# Accuracy of Ground Surface Interpolation from Airborne Laser Scanning (ALS) Data in Dense Forest Cover

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Test Site

^{2}, located in the Latoriței Mountains of Romania, along the Bora River valley (Figure 1). Most of the area is covered with forest vegetation, mainly spruce (90 percent) and pine (10 percent). The majority of tree stands have an age of 90–95 years, with younger stands (55–75 years) also being present to a smaller extent. The most recent forest management plan indicates a high canopy density, with an average value of 0.8. The area is characterised by steep, mountainous relief, with ground slope having an average value of 26 degrees.

#### 2.2. LiDAR Data

^{−2}, which was achieved by scanning the same area two or three times. LiDAR data was delivered in the Universal Transversal Mercator (UTM) projection (zone 35N), with matched scan strips but unclassified point data.

^{−2}, corresponding to an average point spacing of 1.06 m.

#### 2.3. Validation Data and Accuracy Assessment

^{2}and an average distance between pairs of closest points of 2.48 m

^{2}(SD = 1.58 m). It is reasonable to assume that this would ensure a sufficient reliability of the accuracy analysis.

_{(i)}= Z

_{DTM(i)}− Z

_{ALS(i)}

_{(i)}is the vertical error (or residual value) for validation point i, Z

_{DTM(i)}is the elevation value of the raster cell in which point i is located and Z

_{ALS(i)}is the measured elevation for point i.

_{DTM(i)}is the elevation value estimated via interpolation for the centre of the DTM cell in which i is located, not for its exact location. Therefore, an extra error component is added, because of the distance between validation points and the cell centres of the raster model. This effect is evidently exacerbated in rougher terrain (where elevation has a higher local variance) and for lower modelling resolutions (larger cell sizes). Based on the residual values, accuracy estimates such as mean signed/unsigned error, standard deviation and Root Mean Square Error (RMSE) were calculated.

#### 2.4. Interpolation Algorithms

_{i}) is the measured value for point ${x}_{i}$, ${\lambda}_{i}$ is the weight asociated with observation z(x

_{i}) and n is the total number of reference points used for the estimation of $\hat{z}({x}_{0})$.

- Inverse Distance Weighted (IDW) is one of the most widely used algorithms for surface modelling. The weight attributed to a reference point is based on its distance from the unsampled point for which the estimation is made, with reference points further away having a lower weight [60]. IDW is an intuitive method of interpolation, in that it assumes that there is a degree of spatial autocorrelation of measured values; in other words, closer points are likely to have similar values.
- Nearest Neighbour (NeN) is a simple approach: each unsampled point will be given the value of its closest measured point. This is achieved by constructing Thiessen polygons for all reference points. The Thiessen polygon of a point is the area of influence associated with that point; all unsampled points inside a polygon are closer to its centre than to any other reference point. Therefore, all unsampled points inside a Thiessen polygon will be assigned the value of the reference point in the centre of that polygon.
- Natural Neighbour (NN) is similar to Nearest Neighbour, but more complex [61]. The first step is the generation of a network of Thiessen polygons. Unsampled points for which the prediction is made are then overlaid with the network and a new set of Thiessen polygons are generated for them. Each new polygon will overlap with one or more of the original polygons. The centre points of those polygons are called the “natural neighbours” of the unsampled point used to generate the Thiessen polygon. The weight each reference value has in the interpolation of an unsampled point is based on the degree of overlap between initial Thiessen polygons and the polygon associated with the unsampled point.
- Delauney Triangulation (DT) involves the generation of a Triangular Irregular Network (TIN) from the input point data. The general principle of a Delauney triangulation network is that no known point is inside the circumcirle of any of the generated triangles [62]. By following this approach, a Delauney network will maximise the minimum angle of all triangles and ensure that triangle edges are relatively short. Each unsampled point is located inside a triangle of the network and its predicted value is calculated via a simple linear or polynomial interpolation using the known values for that triangle’s vertices.
- Spline-based interpolators are a class of global, non-convex interpolation algorithms, which involve the fitting of a flexible surface (commonly called spline) through a set of measured observations. The interpolation function will pass through (or close to) all measured values. These algorithms work on the assumption that each measured value has an inherent measurement error, which can be reduced by generating a smooth surface of minimal tension [43]. This local smoothing is achieved by generating a spline surface of minimal tension. The main difference between these algorithms is the mathematical form of the function used for surface generation. For this paper, we tested four spline-based algorithms:
- Multilevel B-Spline (BS), which uses a hierarchy of coarse-to-fine control lattices to generate a sequence of B-Spline functions, the sum of which is the final interpolation function [63].
- Cubic Spline (CS), which is based on the construction of a bivariate cubic spline function [64].
- Thin-Plate Spline (TPS), which uses the namesake function to generate surfaces. Thin-plate Spline is the 2D generalization of the cubic spline function [65].
- Thin-Plate Spline by TIN (TPS
_{TIN}) is a variant of TPS that involves the construction of a TIN (Triangular Irregular Network) prior to interpolation. A TIN is a 3D mesh composed of triangles of varying area, using the set of measured points as vertices. Instead of a global Thin-plate Spline function, TPS_{TIN}involves the generation of a separate function for each triangle of the network.

