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Thursday, 2020-Nov-26 — Bicubic vs Bilinear Interpolation of Copernicus DEM

The surface of the earth more often follows the shape of a curve than of a straight 30 meter line. For this reason, bicubic will reduce RMSE values over bilinear interpolation when operating on DEM grids of 30 meters or greater. (For grids of 1 or 2 meters, RMSE differences fall below the noise floor, making bilinear acceptable.)

To demonstrate this, we reran our previous study but with bilinear interpolation on the two relatively uncluttered landcover types, short vegetation and barren, which provide good exposure of the curved earth to the sensing equipment above. The results are below:

LabelDEMGrid SpacingInterpolation
CUBIC30COPernicus DEM30mBicubic
LINEAR30COPernicus DEM30mBilinear
CUBIC90COPernicus DEM90mBicubic
LINEAR90COPernicus DEM90mBilinear

Short Vegetation (Flat terrain)
Graph comparing ICESat-2 terrain heights in flat terrain in short vegetation with ASTER, AW3D30, Copernicus DEM, NASADEM and SRTM
LabelComparisonsRMSEMeanStDev< 2m> 8m
CUBIC3045,115,6401.200.121.2097.0%0.4%
LINEAR3045,122,0561.600.141.6097.0%0.4%
CUBIC9045,115,6401.260.151.2595.8%0.3%
LINEAR9045,121,4521.970.191.9695.1%0.4%
Terrain Slope





Barren (Flat terrain)
Graph comparing ICESat-2 terrain heights in flat barren terrain with ASTER, AW3D30, Copernicus DEM, NASADEM and SRTM
LabelComparisonsRMSEMeanStDev< 2m> 8m
CUBIC306,111,1491.22-0.121.2198.5%0.3%
LINEAR306,113,6832.37-0.072.3798.4%0.3%
CUBIC906,111,1491.30-0.121.2997.6%0.3%
LINEAR906,113,6482.64-0.082.6396.9%0.4%
Terrain Slope






Bilinear interpolation returns a lower RMSE for Copernicus DEM 30 in very rugged terrain (terrain_slope > 0.125). This anomaly might be due to issues with ICESat-2 ATL08 and its 100m segment size.

Bilinear involves far fewer calculations than bicubic. We ran a variety of benchmarks to measure the throughput of both interpolation methods and found bilinear to be as fast to four times faster than bicubic interpolation, depending on the frequency of CPU cache misses. If throughput is important to you, use bicubic interpolation and optimize how you access the DEM grid. CPUs access memory sequentially 100X faster than randomly. Based on the physical layout of most uncompressed DEM grids, start with the highest latitude, work east, and then work south.

Copernicus DEM presents the highest standard in accuracy for a global DEM. Bilinear interpolation will degrade this accuracy, especially for the 90m product. If performance is a concern, use bicubic interpolation and query the DEM in a way that minimizes CPU cache misses.

Wednesday, 2020-Nov-25 — Digital Elevation Model Comparison

Digital Elevation Models (DEM) have many uses, from land and forest management, wireless propagation studies and satellite image orthorectification. One-meter DEMs now let flight simulators deliver state-of-the-art visuals — for only $59.99!

Fortunately we have many DEMs to choose from, including these six which are the subjects of our comparison:


Each DEM has its own strengths, and choosing the the right one depends on evaluating metrics such as

