A Digital Elevation Model (DEM) has bare-earth z-values at regularly spaced intervals in the horizontal directions. Although DEMs all contain a constant grid resolution, the grid spacing, datum, coordinate systems, data formats, and other characteristics may vary widely (Heidemann, 2012). This interval, known as point spacing or grid resolution, becomes pixel size in raster data representations. There is no consistent, clear methodology to determine optimal post spacing from the nominal point spacing (NPS) of raw lidar data. Literature consensus seems to be that the DEM point spacing size (grid resolution) should be at least equal if not greater than the NPS. This presentation reports on methods used to help determine post spacing primarily using open source software. Raw lidar point clouds over the Great Smoky Mountains and Grand Canyon National Parks were used to test DEM spacing scenarios gleaned from literature. Ground points were filtered using LP360. Approximately 5% of these were reclassified to act as control points and create a shapefile(using open source lasthin and las2shp). DEMs were created from the ~95% remaining points using Global Mapper. Then the DEM and shapefile z-values were used to calculate RMSE. The results show a general relationship of RMSE to the optimal post spacing scalar and RMSE to DEM post size follow generally a power law curve. Ongoing analysis continues. We believe this analytical approach can shed light on the NPS/ DEM problem by empirically deriving a recommended scalar from various published techniques and two different study areas.

1. {
U.S. Department of the Interior
U.S. Geological Survey Frank Velasquez / Kristina Yamamoto
GIS in the Rockies
24 September 2015
Point Spacing and Pixel Size

2. Agenda
 Objective
 Literature
 Alpha Table
 Concept of Operations
 Study Areas
 Point Clouds
 Process Flow
 RMSE Results
 Charts
 Discussions

3. Objective
• Attempt to fill in the literature surrounding the optimal
point space (DEMpost space)
• For a given nominal point space (nps) distance (or
point density) of a lidar point cloud, what is the
optimal grid resolution without overly interpolating
pixels or losing data quality
• i.e. What is the scalar to apply to nps to achieve
optimal DEMpost spacing
• α = nps ÷ DEMpost
• Empirical research problem
• Experimental design is complete
• Execution of the designed tests are (mostly) complete
• Analysis of results is ongoing

4. Literature
• Rees and Arnold (2007) created a 2m raster grid from a 0.8 m nps
• Hopkinson et al. (2009) had varying nps between 1 and 4m and created
two DEMs; one at 5m and one at 25m
• Perroy et al. (2010) created a raster grid at resolution equal to the nps
• Gonzalez et al. (2010) suggest creating a grid of 2m or 5m from a 1m
nominal point spacing
• Jones et al. (2010) used a 2m resolution grid from a 1.6m nps
• Dong et al. (2010) suggested a raster resolution of one-third to one-fifth the
nps; i.e. of 3 to 5m, raster resolution would be 1m
• Long et al. (2011) contradicts (or inverted) this suggesting that for spacing
you would construct a 3m resolution grid.
• Keith Clarke (UCSB) suggested a DEMpost of 2 times the min, max, mean,
or median of the nearest neighbor interpoint spacing

5. Alpha Table
Technique No. Name Formula NPS Alpha DEM Post
(cell size m2)
1 Nyquist-Shannon * p ≤ (h-bar [sub ij]/2)
2 Finn et al. (2012) 1.4 .90 1.6
3a Clarke min * DEM Post = 2 x spacing min
3b Clarke max * DEM Post = 2 x spacing max
3c Clarke mean * DEM Post = 2 x spacing mean
3d Clarke median * DEM Post = 2 x spacing median
4 Rees and Arnold (2007) 0.8 .40 2.0
5a Hopkinson et al. (2009) 1.0 .20 5.0
5b Hopkinson et al. (2009) 4.0 .16 25.0
5c Hopkinson et al (2009) 4.0 .80 5.0
6 Perroy et al. (2010) 1.5 1.0 1.5
7a Gonzalez et al. (2010) 1.0 .50 2.0
7b Gonzalez et al (2010) Same as 5a 1.0 .20 5.0
8 Jones et al (2010) Same as 5c 1.6 .80 2.0
9a Dong et al. (2010) 3.0 3.0 1.0
9b Dong et al (2010) 5.0 5.0 1.0
10 Long et al. (2011) 1.0 .33 3.0
11 Heideman 2.0 .67 3.0
* Postgres/PostGIS algorithm developed by Mike Gleason of NREL to calculate nearest neighbor
All other alphas were derived from the literature (α = nps ÷ DEMpost)

6. Concept of Operations
Raw point
cloud
Filter ground
points
Reclassify ~5%
of points for
control group
Generate DEM
Ground
points
Generate
shapefile of
control points
ShapefileDEM
Compare DEM
to shapefile,
calculate Δz
Reclassified
point cloud
Output csv
& generate
charts