- Ordinary Kriging (OK) is a geostatistical interpolation procedure that aims to quantify the degree of spatial auto-correlation of measured values. This is done by generating a semi-variogram, which depicts the overall shape, magnitude and spatial scale for the variation of the measured variable [66]. Over this graph a model is then fitted, which is used to assign weights to observations. These weights used for interpolation are therefore a function of the spatial co-variance of the observations [46].

#### 2.5. Factors Considered for DTM Accuracy

- Spatial resolution, which is the main characteristic of a raster data structure (such as a DTM); four model resolutions were tested: 0.5 m, 1.0 m, 1.5 m and 2.0 m.
- Ground slope, which was determined using a reference DTM, generated from the complete dataset (both prediction and validation points) by Natural Neighbour interpolation at a 0.5 m resolution. Ground slope values were then classified as follows: 0–10, 11–20, 21–30, 31–40, 41–50 and >50 degrees.
- Point density, which was determined in a raster structure at a 1.0 m resolution. To reduce the noise level of the density model, a mean filter with a 5 × 5 kernel size was applied. Point density values were classified as follows: 0–0.25, 0.26–0.50, 0.51–0.75, 0.75–1.00 and >1.00 points m
^{−2}. - Canopy density was determined in a raster structure, using the initial LiDAR point cloud and the formula proposed by [68], implemented in the FUSION LiDAR processing software:$${\delta}_{i}=\frac{n}{N}\xb7100$$
_{i}is the canopy density for cell i of the model, n is the number of LiDAR returns inside cell i that are above a user-established height threshold and N is the total number of returns inside cell i. Height threshold was set to 3 m, assuming that LiDAR returns below this height are likely objects such as understory vegetation, fallen tree trunks, boulders etc.

## 3. Results

#### 3.1. General Acuraccy of Ground Surface Interpolation

_{TIN}) have a large percentage (93–95%) of absolute vertical errors in the first category (<0.20 m).

#### 3.2. Effect of Model Resolution on DTM Accuracy

#### 3.3. Effect of External Conditions on DTM Accuracy

## 4. Discussion

_{TIN}; (2) some algorithms have a decreased performance at higher canopy density/lower ground point density—NeN, OK, IDW and (3) the rest of the algorithms seem to have random changes in terms of accuracy between classes of canopy density/ground point density.

## 5. Conclusions

_{TIN}, results do not highlight any significant difference between their respective DTMs. Therefore, the significantly longer processing time of TPS

_{TIN}(relative to TPS) is not justified.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{ASTER}and DTM

_{SRTM}going forward. The DTM interpolated from ALS data that was chosen for this comparison was generated using the Natural Neighbour (NN) algorithm, and will be identified as DTM

_{N}. This algorithm was chosen not only because of its comparatively good performance (on par with spline-based methods), but also because of its advantages from the end-user perspective: no parameters to be set, easy to understand approach to interpolation and a faster generation of DTMs than most other algorithms. Elevation values of DTM

_{NN}were translated to the EGM96 vertical datum, in order to coincide with the datum of ASTER/SRTM data.

_{ASTER}and DTM

_{SRTM}had an initial resolution of 1 arc-second (a cell-size of approx. 21 × 30 m at the latitude of our test site) so had to be resampled in order to get the appropriate resolutions. It is worth pointing out that resampling does not add any new information to the ASTER/SRTM models, but it is necessary in order to overlay and compare the models. Additionally, in order to test DTM

_{ASTER}and DTM

_{SRTM}at their native resolution, one of the DTM

_{NN}models (initially at a 0.5 m resolution) was downgraded to a resolution of 21.0 m by averaging cell values.

_{NN}offers a much more detailed representation of the ground surface, highlighting local features such as gulleys or forest tracks. Meanwhile, DTM

_{ASTER}and DTM

_{SRTM}only offer a more general perspective, representing larger changes of the landscape.

_{NN}from DTM

_{ASTER}) and for the SRTM dataset (by subtracting cell values of DTM

_{NN}from DTM

_{SRTM}). Since the purpose of this appendix is to compare (rather than validate) SRTM/ASTER data with a DTM interpolated from ALS data, we will refer to the differences of elevation values between these models as vertical deviation or vertical displacement instead of vertical error. General statistics for vertical displacement between ASTER/SRTM data and DTM

_{NN}are presented in Table A1.

_{NN}for both DTM

_{ASTER}and DTM

_{SRTM}, the other statistics (for example the root mean sq. of vertical displacement values) indicate that the two models have an overall similar deviation from DTM

_{NN}. No effect of changes in terms of model resolution on the overall displacement between ground-surface models is noticeable. This is not surprising, as between the original ASTER/SRTM model at a resolution of approx. 21.0 m and the models resampled at higher resolutions there are no functional differences. Going by the root mean sq. of vertical deviations, we can summarize that an overall difference of around 40–42 m is to be expected between ASTER/SRTM data and DTM

_{NN}, regardless of model resolution.

_{SRTM}seems to deviate less from DTM

_{NN}, at least in some areas.

**Figure A1.**DTMs for a subset of the test site, generated from: (

**a**) ASTER data, at the original 21.0 m resolution, (

**b**) SRTM data, at the original 21.0 m resolution and (

**c**) ALS data, interpolated at 0.5 m resolution using the Natural Neighbour algorithm.

**Figure A2.**Raster models of vertical displacement between: (

**a**) DTM

_{NN}and DTM

_{ASTER}and (

**b**) DTM

_{NN}and DTM

_{SRTM}.

Models Compared | Mean Signed Displ. (m) | Mean Unsigned Displ. (m) | Std. Dev. of Signed Displ. | Root Mean sq. of Vert. Displ. (m) |
---|---|---|---|---|

