Modelling lidar waveforms to solve for canopy properties

University College London

Modelling lidar waveforms to solvefor canopy properties

MSc in Remote Sensing

Department of Civil, Environmental

and Geomatic Engineering

Kim Calders

Supervisor: Prof. P. Lewis

August-2010

Acknowledgements

I would like to thank everybody who contributed to the completion of this disser-

tation. In the first place I would like to express my gratitude to prof. P. Lewis

for offering me the opportunity to do this project and the support and advice given

throughout the year.

I’d also like to thank Mat Disney for his tips and help on programming. Many

thanks to Chris Knell for helping me out with computer issues.

Tot slot wil ik ook nog mijn ouders bedanken voor hun onvoorwaardelijke en geweldige

steun het afgelopen jaar.

Kim,

London, August 2010

i

Abstract

Airborne LiDAR (light detection and ranging) can serve as a means to extract struc-

tural information from forests. The research objective of this dissertation was to

describe the LiDAR waveform of three different generic crown shapes (cuboid, con-

ical and prolate spheroid) by developing a new set of analytical solutions based on

the radiative transfer solution for single order scattering in the optical case.

The developed analytical formulae were tested against Monte Carlo ray tracing

simulations of different tree models by using a look up table (LUT) inversion ap-

proach. At first, simple tree models were used for the validation. When canopy

height was assumed to be known, results showed good agreement between the ana-

lytical solutions and the simulated waveforms. When canopy height was unknown,

a simple robust ratio technique was applied to generate the LUT. In this case LAI

values tended to be slightly overestimated. The analytical solutions were also com-

pared with simulated waveforms of more realistic Birch tree models but no satisfying

LAI predictions were obtained. This was mainly due to the fact that these analytical

formulae were developed for the specific case where leaf area density is a constant

and only leaf material is present in the canopy. Unlike the simple tree models, the

realistic Birch tree models fail both of these assumptions since gaps and branches

were present and therefore simple interpretation techniques proved not to work well

for more realistic trees.

This project showed the great potential of these analytical equations to extract

structural information from LiDAR waveforms. Further research should focus on

developing a more robust technique (e.g. curve fitting) to link the information of

LiDAR waveforms with the analytical solutions. An important question to be asked

is what such analytical solutions can really mean for inversion problems and what

practical use they can have. Earlier studies already indicated that a combination of

data from different sensors improved the assessment of structure in forests signifi-

ii

cantly. It is therefore suggested that these analytical equations can serve as a useful

tool to add a physical constraint to this inversion problem.

iii

Contents

Acknowledgements i

Abstract ii

Contents iv

List of Tables vii

List of Figures x

1 Introduction 1

2 Literature review 3

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Airborne laser scanning . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Principles of airborne LiDAR . . . . . . . . . . . . . . . . . . 4

2.2.2 Types of LiDAR system . . . . . . . . . . . . . . . . . . . . . 7

2.2.3 Sensor platforms . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.4 Alternative active remote sensing technologies in

vegetation studies . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 LiDAR waveform simulation via Monte Carlo ray tracing . . . . . . . 9

3 Data and Methodology 11

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Generation of synthetic tree models . . . . . . . . . . . . . . . . . . . 11

3.2.1 Simple tree shapes . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.2 Realistic tree models . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 LiDAR simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Analysis and interpretation of the LiDAR waveform . . . . . . . . . . 18

iv

CONTENTS

3.4.1 Solution to the radiative transfer . . . . . . . . . . . . . . . . 18

3.4.2 Analytical solutions to the inversion problem . . . . . . . . . . 21

3.5 Validation of the analytical solutions using simulated LiDAR wave-

forms of simple tree models . . . . . . . . . . . . . . . . . . . . . . . 30

3.5.1 Tree height is exactly known . . . . . . . . . . . . . . . . . . . 31

3.5.2 Tree height is not exactly known: robust method . . . . . . . 32

3.6 Applying the analytical solution to simulated LiDAR waveforms of

realistic Birch trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Results 34







5 Discussion 53







5.3 Further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6 Conclusion 59

Appendix A:

Optimizing the parameters for LiDAR simulations 66

Appendix B:

Validation of LiDAR waveforms: cuboid canopy 72

Appendix C:

Validation of LiDAR waveforms: conical canopy 76

Appendix D:

v

CONTENTS

Validation of LiDAR waveforms: spheroid canopy 79

Appendix E:

Applying the analytical solution to simulated LiDAR waveforms of

Birch trees 83

Appendix F:

Overview of data folder 85

vi

List of Tables

3.1 Parameter ranges of the synthetic tree data set . . . . . . . . . . . . 13

3.2 LAI values for realistic birch trees . . . . . . . . . . . . . . . . . . . . 15

3.3 Summary of LiDAR camera parameters used for MCRT simulations . 19

4.1 Resulting RMSE values for all LAI combinations between the simu-

lated and the analytical waveforms derived from comparing normalised

canopy height of the 75% fraction of the normalised cumulative I func-

tions for a cuboid canopy: a . . . . . . . . . . . . . . . . . . . . . . . 38



canopy height of four different fractions of the normalised cumulative

I functions for a conical canopy: a . . . . . . . . . . . . . . . . . . . . 42




I functions for a conical canopy: b . . . . . . . . . . . . . . . . . . . . 42




tions for a spheroid canopy: a . . . . . . . . . . . . . . . . . . . . . . 46

4.5 Percentage of true canopy height and backscatter detected after thresh-

olding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.6 Normalised canopy height of the 50%, 75% and their ratio fraction of

normalised I for both the analytical and the MCRT simulated LiDAR

waveform for a spheroid canopy . . . . . . . . . . . . . . . . . . . . . 47

vii

LIST OF TABLES



canopy height of the ratio fraction of the normalised cumulative I func-

tions for a spheroid canopy with unknown canopy height . . . . . . . 48

4.8 Resulting RMSE values derived from comparing Birch MCRT wave-

form simulations with the analytical solutions for a spheroid canopy

(simple robust method) . . . . . . . . . . . . . . . . . . . . . . . . . . 51


form simulations with the analytical solutions for a cuboid canopy

(simple robust method) . . . . . . . . . . . . . . . . . . . . . . . . . . 51


form simulations with the analytical solutions for a spheroid canopy

(based on 4 positions of relative canopy height) . . . . . . . . . . . . 52

1 Normalised canopy height of the 75% fraction of normalised I for a

cuboid canopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2 Resulting RMSE values for all LAI combinations between the simu-



tions for a cuboid canopy: b . . . . . . . . . . . . . . . . . . . . . . . 74




tions for a cuboid canopy: c . . . . . . . . . . . . . . . . . . . . . . . 75

4 Normalised canopy height of selected fractions of normalised I for a

conical canopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77




I functions for a conical canopy: c . . . . . . . . . . . . . . . . . . . . 78

6 Normalised canopy height of the 75% fraction of normalised I for a

spheroid canopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

viii

LIST OF TABLES




tions for a spheroid canopy: b . . . . . . . . . . . . . . . . . . . . . . 81




tions for a spheroid canopy: c . . . . . . . . . . . . . . . . . . . . . . 82

9 Normalised canopy height of the 50%, 75% and ratio fraction of nor-

malised I for MCRT simulations on the Birch trees . . . . . . . . . . 83

10 Normalised canopy height of the 50%, 75% and ratio fraction of nor-

malised I for spheroid and cuboid analytical solutions . . . . . . . . . 83

11 Normalised canopy height of the 20%, 50%, 75% and 80% fraction of

normalised I for MCRT simulations on the Birch trees . . . . . . . . . 84

12 Normalised canopy height of the 20%, 50%, 75% and 80% fraction of

normalised I for spheroid analytical solution . . . . . . . . . . . . . . 84

ix

List of Figures

2.1 LiDAR data collection . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Ranging techniques: pulsed and phase differencing . . . . . . . . . . . 6

2.3 Time-of-flight LiDAR ranging . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Types of LiDAR systems . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 Examples of different canopy shapes used in tree generation . . . . . 13

3.2 Standard reflectance of material used in the generated tree models . . 14

3.3 First order scattering domination in MCRT LiDAR simulations . . . 18

3.4 Overview of plane parallel medium geometry . . . . . . . . . . . . . . 19

3.5 Cone cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.6 Vertical cross section of spheroid canopy (upper hemisphere) . . . . . 26

3.7 Spheroid cross section (lower hemisphere) . . . . . . . . . . . . . . . . 29

3.8 G-functions for different leaf angle distributions. (from Lewis (2010c)) 31

4.1 MCRT LiDAR simulations for a tree with a cuboid canopy: a . . . . 35

4.2 Analytical LiDAR waveforms for a tree with a cuboid canopy . . . . . 36

4.3 Comparison of the normalised cumulative I functions for a tree with

a cuboid canopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.4 MCRT LiDAR simulations for a tree with a conical canopy . . . . . . 39

4.5 Analytical LiDAR waveforms for a tree with a conical canopy . . . . 40


a conical canopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.7 MCRT LiDAR simulations for a tree with a spheroid canopy . . . . . 43

4.8 Analytical LiDAR waveforms for a tree with a spheroid canopy . . . . 44


a spheroid canopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.10 Overview of the normalised cumulative I functions for a tree with a

spheroid canopy. MCRT LiDAR waveform after thresholding . . . . . 47

x

LIST OF FIGURES

4.11 Overview of MCRT Lidar simulations on realistic Birch trees . . . . . 49

4.12 Normalised cumulative I functions for MCRT simulated Birch tree data 50

4.13 Normalised cumulative I functions comparison between MCRT simu-

lated Birch tree data and spheroid canopy analytical waveforms . . . 50

4.14 Normalised cumulative I functions comparison between MCRT simu-

lated Birch tree data and cuboid canopy analytical waveforms . . . . 51

1 LiDAR simulations for different footprint sizes . . . . . . . . . . . . . 66

2 LiDAR simulations with different bin size: a . . . . . . . . . . . . . . 67

3 LiDAR simulations with different bin size: b . . . . . . . . . . . . . . 68

4 LiDAR simulations with different rays per pixel (RPP) . . . . . . . . 69

5 Zoom of LiDAR simulations with different rays per pixel (RPP) . . . 70

6 The effect of leaf size on the MCRT LiDAR simulation . . . . . . . . 71

7 MCRT LiDAR simulations for a tree with a cuboid canopy: b . . . . 72

8 MCRT LiDAR simulations for a tree with a cuboid canopy: c . . . . 73

9 Analytical normalised LiDAR waveforms for a tree with a cuboid canopy 74

10 MCRT LiDAR simulations for a tree with a conical canopy: b . . . . 76

11 MCRT LiDAR simulations for a tree with a conical canopy: c . . . . 77

12 Analytical normalised LiDAR waveforms for a tree with a conical canopy 78

13 MCRT LiDAR simulations for a tree with a spheroid canopy: b . . . 79

14 MCRT LiDAR simulations for a tree with a spheroid canopy: c . . . 80

15 Analytical normalised LiDAR waveforms for a tree with a spheroid

canopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

xi

Nomenclature

ρl: Leaf reflectance

ρs: Soil reflectance

Γ(Ω0→ Ωs): Area scattering phase function

κe: Optical extinction coefficient

Ω0: Direction of incident LiDAR pulse

Ωs: Direction of scattering

Al: Leaf area [m2]

a: Semi-minor radius of prolate spheroid canopy [m]

AGCMs: Atmospheric general circulation models

BRDF: Bidirectional reflectance distribution function

C: Canopy cover

c: Semi-major radius of prolate spheroid canopy [m]

H: Tree height [m]

h: Canopy height [m]

I(Ωs, z): Received backscatter by the sensor

I: Intensity (radiance or brightness) [W m−2 sr−1]

IFOV: Instantaneous field of view

InSAR: Interferometric synthetic aperture radar

xii

Nomenclature

l: Length of cuboid canopy [m]

LAI: Leaf area index

LiDAR: Light detection and ranging

LUT: Look up table

LVIS: Laser vegetation imaging sensor

MCRT: Monte Carlo ray tracing

Nv: Leaf number density [ leaves / m2]

NA: Not Available

P(Ω0→ Ωs): Volume scattering phase function

r’: An intermediate position along the radius axis

R: Base radius of conical canopy [m]

RMSE: Root mean squared error

RPP: Rays per pixel

SLICER: Scanning LiDAR imager of canopies by echo recovery

TOC: Top of canopy

ul: Leaf area density function [m2 / m3]

VCL: Vegetation canopy lidar

w: Width of cuboid canopy [m]

z’: Effective canopy penetration

z: Penetration depth with top of the canopy as reference (TOC = 0)

xiii

Chapter 1

Introduction

Forests play an important role in today’s society, fulfilling production, recreational

and ecological functions. To keep these functions in balance, accurate and precise

information about the structure of the forest and its biophysical parameters is es-

sential. Data retrieved via remote sensing methods, particularly via satellite and

airborne acquisition methods, can provide a useful tool to obtain this information

due to their synoptic view over large or inaccessible areas.

According to Sellers et al. (1997), vegetation makes a large contribution towards

the control of the heat and mass fluxes in the terrestrial biosphere. Previous studies

have suggested that this terrestrial biosphere functions as a large sink for atmospheric

carbon dixode (CO2) and that the carbon cycle is connected to the physical climate

system. Therefore, in times where the debate about climate change and global warm-

ing is a hot topic, more accurate and precise assessment of these biophysical variables

within forests are important for the land surface parametrization of the current global

atmospheric general circulation models (AGCMs). These AGCMs have evolved from

simple unrealistic schemes in the late 1960s and 1970s (the first generation mod-

els) into realistic soil-vegetation-atmosphere transfer systems (the third generation

models). The latest models apply modern theories such as photosynthesis and plant

water relations and result in a systematic assessment of evaporation, energy exchange

and carbon exchange by vegetation (Sellers et al., 1997). Leaf area index (LAI ) is

an important structural parameter in forest ecosystems, since several biological and

physical processes are related to the total leaf surface (Wulder et al., 1998). For

example, photosynthesis, respiration, transpiration, carbon and nutrient cycle, and

rainfall interception are closely related to forest structure and LAI.

1

1. Introduction

Most optical, passive, sensors measure the integrated response for some set of

viewing and illumination angles, also called the bidirectional reflectance distribution

function (BRDF). Airborne LiDAR can measure something approximating the nadir

retro-reflectance as a time or distance resolved signal over forest canopies and can

serve as an excellent tool to asses forest structure and the three-dimensional distri-

bution of plant canopies. Although a variety of studies focused on the possibilities

LiDAR (light detection and ranging) offers in structure assessment (Drake et al.,

2002; Dubayah et al., 2000; Harding et al., 2001; Lefsky et al., 1999, 2002; Nelson,

1997; Ni-Meister et al., 2001) little work has been conducted to reconcile measure-

ment of optical sensors and airborne LiDAR to provide an optical description of

the vegetation canopy. The hypothesis that the synergistic effect of both types of

measurement will lead to an improved retrieval of the canopy structure has been

supported by the results obtained by Koetz et al. (2007).

The research objective of this dissertation is to describe the LiDAR waveform of

three different generic crown shapes (cuboid, conical and spheroid) by developing a

new set of analytical solutions based on the solution of the radiative transfer equation

for single order scattering in the optical case. These analytical equations can be used

to analyse and interpret LiDAR waveforms and can therefore serve as a means to

generate a look up table (LUT) to extract crown structure and LAI from these LiDAR

waveforms. This research was carried out in four steps:

(i) Deriving the analytical solutions for generic tree shapes that describe the

canopy structure based on the radiative transfer equation

(ii) Simulating LiDAR measurements over different trees by using Monte Carlo ray

tracing (MCRT)

(iii) Validating the analytical solutions using the simulated LiDAR measurements

(iv) Applying the analytical solutions to realistic tree models

In chapter 2, a literature review of the airborne LiDAR data acquisition technique

and the simulation models used in this project is given as well as results obtained

in other studies. The data and methodology used in this research are described in

chapter 3. The relevant results can be found in chapter 4 and are discussed in chapter

5. Finally a conclusion of the research and its results is given in chapter 6.

2

Chapter 2

Literature review

2.1 Introduction

In the previous chapter it was mentioned that a variety of studies explored the

possibilites of LiDAR in canopy structure assessment. Several studies (Dubayah

et al., 2000; Nelson, 1997) focused on extracting tree height from LiDAR waveforms

by differencing the distance to the ground and that to the first detectable return

(or a threshold above that return). Lefsky et al. (1999) derived canopy cover from

waveform data by interpreting the ratio of the ground return with the total signal

power. A correction needed to be applied for differences in ground and canopy

reflectance at the used wavelength (1064 nm).

In this study analytical solutions based on the radiative transfer solution for first

orer scattering in the optical case will be developed for different crown shapes. These

will serve as a means to describe canopy LiDAR waveforms and will be validated with

simulated LiDAR data generated by a Monte Carlo ray tracing model. A similar

approach was used in earlier studies. Ni-Meister et al. (2001) used a hybrid geometric

optical and radiative transfer (GORT) model to interpret the LiDAR waveforms with

respect to canopy structure and validated their findings using SLICER data. Gap

probabilty was identified as the most important link between canopy structure and

modelling LiDAR waveforms.

Koetz et al. (2007) used information derived from LiDAR waveforms to constrain

the outcome of a LUT inversion. The LUT was generated with both spectral and

LiDAR data. In the first step of the LUT inversion, a selection of possible solutions

3

2. Literature review

was made based on the LiDAR data. In the second step the spectral information

was used to come to the final solution. Results proved that the introduction of

LiDAR data as a physical constraint significantly improved the retrieval performance

compared to inversion based only on spectral information.

2.2 Airborne laser scanning

Airborne LiDAR is generally used to collect elevation heights (i.e. the distance be-

tween the sensor and the illuminated spot on the ground) and therefore this data ac-

quisition method can be seen as an alternative to photogrammetric mapping, InSAR

(interferometric synthetic aperture radar) or in situ data collection (Jensen, 2007).

2.2.1 Principles of airborne LiDAR

Like many spaceborne multispectral scanning systems (e.g. LANDSAT MSS and

Thematic Mapper MP), LiDAR systems use opto-mechanical scanning assemblies.

However, because laser scanners are active remote sensing systems, two optical beams

(i.e. the emitted pulse towards the surface and the proportion of that beams that

returns to the sensor) need to be taken into account. According to Wehr and Lohr

(1999), a typical LiDAR system can be divided into three key units: a control and

processing unit, an opto-mechanical scanner and a laser ranging unit. Pulse emission

and reception of the returned signal take place in the ranging unit. The laser beam

divergence is related to the instantaneous field of view (IFOV) which is related to

the instantaneous laser footprint (i.e. the illuminated area on the ground surface).

A condition that must be fulfilled when designing a LiDAR system is that the IFOV

of the receiving instrument must not be smaller than the IFOV of the transmitted

beam.

In the first instance, no real image is created but only a point cloud which is

obtained in the local coordinate system of the sensor (Pfeifer and Bohm, 2008; Wehr

and Lohr, 1999). The basic principle used in laser altimetry is to measure, by some

means, the distance between the LiDAR instrument. This distance between the

LiDAR sensor and the object is also called the range. In general, ranging can be done

either by pulsed ranging or by phase differencing. Pulsed LiDAR systems can emit

pulses at rates (also called pulse repetition frequency) higher than 100,000 pulses per

4


Figure 2.1: LiDAR data collection: A pulse of laser light is transmitted from theLiDAR instrument by the use of a rotating mirror. A part of the energy is scat-tered back to the sensor and recorded in the ranging unit of the LiDAR system.The on-board global positioning system (GPS) and inertial measurement unit (IMU)determine the exact location of the aircraft as well as the roll, pitch and yaw at themoment the laser pulse is emitted and received. (from Jensen (2007))

second (Jensen, 2007). Scanning systems that apply the phase differencing method

continuously emit light and are therefore often referred to as continuous wave (CW)

scanners. In order to derive the range, the emitted signal can be modulated with

a trigonometric function (Wehr and Lohr, 1999). Both principles were illustrated

in figure 2.2 where the continuous signal in the phase differencing technique was

modulated with a sinus function.