7. Great Smoky Mtn. Study Area

8. Grand Canyon Study Area

9. Great Smoky Mtn. Point Clouds
All Points Ground Points Control Points
5,614,743 879,891 46,210 (5.25%)

10. Grand Canyon Point Clouds
All Points Ground Points Control Points
37,975,517 12,003,692 558,699 (4.65%)

11. Process Flow
• LP360 to filter ground points from raw point cloud (Kim Mantey)
• lasthin to filter/reclassify ~5% control points
• ArcMap to create las dataset, calculate point statistics, verify reclassified point %
• las2shp to generate shapefile of control points
• Global Mapper to generate GeoTIFF DEM at various grid resolutions
• ArcMap to verify DEM grid size
• Global Mapper to compare DEM pixel z values to control group z values
 Select all points in shapefile
 Rename “elevation” attribute to “point elevation”
 Add coordinates attributes to control points
 Apply elevation attributes from terrain layer (DEM) to control points
 Verify points now have two elevation attributes
 Calculate elevation delta
 Create a new attribute (“elevation delta”)
 Subtract “elevation” from “point elevation”
 Generate statistics report (included slope attributes for possible future
analysis)
 Calculate RMSE and generate charts

12. Great Smoky Mtn.
Technique No. Name Alpha DEM Post (Cell size m2) Columns and Rows RMSE
1 Nyquist-Shannon * .599 1.15 1305 x1305 0.35286
2 Finn et al. .90 .7667 1957 x 1957 0.19809
3a Clarke min * 3.383 .2039 7356 x 7356 0.17018
3b Clarke max * .077 8.9833 167 x 167 1.08971
3c Clarke mean * .288 2.3964 626 x 626 0.53171
3d Clarke median * .285 2.4252 619 x 619 0.54222
4 Rees and Arnold .40 1.725 870 x 870 0.47653
5a Hopkinson et al. .20 3.45 435 x 435 0.80557
5b Hopkinson et al. .16 4.3125 348 x 348 1.06316
5c Hopkinson et al. .80 .8625 1740 x 1740 0.18409
6 Perroy et al. 1.0 .69 2174 x 2174 0.18965
7a Gonzalez et al. .50 1.38 1087 x 1087 0.35112
9a Dong et al. 3.0 .23 6523 x 6523 0.17358
9b Dong et al. 5.0 .1380 10872 x 10872 0.17227
10 Long et al. .33 2.0909 718 x 718 0.59202
11 Heideman .67 1.0299 1457 x 1457 0.34911
Yamamoto .069 10.0 150 x 150 1.10687
1 meter cell .69 1.0 1500 x 1500 0.33047
• NPS .69
• Gonzalez 7b alpha same as Hopkinson 5a
• Jones 8 alpha same as Hopkinson 5c
* Gleason algorithm nearest neighbor results:
Min: 0.1019 m Mean: 1.1982 m
Max: 4.4916 m Median: 1.2126 m

13. Grand Canyon
Technique No. Name Alpha DEM Post (Cell size m2) Columns and Rows RMSE
2 Finn et al. .90 .3522 4259 x 4259 0.0890
4 Rees and Arnold .40 .7925 1893 x 1893 0.6000
5a Hopkinson .20 1.59 943 x 943 0.9363
5b Hopkinson .16 1.9813 757 x 757 0.9603
5c Hopkinson .80 .3963 3785 x 3785 0.4815
6 Perroy et al. 1.0 .317 4688 x 4688 0.3395
7a Gonzalez et al. .50 .63 2381 x 2381 0.4714
9a Dong et al. 3.0 .11 13638 x 13638 0.3430
9b Dong et al. 5.0 0.0634 23662 x 23662 0.3456
10 Long et al. .33 .9606 1562 x 1562 0.6401
11 Heideman .67 .4731 3171 x 3171 0.4699
Yamamoto .069 10.0 150 x 150 1.5501
1 meter cell .69 1.0 1500 x 1500 0.6350
• NPS .317 (Reported point density 9.97m2)
• Nyquist-Shannon and Clarke inter-point spacing data unavailable at this time
• Gonzalez 7b alpha same as Hopkinson 5a
• Jones 8 alpha same as Hopkinson 5c

14. Great Smoky Mtn. Alpha / RMSE
y = 0.2766x-0.531
R² = 0.8711
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0.00 1.00 2.00 3.00 4.00 5.00
RMSE
Alpha

15. Great Smoky Mtn. DEM Post / RMSE
y = 0.3369x0.5311
R² = 0.8711
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
RMSE
DEM Post

16. Grand Canyon Alpha / RMSE
y = 0.3886x-0.38
R² = 0.5059
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
0.00 1.00 2.00 3.00 4.00 5.00
RMSE
Alpha

17. Grand Canyon DEM Post / RMSE
y = 0.6015x0.3803
R² = 0.50590.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
RMSE
DEM Post

18. {
U.S. Department of the Interior
U.S. Geological Survey Frank Velasquez / Kristina Yamamoto
GIS in the Rockies
24 September 2015
Point Spacing and Pixel Size
Discussions