DTM_{NN} vs. DTM_{ASTER} (0.5 m) | −1.57 | 33.95 | 24.30 | 40.90 |

DTM_{NN} vs. DTM_{SRTM} (0.5 m) | −11.57 | 31.54 | 26.91 | 40.61 |

DTM_{NN} vs. DTM_{ASTER} (1.0 m) | −1.57 | 33.95 | 24.30 | 41.75 |

DTM_{NN} vs. DTM_{SRTM} (1.0 m) | −11.57 | 31.54 | 26.90 | 41.45 |

DTM_{NN} vs. DTM_{ASTER} (1.5 m) | −1.57 | 33.95 | 24.31 | 41.75 |

DTM_{NN} vs. DTM_{SRTM} (1.5 m) | −11.57 | 31.53 | 26.92 | 41.45 |

DTM_{NN} vs. DTM_{ASTER} (2.0 m) | −1.56 | 33.93 | 24.28 | 41.72 |

DTM_{NN} vs. DTM_{SRTM} (2.0 m) | −11.56 | 31.52 | 26.89 | 41.43 |

DTM_{NN} vs. DTM_{ASTER} (21.0 m) | −1.59 | 34.05 | 24.22 | 41.79 |

DTM_{NN} vs. DTM_{SRTM} (21.0 m) | −11.67 | 31.54 | 27.05 | 41.55 |

_{NN}) with deviations seemingly increasing with altitude. Deviations for DTM

_{ASTER}reach values as low as −82 m in the highest areas, while DTM

_{SRTM}deviations are as low as −90 m in the same areas. Meanwhile, slopes facing eastward (mostly on the western side of the test plot) have positive deviations (ASTER/SRTM models are above DTN

_{NN}) with deviations also seemingly increasing with altitude. On this side, deviations for DTM

_{ASTER}go up to values as high as 80 m in the highest areas, while DTM

_{SRTM}deviations are as high as 66 m in the same areas.

Models Compared | No. of Cells (% of Total no. Cells) | |||
---|---|---|---|---|

Abs. Vertical Displ. under 1.0 m | Abs. Vertical Displ. between 1.0 and 5.0 m | Abs. Vertical Displ. between 5.0 and 10.0 m | Abs. Vertical Displ. over 10.0 m | |

DTM_{NN} vs. DTM_{ASTER} | 1.8 | 6.0 | 7.8 | 84.4 |

DTM_{NN} vs. DTM_{SRTM} | 2.8 | 10.7 | 12.2 | 74.3 |

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**Figure 2.**Hillshade models derived from interpolated Digital Terrain Models (DTMs), for a subset of the test site (500 × 500 m).

**Figure 4.**Variation of interpolation accuracy by ground slope class (model resolution: 0.5 m). For clarity, points are horizontally offset.

**Figure 5.**Variation of interpolation accuracy by point density class (model resolution: 0.5 m). For clarity, points are horizontally offset.

**Figure 6.**Variation of interpolation accuracy by canopy density class (model resolution: 0.5 m). For clarity, points are horizontally offset.

Laser scanner parameters | |

Pulse repetition rate [kHz] | 100 |

Wavelength | near infrared |

Field of view [deg] | 60 |

Returns [reflections pulse^{−1}] | unlimited ^{1} |

Scan pattern | parallel lines |

Scan rate [lines sec^{−1}] | 58.9 |

Accuracy for distance measurement [mm] | 20 |

Precision for distance measurement [mm] | 10 |

Angular measurement resolution [deg] | 0.001 |

Flight data | |

Flight altitude [m] | 750 |

Groundspeed [m s^{−1}] | 54 |

Scan pattern and point data | |

Scan line spacing [m] | 0.92 |

In-line point spacing [m] | 0.69 |

Avg. stripe width [m] | 866 |

Total no. of point returns (all classes) | approx. 8.18 million |

Total no. of point returns (ground class) | approx. 1.06 million |

^{1}practically, the number of reflections per pulse is limited by the transfer rate of the data storage system.

**Table 2.**General characteristics of tested interpolation algorithms. Detailed explanation of terms can be found in [67].

Interpolation Algorithm | Category | Scale of Analysis | Smoothing | Shape |
---|---|---|---|---|

Inverse Distance Weighted (IDW) | Deterministic | Local | Exact | Convex |

Nearest Neighbour (Nen) | Deterministic | Local | Exact | Convex |

Natural Neighbour (NN) | Deterministic | Local | Exact | Convex |

Delauney Triangulation (DT) | Deterministic | Local | Exact | Convex |

Multilevel B-Spline (BS) | Deterministic | Global | Approximate | Non-convex |

Cubic Spline (CS) | Deterministic | Global | Approximate | Non-convex |

Thin-Plate Spline (TPS) | Deterministic | Global | Approximate | Non-convex |

Thin-Plate Spline by TIN (TPS_{TIN}) | Deterministic | Global | Approximate | Non-convex |

Ordinary Kriging (OK) | Geostatistical | Global | Approximate | Convex |

Interpolation Algorithm | Mean signed Error (m) | Mean Unsigned Error (m) | Std. Dev. of Signed Errors | Error Interval (m) | RMSE |
---|---|---|---|---|---|

IDW | −0.0042 | 0.1401 | 0.20 | 7.48 | 0.20 |

NeN | −0.0030 | 0.2022 | 0.28 | 10.77 | 0.28 |

NN | 0.0005 | 0.0835 | 0.12 | 5.88 | 0.11 |

DT | −0.0003 | 0.0840 | 0.12 | 5.88 | 0.12 |

BS | −0.0037 | 0.0825 | 0.11 | 6.32 | 0.11 |

CS | 0.0061 | 0.1149 | 0.16 | 6.52 | 0.16 |

TPS | −0.0024 | 0.0818 | 0.11 | 5.88 | 0.11 |

TPS_{TIN} | −0.0022 | 0.0818 | 0.11 | 5.90 | 0.11 |

OK | −0.0015 | 0.2011 | 0.28 | 8.09 | 0.28 |

Interpolation Algorithm | No. of Validation Points (% of Total No. Validation Points) | |||
---|---|---|---|---|

Very Low Errors ^{1} | Low Errors ^{2} | Significant Errors ^{3} | Extreme Errors ^{4} | |

IDW | 77.73 | 19.87 | 2.24 | 0.17 |

NeN | 62.20 | 30.93 | 6.32 | 0.54 |

NN | 93.78 | 6.05 | 0.15 | 0.02 |

DT | 93.59 | 6.22 | 0.17 | 0.02 |

BS | 94.04 | 5.82 | 0.13 | 0.02 |

CS | 84.80 | 14.42 | 0.73 | 0.05 |

TPS | 94.17 | 5.68 | 0.13 | 0.02 |

TPS_{TIN} | 94.17 | 5.68 | 0.13 | 0.02 |

OK | 63.45 | 29.15 | 6.78 | 0.62 |

^{1}between 0.0 and 0.20 m;

^{2}between 0.21 and 0.50 m;

^{3}between 0.51 and 1.00 m;

^{4}over 1.00 m.

Interpolation Algorithm | Correlation Coefficient (r) | ||
---|---|---|---|

Ground Slope | ALS Ground Point Density | Canopy Cover Density | |

IDW | 0.91 ** | −0.99 *** | 0.59 * |

NeN | 0.94 *** | −1.00 *** | 0.66 * |

NN | 0.83 ** | −0.93 ** | −0.62 * |

DT | 0.84 ** | −0.93 ** | −0.60 ** |

BS | 0.81 * | −0.93 ** | −0.63 * |

CS | 0.81 * | −0.88 ** | −0.86 * |

TPS | 0.81 * | −0.87 * | −0.63 * |

TPS_{TIN} | 0.81 * | −0.93 ** | −0.63 * |

OK | 0.92 *** | −0.98 *** | 0.95 ** |

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## Share and Cite

**MDPI and ACS Style**

Cățeanu, M.; Ciubotaru, A. Accuracy of Ground Surface Interpolation from Airborne Laser Scanning (ALS) Data in Dense Forest Cover. *ISPRS Int. J. Geo-Inf.* **2020**, *9*, 224.
https://doi.org/10.3390/ijgi9040224

**AMA Style**

Cățeanu M, Ciubotaru A. Accuracy of Ground Surface Interpolation from Airborne Laser Scanning (ALS) Data in Dense Forest Cover. *ISPRS International Journal of Geo-Information*. 2020; 9(4):224.
https://doi.org/10.3390/ijgi9040224

**Chicago/Turabian Style**

Cățeanu, Mihnea, and Arcadie Ciubotaru. 2020. "Accuracy of Ground Surface Interpolation from Airborne Laser Scanning (ALS) Data in Dense Forest Cover" *ISPRS International Journal of Geo-Information* 9, no. 4: 224.
https://doi.org/10.3390/ijgi9040224