MetricDetails
Vertical accuracy (absolute)DEM elevation compared to geodetic vertical control points, located across a range of terrain slopes and landcover types. A DEM might perform well in flat, uncluttered spaces but not on forested hillsides
Vertical accuracy (relative)Elevation difference between adjacent DEM cells constitute a slope's y component
Horizontal uncertaintyAdds uncertainty to a slope's x component
Grid spacingA 90m DEM captures fewer terrain details than a 30m DEM
Storage / bandwidthA 90m DEM uses 1/9th the storage or bandwidth of a 30m DEM
Co-registrationHorizontal bias may prevent integration with landcover, road network, and other spatial databases
Terrain and / or SurfaceDigital Terrain Model (DTM) helps analyze water flow; Digital Surface Model (DSM) helps analyze wireless signal propagation. Combine both to measure tree canopy height or forest volume
Coverage areaGlobal or local? 1, 2 or 5m DEMs might be available only for regional areas of interest
AgeSome DEMs (not listed above) are 30+ years old
VoidsPeaks can obscure mountain side creating nulls or voids in data. How prevalent?
SecurityWas DEM corrupted during download, or otherwise tampered with? Are checksums available?
AccessibilityHow easy is it to download DEM? Ftp, batch wget or cumbersome website?
LicenseWhat use cases are allowed or prohibited? Can you make derivative works?
CostPrice can vary from free to thousands of dollars for a regional area of interest
MetadataIs ancillary data provided, eg. waterbody mask?
WaterbodyAre lakes flattened to one elevation, river elevations adjusted by a constant interval and shorelines elevated above adjacent waterbody?
UpdatesAll DEMs have errors. How do you report them? How often are updates published?
FormatA binary file with 1 arcsecond grid spacing (eg. SRTM) is easy to use via your own bespoke software, but a TIF with a more efficient projection might be easier to use with packaged software
Coordinates / datumDoes DEM use an obscure, local projection or vertical datum?

Our study addresses only the first metric above, Vertical accuracy (absolute). We loosely followed the approach of this study which used ICESat-2 ATL08 segments as vertical control points, but with a few differences:

Our study compared 280 million ATL08 segments captured from 2018-Oct-14 to 2020-Sep-30 across North America to the six DEMs above. NALCMS 2015 v2 provided landcover type:

# ComparisonsPercentageNALCMS Landcover Type
174,238,42762.2%Short Vegetation
48,308,78017.3%Needleleaf Forest
28,604,99110.2%Barren
12,895,1844.6%Broadleaf Forest
9,683,6313.5%Mixed Forest
6,208,4902.2%Urban
279,939,503100.0%ALL
# ComparisonsPercentageATL08 terrain_slope
87,319,33731.2%Mild (0.005 - 0.02)
71,503,95725.6%Flat (< 0.005)
61,059,04421.8%Moderate (0.02 - 0.05)
39,805,80814.2%Rugged (0.05 - 0.125)
20,251,3577.2%Very Rugged (> 0.125)
279,939,503100.0%ALL

An ICESat-2 ATL08 segment reports one latitude, one longitude, one terrain and one canopy elevation from where 100s of photons struck a track 14m wide and 100m long, at ground level, up in a pervious vegetative canopy — or infrequently way up in the clouds! This process is at the forefront of remote sensing technology and offers an unbelievable volume of data with an unbelievable level of precision. But, what about errors and their distribution? Are they random or have some systemic properties? If the 100m track is flat terrain, we have confidence in the values reported. For rough terrain, our confidence drops. For this reason, we caution you when interpreting any results below where terrain_slope > 0.02 (ie. > 2m / 100m).

RMSE, Mean, StDev and x-axis are in meters. Graphs omit ASTER as its curves ride near the x-axis (it's not that ASTER is bad; it's that Copernicus DEM is very good!) Toggle the Terrain Slope buttons to see results at various slopes (for uncluttered landcover types only).

Short Vegetation (Flat terrain)
Graph comparing ICESat-2 terrain heights in flat terrain in short vegetation with ASTER, AW3D30, Copernicus DEM, NASADEM and SRTM
DEMComparisonsRMSEMeanStDev< 2m> 8m
ASTER45,115,6239.44-3.918.5920.6%32.1%
AW3D3045,115,5962.681.092.4569.6%0.9%
COP3045,115,6401.200.121.2097.0%0.4%
COP9045,115,6401.260.151.2595.8%0.3%
NASADEM29,583,5482.38-0.232.3775.9%1.2%
SRTMV328,703,7992.820.152.8157.3%1.2%
Terrain Slope





