correlation length of inpainted offsets.

Approach | $\overline{\mathit{d}\mathit{h}}\text{}(\mathbf{m})$ | ${\overline{\mathit{d}\mathit{h}}}^{\mathit{L}}\text{}(\mathbf{m})$ | ${\overline{\mathit{d}\mathit{h}}}^{\mathit{U}}\text{}(\mathbf{m})$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{h}}\text{}\left(\mathbf{m}\right)$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{h}}^{\mathit{L}}\text{}\left(\mathbf{m}\right)$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{h}}^{\mathit{U}}\text{}\left(\mathbf{m}\right)$ | ${\mathit{d}}_{\mathit{c}\mathit{o}\mathit{r}}\text{}\left(\mathbf{pixel}\right)$ |
---|---|---|---|---|---|---|---|

NS-02 | −0.0538 | −0.0947 | −0.0128 | 2.4515 | 2.4286 | 2.4748 | 5 |

NS-05 | −0.0503 | −0.0931 | −0.0074 | 2.5643 | 2.5403 | 2.5887 | 5 |

NS-08 | −0.0497 | −0.0928 | −0.0066 | 2.5811 | 2.5570 | 2.6057 | 5 |

NS-10 | −0.0496 | −0.0928 | −0.0064 | 2.5876 | 2.5634 | 2.6122 | 5 |

NS-15 | −0.0489 | −0.0922 | −0.0056 | 2.5943 | 2.5700 | 2.6189 | 5 |

NS-20 | −0.0481 | −0.0915 | −0.0047 | 2.5990 | 2.5747 | 2.6237 | 5 |

Telea-02 | −0.0480 | −0.0901 | −0.0058 | 2.5222 | 2.4986 | 2.5461 | 5 |

Telea-05 | −0.0758 | −0.1315 | −0.0200 | 2.7893 | 2.7632 | 2.8158 | 6 |

Telea-08 | −0.0774 | −0.1377 | −0.0171 | 3.0168 | 2.9886 | 3.0455 | 6 |

Telea-10 | −0.0773 | −0.1400 | −0.0145 | 3.1387 | 3.1093 | 3.1685 | 6 |

Telea-15 | −0.0632 | −0.1548 | 0.0284 | 3.4256 | 3.3936 | 3.4582 | 8 |

Telea-20 | −0.0521 | −0.1469 | 0.0427 | 3.5475 | 3.5143 | 3.5812 | 8 |

SL-5 | −0.0090 | −0.0354 | 0.0175 | 1.9837 | 1.9651 | 2.0025 | 4 |

SL-6 | −0.0239 | −0.0518 | 0.0039 | 2.0891 | 2.0695 | 2.1089 | 4 |

SL-7 | −0.0113 | −0.0520 | 0.0293 | 2.4329 | 2.4101 | 2.4560 | 5 |

CL-BL | −0.0054 | −0.0334 | 0.0226 | 2.1007 | 2.0811 | 2.1207 | 4 |

CL-GL | −0.3240 | −0.7269 | 0.0788 | 12.1213 | 12.0080 | 12.2365 | 10 |

CL-LO | −0.0440 | −0.2474 | 0.1594 | 6.1212 | 6.0640 | 6.1794 | 10 |

**Table 4.**Summary of mean and standard deviation of X, p-value of t-test and revealed value of $\frac{{\mathsf{\sigma}}_{\mathrm{X}}}{{\mathrm{d}}_{\mathrm{cor}}\cdot {\mathsf{\sigma}}_{\mathrm{dh}}}$ for the three tested void filling techniques.

Approach | Mean of X (m) | p-Value | ${\mathit{\sigma}}_{\mathit{X}}\text{}\left(\mathbf{m}\right)$ | $\frac{{\mathit{\sigma}}_{\mathit{X}}}{{\mathit{d}}_{\mathit{c}\mathit{o}\mathit{r}}\cdot {\mathit{\sigma}}_{\mathit{d}\mathit{h}}}$ |
---|---|---|---|---|

SL-5 | −0.799 | 0.211 | 9.285 | 1.17 |

CL-BL | −0.944 | 0.140 | 9.289 | 1.11 |

CL-LO | 20.760 | 0.371 | 125.142 | 1.02 |

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**MDPI and ACS Style**

Seehaus, T.; Morgenshtern, V.I.; Hübner, F.; Bänsch, E.; Braun, M.H. Novel Techniques for Void Filling in Glacier Elevation Change Data Sets. *Remote Sens.* **2020**, *12*, 3917.
https://doi.org/10.3390/rs12233917

**AMA Style**

Seehaus T, Morgenshtern VI, Hübner F, Bänsch E, Braun MH. Novel Techniques for Void Filling in Glacier Elevation Change Data Sets. *Remote Sensing*. 2020; 12(23):3917.
https://doi.org/10.3390/rs12233917

**Chicago/Turabian Style**

Seehaus, Thorsten, Veniamin I. Morgenshtern, Fabian Hübner, Eberhard Bänsch, and Matthias H. Braun. 2020. "Novel Techniques for Void Filling in Glacier Elevation Change Data Sets" *Remote Sensing* 12, no. 23: 3917.
https://doi.org/10.3390/rs12233917