# Extracting More Data from LiDAR in Forested Areas by Analyzing Waveform Shape

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## Abstract

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## 1. Introduction

#### 1.1. Potential Benefits of Identifying LiDAR Pulses for Forestry

#### 1.2. Waveform LiDAR

## 2. Method

#### 2.1. Data Collection

^{2}were obtained over the sample plots. Whilst the conventional Optech ALTM 3100EA LiDAR system collected discrete return information, the waveform digitizer simultaneously recorded the waveform of the same laser pulses. Raw GPS data and discrete LiDAR information were processed with Optech’s proprietary data-extraction software REALM into a Corrected Sensor Data (CSD) file. The waveform data were measured at 1 ns intervals and provided as five swathes in Optech’s NDF binary format with an IDX index file. The CSD file was subsequently read by the authors (using Matlab) to obtain discrete return information, as well as positioning information that could be used to georeference each waveform sample in the NDF file using an adapted version of the Matlab code in [21].

#### 2.2. Waveform Analysis

- The forest waveform consists of multiple peaks (returns)
- The peak intensities (in arbitrary units defined by the hardware) of the forest waveform are less than the open pasture.

- Georeference every 1 ns waveform sample to determine which waveforms (or part waveforms) fall into the field plots.
- Determine a ground surface for the field plots.
- Employ Gaussian fitting to describe each peak with a peak height and half-height width.
- Employ exponential curve-fitting to determine the exponential decay rate of each peak.
- Determine the above ground height of each peak (for comparison with field measured foliage data).

- was achieved using the methods and code in Parrish [21], adapted for the New Zealand NZGD2000 coordinate system. The field plots were located using a high-grade, differentially-corrected GPS unit, and the selection of waveform samples simply consisted of any that fell into the vertical space above the field plots.
- was achieved by using the discrete LiDAR data set for the area flown. The ground-filtering algorithm (GroundFilter.exe) in FUSION [24] was used to select only ground returns (based on the linear filtering method of Kraus and Pfeifer [25]). These ground returns were interpolated into a raster using FUSION’s GridSurfaceCreate.exe algorithm.
- As we are interested in decay rates of waveform peaks in (4), we do not need to search for additional ‘hidden’ peaks as other authors have. Peaks that are hidden through close proximity to another or low peak amplitude will not have sufficient data points after the peak to give a good decay. As such, we only select peaks in the waveform data with a corresponding discrete return. Gaussians were fit using Matlab’s fminsearch function, a simplex search method given in [26]. This is a direct search method that does not use numerical or analytic gradients. The number of Gaussians and a first-guess of their locations could be specified by the number and location of returns in the discrete datset. The peak height and half-height width of each Gaussian is recorded, as well as the R
^{2}value of the fit. - In (3), the location of each peak to be analyzed will have been determined. An exponential curve fit of the type y = Ce
^{−}^{λx}was applied from the peak maximum to the following local minima (either before the end of the waveform or before the next peak). The exponential curve was determined, also using Matlab’s fminsearch algorithm, taking the peak maximum as a start point for C, and 0.2 as a start point for λ. The final value for λ is recorded along with the R^{2}value of the fit. - As this study is a proof of concept, only peaks that corresponded to a return in the discrete data set were analyzed (see step 3). Ground height was obtained by linear interpolation of the ground surface generated in (2) to the discrete return’s x, y coordinates. Height above ground was the discrete return height, minus the interpolated ground height. We made the assumption that the waveform peak was coincident with the discrete return, which is appropriate, as the ALTM discrete return data and waveform data were generated from the same input signal. Using the coordinates from the well-calibrated discrete return data eliminated small potential errors from georeferencing the waveform, as the ranging calculation was calibrated on the convolved peak, and not the translated peak observed in the deconvolution. Waveform LiDAR is known to give many more returns than discrete LiDAR, so by only selecting returns with a corresponding discrete return we are limiting our results to only the strongest peaks available. This should demonstrate any correlations most clearly, which may be hidden when including additional peaks with limited data due to close proximity to another peak or very low amplitudes.