Barren (Flat terrain)
Graph comparing ICESat-2 terrain heights in flat barren terrain with ASTER, AW3D30, Copernicus DEM, NASADEM and SRTM
DEMComparisonsRMSEMeanStDev< 2m> 8m
ASTER6,111,14712.13-6.929.9613.3%49.5%
AW3D306,111,1433.341.073.1672.1%1.3%
COP306,111,1491.22-0.121.2198.5%0.3%
COP906,111,1491.30-0.121.2997.6%0.3%
NASADEM1,183,2424.72-0.084.7265.5%3.3%
SRTMV31,144,2674.300.554.2752.9%3.1%
Terrain Slope






Needleleaf Forest (trees > 5m tall)
Graph comparing ICESat-2 terrain heights in needleleaf forest with ASTER, AW3D30, Copernicus DEM, NASADEM and SRTM
DEMComparisonsRMSEMeanStDev< 2m> 8m
ASTER13,035,2909.05-3.448.3817.7%35.4%
AW3D3013,035,2903.611.843.1059.2%3.8%
COP3013,035,2903.261.502.9074.3%3.9%
COP9013,035,2903.231.562.8272.2%3.6%
NASADEM9,574,4303.160.133.1673.3%3.1%
SRTMV38,724,6113.140.063.1459.1%2.5%
Terrain Slope
< 0.5m / 100m  
Broadleaf Forest (trees > 5m tall)
Graph comparing ICESat-2 terrain heights in broadleaf forest with ASTER, AW3D30, Copernicus, NASADEM and SRTM
DEMComparisonsRMSEMeanStDev< 2m> 8m
ASTER3,181,0849.040.809.0018.4%34.6%
AW3D303,181,0846.183.874.8235.4%15.9%
COP303,181,0846.123.994.6442.9%16.2%
COP903,181,0845.994.084.3838.3%15.5%
NASADEM2,871,9744.252.003.7549.1%6.7%
SRTMV32,826,9305.062.644.3240.6%11.0%
Terrain Slope
< 0.5m / 100m  

Mixed Forest (trees > 5m tall)
Graph comparing ICESat-2 terrain heights in mixed forest with ASTER, AW3D30, Copernicus, NASADEM and SRTM
DEMComparisonsRMSEMeanStDev< 2m> 8m
ASTER2,170,6359.24-0.019.2418.3%35.2%
AW3D302,170,6355.953.834.5637.1%15.2%
COP302,170,6356.164.024.6742.9%17.1%
COP902,170,6356.054.124.4438.9%16.7%
NASADEM1,726,0604.001.713.6253.2%5.3%
SRTMV31,682,9474.371.993.8947.9%7.3%
Terrain Slope
< 0.5m / 100m  
Urban / built-up areas
Graph comparing ICESat-2 heights in urban areas with ASTER, AW3D30, Copernicus, NASADEM and SRTM
DEMComparisonsRMSEMeanStDev< 2m> 8m
ASTER1,890,1598.24-0.798.2023.9%24.2%
AW3D301,890,1593.611.913.0750.2%2.6%
COP301,890,1592.861.262.5776.4%2.5%
COP901,890,1592.861.382.5073.3%2.3%
NASADEM1,877,6953.140.683.0762.6%2.1%
SRTMV31,875,5764.071.663.7245.1%4.7%
Terrain Slope
< 0.5m / 100m  

In broadleaf and mixed forests, all DEMs had a positive bias, returning elevations above the forest floor. SRTM / NASADEM had less bias (ie. lower Mean) than Copernicus DEM, as its lower frequency C-band radar penetrated farther into the canopy and closer to the forest floor than Copernicus DEM's higher frequency X-band radar.

We use DEMs for wireless propagation studies, where a tree can act like a hill, blocking all signals. So, Copernicus DEM measurements closer to the top of the forest canopy is a bonus, and its higher RMSE value (6.16 for COP30 vs 4.00 for NASADEM in mixed forests) is not a concern. For you, this higher RMSE value might be a concern. It all depends on your use case.

Methodology:

Geoid DEM elevations were first converted to WGS84 ellipsoid elevations before being compared to ICESat-2 elevations. Gridded data (ie. DEM and vertical datum conversion) used 16-value bicubic interpolation. SRTM / NASADEM had fewer comparisons because their coverage does not extend north of 60N and 61N latitude, respectively.

Please contact us if you you have any questions about our study.

Citations: