# Using Deep Learning to Model Elevation Differences between Radar and Laser Altimetry

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data

## 3. Method

#### 3.1. Data Filtering and Preparation

#### 3.1.1. CryoSat-2 Data

^{i}and relative-elevation-mean

^{j}. For a given waveform sample number n, the relative-elevation

^{i}is defined as elevation

^{n+i}–elevation

^{n}where i = {−3, −2, −1, 1, 2, 3}. For the same sampled waveform point, n, the relative-elevation-mean

^{j}is the arithmetic mean of relative-elevation

^{i}selecting the neighbouring j points where −$\frac{\mathrm{j}}{2}$ ≤ i < $\frac{\mathrm{j}}{2}$ and j = {10, 20}. The direct elevation measurement was omitted from model inputs to avoid spatial overfitting with the aim of improved spatial transferability.

#### 3.1.2. ICESat-2 Data

#### 3.1.3. ArcticDEM Data

#### 3.1.4. Spatio-Temporal Join

#### 3.2. Statistical Assumptions

#### 3.3. CryoSat-2 SARIn vs. ICESat-2 LIDAR Maps

#### 3.4. Models

## 4. Results

^{−13}—i.e., significantly less than 0.05. Lastly, a 34% improvement in the Pearson correlation coefficient (r) demonstrates the NN tends to better predict the observations.

## 5. Discussion

#### 5.1. Observations

#### 5.1.1. LIDAR-SARIn Differences-Observed Contribution Drivers

^{1}would be 10 m in this case. This would require a LIDAR-SARIn correction of −10 m to be accounted for as well as additional physical and systematic contributors. If relative-elevation

^{i}captured only stochastic noise, one would expect the gradient of adjustment vs. relative elevation to be −1. A gradient of circa −0.55 is observed for relative elevations beneath its modal value and circa −0.90 otherwise. One interpretation is that relative elevations above an expected amount are more likely to be the result of stochastic drivers. In contrast, those less than the expected amount are made up less of stochastic noise and more of other physical or systematic contributions. The behaviour of adjustment as a function of relative-elevation

^{−1}is an x-shifted equivalent of that of relative-elevation

^{1}, which is to be expected on an aggregated basis. The relative mean elevation quantities can also be interpreted in a similar way and exhibit similar behaviour. However, the pre-modal gradients for the adjustment as a function of relative-mean-elevation

^{10}and relative-mean-elevation

^{20}are circa −0.63 and −0.70, respectively, suggesting they are even stronger stochastic correction indicators.