^{2}value of both the Gaussian fit and the exponential curve fit is calculated relative to their recorded values. Our variable of interest—the vegetation type causing each reflection—is not known for each return, but is known to be a function of height. In each plot the ground existed at 0 m by default (±1 m to account for the 1 m × 1 m resolution of the DTM and inaccuracies in the filtering process), broadleaved understorey and ferns existed in the first 4 m above ground (as measured in the field), and radiata pine foliage existed above 4 m (except in one plot where a small amount was found down to 2 m). Figure 4 gives two examples of waveforms that have had the Gaussian and exponential curve fits performed. Note that the red line is the Gaussian curve fit, not the original convolved waveform, hence the reason why there is no shift in the peak.

#### 2.3. Metric Analysis

^{2}value of less than 0.75 were excluded from the final results. If a distinguishing metric can be found for these peaks, then it can subsequently be tried on increasingly ambiguous peaks that are harder to extract from the background noise, to find the point at which the correlation is no longer tenable. Once all metrics were collected for all suitable peaks, they were each plotted against height in bivariate frequency distributions. These are effectively two-dimensional histograms, and show how the distribution of results for each metric varies with height. After checking for any height trends (which may be used to distinguish individual returns as foliage, understory or ground), the peaks in each 2 m height band were compared—as an average—against field sampled LAD.

## 3. Results and Discussion

^{2}values of either (or both) the Gaussian and exponential curve fits were less than 0.75 (Table 1). As this study is a proof of concept, only the cleanest, clearest peaks were used. If a metric of interest can be found to clearly relate peaks to their targets, then it can then be trialed on peaks with greater ambiguity.

#### 3.1. Gaussian Peak Height

#### 3.2. Gaussian Half-Height Width

#### 3.3. Pulse Decay Rate

#### 3.4. Comparison with Foliage Density

^{2}per m

^{3}because the foliage is so sparse that most LiDAR pulses will simply miss it. Figure 11(a–d) shows the average values for these LiDAR metrics in each height band vs. the LAD for the respective height bands, along with a line of best fit and an R

^{2}value for the fit.

^{3}, but varies substantially in localized regions (i.e., close to a branch vs. far from a branch). As the reflection from a LiDAR footprint (and any metrics derived from it) is defined over a volume of around ∼0.03 m

^{3}, it will be determined by the local value not the wider-scale average. Hence, individual values will vary, but if the sampling were fair, the average values of any metrics indicative of LAD should show strong correlations with the average value of LAD. However, sampling was not fair, as a return will only be registered if the pulse encounters sufficient material in one sample volume, thereby creating a tendency to overestimate LAD in more vegetated volumes. However, the fact that there is a moderate correlation between average decay rate and LAD is scientifically interesting. Note that as mentioned in Section 2.3, this correlation is only for the peaks with high intensities and good curve fits (R

^{2}values > 0.75). It is interesting to see what happens to this correlation when we allow more peaks into the analysis. If we drop the R

^{2}requirement for curve-fitting to >0.65, the correlation between the average decay rate and LAD drops to an R

^{2}of 0.17, whilst the number of peaks analyzed rises from 52,078 to 80,957. If we drop the requirement that the waveforms analyzed have an intensity >25 and instead take all waveforms with an intensity >15 and curve-fitting R

^{2}> 0.65, the R

^{2}for average decay rate to LAD drops to 0.16. This demonstrates how sensitive the metrics are to curve-fitting quality, and justifies the subset of peaks analyzed in this study. So, although the relationships are weak and only visible with an optimum carefully selected subset of the data, the fact that a relationship between LAD and decay rate can be observed even with heavy caveats is interesting and points towards a new way of thinking about LiDAR returns.

#### 3.5. Models for Interpreting Waveform Shape

- The standard interpretation is as the result of a series of discrete ‘hard’ returns, such as may be achieved by well-separated layers of foliage. These supposed layers are of a thickness equal to or less than the return distance travelled by the laser pulse in one sampling period (0.15 m for 1 ns sampling).
- As the result of transmission and reflectance from a volume of semi-transparent foliage which attenuates the radiation exponentially.

_{T}is the transmitted intensity, I

_{0}is the initial intensity, α is the absorption coefficient and x is the distance travelled through the gas. At a depth x into the canopy, a constant proportion (R) of the incident light will be reflected. We assume Lambertian reflectance, and that this is further attenuated by the gas as it leaves. I

_{R}at the surface is then given by

## 4. Conclusions

^{2}> 0.75) were used to remove additional variability in the metrics, justified, as this study is a proof of concept at this stage. Due to the complexity of the surfaces and multitude of angles, textures and paths etc., each waveform shape metric showed more potential variation within a surface type than it did between surface types. This negates the possibility of identifying the source of individual returns.

^{2}value of 0.37 was obtained. Although moderate, this correlation indicates that the spatially-averaged decay rate may be beneficial in estimating LAD, especially if it can be combined with other (independent) metrics in a multiple regression analysis. When less stringent criteria for selecting peaks were used, the strength of this correlation dropped.

## Acknowledgments

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**Figure 3.**Convolved and deconvolved forms of the waveforms in Figure 2 from pasture and from forest.

**Figure 4.**Deconvolved LiDAR data (blue) with multiple Gaussian fitting (red, 1 on the left, two on the right), and decay rates of each peak estimated (green).

**Figure 5.**Peak height of Gaussian curves fitted to deconvolved waveform LiDAR vs. corresponding discrete return intensities recorded by the discrete-return LiDAR unit.

**Figure 6.**Bivariate frequency distribution of peak height of Gaussian curves fitted to deconvolved waveform LiDAR vs. height above ground.

**Figure 7.**Bivariate frequency distribution of half-height width of Gaussian curves fitted to deconvolved waveform LiDAR vs. height above ground.

**Figure 8.**Bivariate frequency distribution of decay rate of exponential curves fitted to deconvolved waveform LiDAR vs. height above ground.

**Figure 9.**Scatter plot of half-height width vs. decay rate for curve fits on deconvolved waveform LiDAR.

**Figure 11.**Comparison of the average value in a set of 1,600 m

^{2}× 2 m sample plots for (

**a**) intensity of discrete LiDAR returns, (

**b**) peak height of deconvolved waveform LiDAR returns, (

**c**) half-height width of deconvolved waveform LiDAR and (

**d**) exponential decay rate of deconvolved waveform LiDAR returns, relative to the average field-measured leaf area density over the corresponding height band above ground.

Plot | Number of Waveforms with Peak Intensity > 25 | Number of Returns | Number of Returns with Good Quality Fits and Intensity > 25 | Percentage Used |
---|---|---|---|---|

1 | 12,558 | 11,616 | 4,570 | 28% |

2 | 16,150 | 14,939 | 5,608 | 27% |

3 | 12,305 | 11,261 | 4,942 | 33% |

4 | 11,332 | 10,286 | 4,207 | 29% |

5 | 12,326 | 11,075 | 4,541 | 31% |

6 | 13,810 | 13,130 | 5,488 | 32% |

7 | 12,687 | 11,992 | 4,598 | 27% |

8 | 14,849 | 14,210 | 4,847 | 25% |

9 | 15,862 | 15,145 | 8,746 | 45% |

10 | 14,312 | 13,513 | 4,531 | 24% |

Total | 136,191 | 127,167 | 52,078 | 30% |

## Share and Cite

**MDPI and ACS Style**

Adams, T.; Beets, P.; Parrish, C.
Extracting More Data from LiDAR in Forested Areas by Analyzing Waveform Shape. *Remote Sens.* **2012**, *4*, 682-702.
https://doi.org/10.3390/rs4030682

**AMA Style**

Adams T, Beets P, Parrish C.
Extracting More Data from LiDAR in Forested Areas by Analyzing Waveform Shape. *Remote Sensing*. 2012; 4(3):682-702.
https://doi.org/10.3390/rs4030682

**Chicago/Turabian Style**

Adams, Thomas, Peter Beets, and Christopher Parrish.
2012. "Extracting More Data from LiDAR in Forested Areas by Analyzing Waveform Shape" *Remote Sensing* 4, no. 3: 682-702.
https://doi.org/10.3390/rs4030682