#### 5.1.2. LIDAR-SARIn Differences–Spatial and Temporal Trends

#### 5.2. OLS and NN Models-Strengths

#### 5.3. OLS and NN Models-Limitations

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Zwally, H.J.; Bindschadler, R.A.; Brenner, A.C.; Major, J.A.; Marsh, J.G. Growth of Greenland Ice Sheet: Measurement. Science
**1989**, 246, 1587–1589. [Google Scholar] [CrossRef] [PubMed] - Wingham, D.J.; Ridout, A.J.; Scharroo, R.; Arthern, R.J.; Shum, C.K. Antarctic Elevation Change from 1992 to 1996. Science
**1998**, 282, 456–458. [Google Scholar] [CrossRef] - Shepherd, A.; Wingham, D.; Payne, T.; Skvarca, P. Larsen Ice Shelf Has Progressively Thinned. Science
**2003**, 302, 856–859. [Google Scholar] [CrossRef] [Green Version] - Shepherd, A.; Wingham, D.J.; Mansley, J.A.D.; Corr, H.F.J. Inland Thinning of Pine Island Glacier, West Antarctica. Science
**2001**, 291, 862–864. [Google Scholar] [CrossRef] - Zwally, H.J.; Giovinetto, M.B.; Li, J.; Cornejo, H.G.; Beckley, M.A.; Brenner, A.C.; Saba, J.L.; Yi, D. Mass Changes of the Greenland and Antarctic Ice Sheets and Shelves and Contributions to Sea-Level Rise: 1992–2002. J. Glaciol.
**2005**, 51, 509–527. [Google Scholar] [CrossRef] [Green Version] - Wingham, D.J.; Shepherd, A.; Muir, A.; Marshall, G.J. Mass Balance of the Antarctic Ice Sheet. Philos. Trans. R. Soc. Math. Phys. Eng. Sci.
**2006**, 364, 1627–1635. [Google Scholar] [CrossRef] - Fricker, H.A.; Scambos, T.; Bindschadler, R.; Padman, L. An Active Subglacial Water System in West Antarctica Mapped from Space. Science
**2007**, 315, 1544–1548. [Google Scholar] [CrossRef] [Green Version] - Pritchard, H.D.; Arthern, R.J.; Vaughan, D.G.; Edwards, L.A. Extensive Dynamic Thinning on the Margins of the Greenland and Antarctic Ice Sheets. Nature
**2009**, 461, 971–975. [Google Scholar] [CrossRef] - Kääb, A.; Berthier, E.; Nuth, C.; Gardelle, J.; Arnaud, Y. Contrasting Patterns of Early Twenty-First-Century Glacier Mass Change in the Himalayas. Nature
**2012**, 488, 495–498. [Google Scholar] [CrossRef] - Bamber, J.L.; Griggs, J.A.; Hurkmans, R.T.W.L.; Dowdeswell, J.A.; Gogineni, S.P.; Howat, I.; Mouginot, J.; Paden, J.; Palmer, S.; Rignot, E.; et al. A New Bed Elevation Dataset for Greenland. Cryosphere
**2013**, 7, 499–510. [Google Scholar] [CrossRef] - McMillan, M.; Shepherd, A.; Sundal, A.; Briggs, K.; Muir, A.; Ridout, A.; Hogg, A.; Wingham, D. Increased Ice Losses from Antarctica Detected by CryoSat-2. Geophys. Res. Lett.
**2014**, 41, 3899–3905. [Google Scholar] [CrossRef] - Gourmelen, N.; Goldberg, D.N.; Snow, K.; Henley, S.F.; Bingham, R.G.; Kimura, S.; Hogg, A.E.; Shepherd, A.; Mouginot, J.; Lenaerts, J.T.M.; et al. Channelized Melting Drives Thinning under a Rapidly Melting Antarctic Ice Shelf. Geophys. Res. Lett.
**2017**, 44, 9796–9804. [Google Scholar] [CrossRef] - Gourmelen, N.; Escorihuela, M.J.; Shepherd, A.; Foresta, L.; Muir, A.; Garcia-Mondéjar, A.; Roca, M.; Baker, S.G.; Drinkwater, M.R. CryoSat-2 Swath Interferometric Altimetry for Mapping Ice Elevation and Elevation Change. Adv. Space Res.
**2018**, 62, 1226–1242. [Google Scholar] [CrossRef] [Green Version] - Rémy, F.; Parouty, S. Antarctic Ice Sheet and Radar Altimetry: A Review. Remote Sens.
**2009**, 1, 1212–1239. [Google Scholar] [CrossRef] [Green Version] - Nilsson, J.; Vallelonga, P.; Simonsen, S.B.; Sørensen, L.S.; Forsberg, R.; Dahl-Jensen, D.; Hirabayashi, M.; Goto-Azuma, K.; Hvidberg, C.S.; Kjær, H.A.; et al. Greenland 2012 Melt Event Effects on CryoSat-2 Radar Altimetry. Geophys. Res. Lett.
**2015**, 42, 3919–3926. [Google Scholar] [CrossRef] - Slater, T.; Shepherd, A.; Mcmillan, M.; Armitage, T.W.K.; Otosaka, I.; Arthern, R.J. Compensating Changes in the Penetration Depth of Pulse-Limited Radar Altimetry over the Greenland Ice Sheet. IEEE Trans. Geosci. Remote Sens.
**2019**, 57, 9633–9642. [Google Scholar] [CrossRef] [Green Version] - Gray, L. Brief Communication: Glacier Run-off Estimation Using Altimetry-Derived Basin Volume Change: Case Study at Humboldt Glacier, Northwest Greenland. Cryosphere
**2021**, 15, 1005–1014. [Google Scholar] [CrossRef] - Slater, T.; Shepherd, A.; McMillan, M.; Leeson, A.; Gilbert, L.; Muir, A.; Munneke, P.K.; Noël, B.; Fettweis, X.; van den Broeke, M.; et al. Increased Variability in Greenland Ice Sheet Runoff from Satellite Observations. Nat. Commun.
**2021**, 12, 6069. [Google Scholar] [CrossRef] [PubMed] - Arthern, R.J.; Wingham, D.J.; Ridout, A.L. Controls on ERS Altimeter Measurements over Ice Sheets: Footprint-Scale Topography, Backscatter Fluctuations, and the Dependence of Microwave Penetration Depth on Satellite Orientation. J. Geophys. Res. Atmos.
**2001**, 106, 33471–33484. [Google Scholar] [CrossRef] [Green Version] - Gray, L.; Burgess, D.; Copland, L.; Langley, K.; Gogineni, P.; Paden, J.; Leuschen, C.; van As, D.; Fausto, R.; Joughin, I.; et al. Measuring Height Change around the Periphery of the Greenland Ice Sheet with Radar Altimetry. Front. Earth Sci.
**2019**, 7, 146. [Google Scholar] [CrossRef] - Recchia, L.; Scagliola, M.; Giudici, D.; Kuschnerus, M. An Accurate Semianalytical Waveform Model for Mispointed SAR Interferometric Altimeters. IEEE Geosci. Remote Sens. Lett.
**2017**, 14, 1537–1541. [Google Scholar] [CrossRef] - Snauffer, A.M.; Hsieh, W.W.; Cannon, A.J.; Schnorbus, M.A. Improving Gridded Snow Water Equivalent Products in British Columbia, Canada: Multi-Source Data Fusion by Neural Network Models. Cryosphere
**2018**, 12, 891–905. [Google Scholar] [CrossRef] [Green Version] - Tollenaar, V.; Zekollari, H.; Lhermitte, S.; Tax, D.M.J.; Debaille, V.; Goderis, S.; Claeys, P.; Pattyn, F. Unexplored Antarctic Meteorite Collection Sites Revealed through Machine Learning. Sci. Adv.
**2022**, 8, eabj8138. [Google Scholar] [CrossRef] [PubMed] - Braakmann-Folgmann, A.; Donlon, C. Estimating Snow Depth on Arctic Sea Ice Using Satellite Microwave Radiometry and a Neural Network. Cryosphere
**2019**, 13, 2421–2438. [Google Scholar] [CrossRef] [Green Version] - Memarian Sorkhabi, O.; Asgari, J.; Amiri-Simkooei, A. Wavelet Decomposition and Deep Learning of Altimetry Waveform Retracking for Lake Urmia Water Level Survey. Mar. Georesources Geotechnol.
**2022**, 40, 361–369. [Google Scholar] [CrossRef] - Gray, L.; Burgess, D.; Copland, L.; Cullen, R.; Galin, N.; Hawley, R.; Helm, V. Interferometric Swath Processing of Cryosat Data for Glacial Ice Topography. Cryosphere
**2013**, 7, 1857–1867. [Google Scholar] [CrossRef] [Green Version] - Hawley, R.L.; Shepherd, A.; Cullen, R.; Helm, V.; Wingham, D.J. Ice-Sheet Elevations from across-Track Processing of Airborne Interferometric Radar Altimetry. Geophys. Res. Lett.
**2009**, 36, L22501. [Google Scholar] [CrossRef] [Green Version] - Krabill, W. IceBridge ATM L2 Icessn Elevation, Slope, and Roughness, Version 2; NASA Distribution Active Archive Center, National Snow Ice Data Center: Boulder, CO, USA, 2014.
- Smith, B. ATLAS/ICESat-2 L3A Land Ice Height, Version 3; National Snow and Ice Data Center: Boulder, CO, USA, 2020. [Google Scholar]
- Porter, C.; Morin, P.; Howat, I.; Noh, M.-J.; Bates, B.; Peterman, K.; Keesey, S.; Schlenk, M.; Gardiner, J.; Tomko, K.; et al. ArcticDEM, Version 3. 2019. Available online: https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi:10.7910/DVN/OHHUKH (accessed on 20 February 2022).
- Legrésy, B.; Rémy, F. Altimetric Observations of Surface Characteristics of the Antarctic Ice Sheet. J. Glaciol.
**1997**, 43, 265–275. [Google Scholar] [CrossRef] [Green Version] - Virtanen, P.; Gommers, R.; Oliphant, T.E.; Haberland, M.; Reddy, T.; Cournapeau, D.; Burovski, E.; Peterson, P.; Weckesser, W.; Bright, J.; et al. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nat. Methods
**2020**, 17, 261–272. [Google Scholar] [CrossRef] [Green Version] - McNabb, R.; Nuth, C.; Kääb, A.; Girod, L. Sensitivity of Glacier Volume Change Estimation to DEM Void Interpolation. Cryosphere
**2019**, 13, 895–910. [Google Scholar] [CrossRef] - Paszke, A.; Gross, S.; Massa, F.; Lerer, A.; Bradbury, J.; Chanan, G.; Killeen, T.; Lin, Z.; Gimelshein, N.; Antiga, L.; et al. PyTorch: An Imperative Style, High-Performance Deep Learning Library. In Advances in Neural Information Processing Systems 32; Curran Associates, Inc.: Nice, France, 2019; pp. 8024–8035. [Google Scholar]
- Maas, A.L.; Hannun, A.Y.; Ng, A.Y. Rectifier Nonlinearities Improve Neural Network Acoustic Models. Proc. Icml.
**2013**, 30, 3. [Google Scholar] - Klambauer, G.; Unterthiner, T.; Mayr, A.; Hochreiter, S. Self-Normalizing Neural Networks. arXiv
**2017**, arXiv:1706.02515. [Google Scholar] [CrossRef] - Srivastava, N.; Hinton, G.; Krizhevsky, A.; Sutskever, I.; Salakhutdinov, R. Dropout: A Simple Way to Prevent Neural Networks from Overfitting. J. Mach. Learn. Res.
**2014**, 15, 1929–1958. [Google Scholar] - Ioffe, S.; Szegedy, C. Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift. In Proceedings of the 32nd International Conference on Machine Learning; Bach, F., Blei, D., Eds.; PMLR: Lille, France, 2015; Volume 37, pp. 448–456. [Google Scholar]
- Ba, J.L.; Kiros, J.R.; Hinton, G.E. Layer Normalization. arXiv
**2016**, arXiv:1607.06450. [Google Scholar] - Sutskever, I.; Martens, J.; Dahl, G.; Hinton, G. On the Importance of Initialization and Momentum in Deep Learning. In Proceedings of the 30th International Conference on Machine Learning, Atlanta, GA, USA, 16–21 June 2013; Dasgupta, S., McAllester, D., Eds.; PMLR: Atlanta, GA, USA, 2013; Volume 28, pp. 1139–1147. [Google Scholar]
- Kingma, D.P.; Ba, J. Adam: A Method for Stochastic Optimization. arXiv
**2014**, arXiv:1412.6980. [Google Scholar] - Huber, P.J. Robust Estimation of a Location Parameter. Ann. Math. Stat.
**1964**, 35, 73–101. [Google Scholar] [CrossRef] - Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; et al. Scikit-Learn: Machine Learning in Python. J. Mach. Learn. Res.
**2011**, 12, 2825–2830. [Google Scholar] - Earthwave. The University of Edinburgh. isardSAT CryoTEMPO-EOLIS—Elevation over Land Ice from Swath—Product Handbook. 2020. Available online: https://Earth.Esa.Int/Eogateway/Documents/20142/37627/CryoTEMPO-Thematic-Product-Handbook.Pdf (accessed on 20 February 2022).
- Gray, L.; Burgess, D.; Copland, L.; Dunse, T.; Langley, K.; Moholdt, G. A Revised Calibration of the Interferometric Mode of the CryoSat-2 Radar Altimeter Improves Ice Height and Height Change Measurements in Western Greenland. Cryosphere
**2017**, 11, 1041–1058. [Google Scholar] [CrossRef] [Green Version] - Davis, C.H.; Moore, R.K. A Combined Surface-and Volume-Scattering Model for Ice-Sheet Radar Altimetry. J. Glaciol.
**1993**, 39, 675–686. [Google Scholar] [CrossRef] [Green Version] - Ridley, J.K.; Partington, K.C. A Model of Satellite Radar Altimeter Return from Ice Sheets. Int. J. Remote Sens.
**1988**, 9, 601–624. [Google Scholar] [CrossRef] - Wingham, D.J.; Francis, C.R.; Baker, S.; Bouzinac, C.; Brockley, D.; Cullen, R.; de Chateau-Thierry, P.; Laxon, S.W.; Mallow, U.; Mavrocordatos, C.; et al. CryoSat: A Mission to Determine the Fluctuations in Earth’s Land and Marine Ice Fields. Adv. Space Res.
**2006**, 37, 841–871. [Google Scholar] [CrossRef] - Krabill, W. Greenland Ice Sheet: Increased Coastal Thinning. Geophys. Res. Lett.
**2004**, 31, L24402. [Google Scholar] [CrossRef] [Green Version] - Bingham, A.W.; Drinkwater, M.R. Recent Changes in the Microwave Scattering Properties of the Antarctic Ice Sheet. IEEE Trans. Geosci. Remote Sens.
**2000**, 38, 1810–1820. [Google Scholar] [CrossRef] - Brunt, K.M.; Neumann, T.A.; Smith, B.E. Assessment of ICESat-2 Ice Sheet Surface Heights, Based on Comparisons over the Interior of the Antarctic Ice Sheet. Geophys. Res. Lett.
**2019**, 46, 13072–13078. [Google Scholar] [CrossRef] [Green Version] - Luthcke, S.B.; Thomas, T.C.; Pennington, T.A.; Rebold, T.W.; Nicholas, J.B.; Rowlands, D.D.; Gardner, A.S.; Bae, S. ICESat-2 Pointing Calibration and Geolocation Performance. Earth Space Sci.
**2021**, 8, e2020EA001494. [Google Scholar] [CrossRef] - Fettweis, X.; Hofer, S.; Krebs-Kanzow, U.; Amory, C.; Aoki, T.; Berends, C.J.; Born, A.; Box, J.E.; Delhasse, A.; Fujita, K.; et al. GrSMBMIP: Intercomparison of the Modelled 1980–2012 Surface Mass Balance over the Greenland Ice Sheet. Cryosphere
**2020**, 14, 3935–3958. [Google Scholar] [CrossRef] - Kern, M.; Cullen, R.; Berruti, B.; Bouffard, J.; Casal, T.; Drinkwater, M.R.; Gabriele, A.; Lecuyot, A.; Ludwig, M.; Midthassel, R.; et al. The Copernicus Polar Ice and Snow Topography Altimeter (CRISTAL) High-Priority Candidate Mission. Cryosphere
**2020**, 14, 2235–2251. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) The geographical distribution of the ~29 million collocated SARIn vs. LIDAR observations in this study and (

**b**) the statistical distribution of the same observations with a fitted normal inverse Gaussian distribution trend line. (c.f. Section 3.2 for details on the standard error calculation).

**Figure 2.**Summary of the observed and modelled LIDAR-SARIn differences as a function of a selection of input parameters for all 3.6 million unseen test data points across GrIS. The distribution of observation counts per parameter is shown in the grey histogram for each plot.

**Figure 3.**Gridded maps of observed LIDAR-SARIn differences for February 2020 (

**a**) and August 2020 (

**d**) and their difference (

**g**), and equivalent model predictions from the ordinary least squares model (

**b**,

**e**,

**h**), and from the neural network model (

**c**,

**f**,

**i**). Note that the scale bars have been adjusted between observations and model predictions to allow for a clearer visual comparison of broad trends.

**Figure 4.**LIDAR-SARIn adjustment as a function of distance to POCA broken down by 6 quantile bands of coherence, power (dB), and slope across.

**Figure 5.**Spatial plots of MAR model outputs across GrIS for (

**a**) cumulative snowfall in February 2020 and the prior five months; (

**b**) cumulative snowfall in August 2020 and the prior five months; (

**c**) the change in accumulative snowfall between August 2020 and February 2020: i.e., the difference between (

**b**) and (

**a**); (

**d**) the approximate average snow density in the top 2 m of the snowpack for February 2020; (

**e**) the average surface mass balance over 12 months prior to and including February 2020; (

**f**) the total summer melt for June, July and August 2020.

Model Input Parameters | Source |
---|---|

Distance to POCA | CryoSat2-SARIn |

Power | CryoSat2-SARIn |

Coherence | CryoSat2-SARIn |

Leading Edge Width | CryoSat2-SARIn |

Relative Elevation^{i} (i = {−3, −2, −1, 1, 2, 3}) | CryoSat2-SARIn |

Relative Elevation Mean^{j} (j = {10, 20}) | CryoSat2-SARIn |

Across Track Slope | ArcticDEM |

Along Track Slope | ArcticDEM |

South/North Aspect Vector Component | ArcticDEM |

West/East Aspect Vector Component | ArcticDEM |

**Table 2.**Summary of OLS and NN accuracy on the unseen test dataset. All values relate to the difference between the model-based estimate of the LIDAR-SARIn adjustment and the observations.

Model | Mean (m) | Standard Error (m) | Median (m) | RMSE (m) | MAD (m) | r |
---|---|---|---|---|---|---|

Ordinary Least Squares (OLS) | −0.175 | 0.020 | −0.105 | 3.183 | 1.102 | 0.372 |

Neural Network (NN) | −0.048 | 0.019 | −0.053 | 2.958 | 0.987 | 0.500 |

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**MDPI and ACS Style**

Horton, A.; Ewart, M.; Gourmelen, N.; Fettweis, X.; Storkey, A.
Using Deep Learning to Model Elevation Differences between Radar and Laser Altimetry. *Remote Sens.* **2022**, *14*, 6210.
https://doi.org/10.3390/rs14246210

**AMA Style**

Horton A, Ewart M, Gourmelen N, Fettweis X, Storkey A.
Using Deep Learning to Model Elevation Differences between Radar and Laser Altimetry. *Remote Sensing*. 2022; 14(24):6210.
https://doi.org/10.3390/rs14246210

**Chicago/Turabian Style**

Horton, Alex, Martin Ewart, Noel Gourmelen, Xavier Fettweis, and Amos Storkey.
2022. "Using Deep Learning to Model Elevation Differences between Radar and Laser Altimetry" *Remote Sensing* 14, no. 24: 6210.
https://doi.org/10.3390/rs14246210