Most operating LiDAR systems use the pulsed ranging method. Also in this

project a pulsed LiDAR was used and therefore that technique will be discussed

further below (continuous wave scanners use different physical principles to derive

the range). A different name for for the ranging technique used by pulsed LiDAR

sensors is time-of-flight ranging because the technique calculates the range accurately

by measuring the travel time from the transmitter to the object on the ground and

5


Figure 2.2: Ranging techniques: pulsed (left) and phase differencing (right). AT

is the transmitted pulse amplitude and AR the received pulse amplitude; tL is thetravelling time. (modified from Wehr and Lohr (1999))

back to the receiver (Jensen, 2007; Wehr and Lohr, 1999). The principle is illustrated

in figure 2.3 .

Figure 2.3: Time-of-flight LiDAR ranging: AT is the transmitted pulse amplitudeand AR the received pulse amplitude; tL is the travelling time. (from Wehr and Lohr(1999))

According to Jensen (2007), Baltsavias (1999) and Wehr and Lohr (1999) the

6


round trip time of travel, tL, for a pulse of light is:

tL = 2R

c(2.1)

where R is the range and c the speed of light (approximately 3 x 108 m s−1). Based on

this equation the distance between the LiDAR sensor and the object can be derived:

R =1

2ctL (2.2)

and therefore the range resolution ∆R is:

∆R =1

2c∆tL (2.3)

which is directly proportional to ∆ tL.

The returned LiDAR signal does not only give information about the range but

also contains intensity information, which indicates the strength of the return. In

general, objects with high reflectance values (i.e. bright objects) will have a higher

intensity return than darker objects with lower reflectance values. Hence, in forestry

studies the radiometric properties of the different materials present in the forest

environment (leaf, trunk, soil, branch, ...) will influence the returned intensity.

Another aspect to keep in mind is the triggering mechanism of the LiDAR instru-

ment. The first return of the LiDAR waveform, which in vegetation studies generally

indicates the top of the canopy, will always tend to be smaller than the true canopy

height. This is because the return signal can not be instantaneous or uniform across

the footprint of the instrument. For commercial confidentiality motives it is gener-

ally impossible to obtain the specific information on how the implemented triggering

mechanism on a certain sensor exactly works. Disney et al. (2010) provides a detailed

overview of the impact of the signal triggering mechanism on estimates of the canopy

height derived from LiDAR simulations.

2.2.2 Types of LiDAR system

Two generic types of LiDAR systems are used these days: the first/last return LiDAR

and the full-waveform LiDAR. The former can use the first pulse to produce a surface

model (DSM) and the last return to produce a digital terrain model (DTM). The

7


latter also provides extra information about the structure of features between the first

and last return. This, for example, can be useful in vegetation studies and therefore

this type of LiDAR was used in this project. A third, intermediate, type of LiDAR

system is the discrete multiple return LiDAR, through which not only the first and

last return are recorded but also the intermediate peaks within the returned LiDAR

signal. An overview of these LiDAR types is given in figure 2.4

Figure 2.4: An overview of the types of LiDAR systems available: first/last re-turn, discrete multiple return and full-waveform LiDAR (from Penn State University(2010); source ASPRS)

2.2.3 Sensor platforms

There are currently no spaceborne LiDAR systems which are optimised to obtain

information about vegetation canopies. The vegetation canopy lidar (VCL) was

specifically designed as such an instrument but has never made it into space (Lewis,

2010a).

SLICER (scanning LiDAR imager of canopies by echo recovery) was developed

in the 1990s as an airborne LiDAR system with a 10 m footprint (Means et al.,

1999). Another important airborne system is LVIS (laser vegetation imaging sensor)

which records the full waveform for a 25 m footprint (Blair et al., 1999). Unlike most

8


passive sensors (e.g. Landsat TM), SLICER and LVIS both showed great capability

to retrieve structural information from forest stands (Dubayah et al., 2000).

2.2.4 Alternative active remote sensing technologies in

vegetation studies

Like LiDAR, also InSAR has the potential to provide accurate information relat-

ing to the forest canopy structure. Andersen et al. (2003) compared canopy models

generated from X-band InSAR data with airborne LiDAR data over conifer forests.

Results have shown that both methods are a valuable tool for measuring the canopy

dimensions. LiDAR showed the advantage of assessing more accurately the complex

morphology of the canopy due to its more dense data whereas InSAR offers a more

generalised description of the canopy structure. However, InSAR has some advan-

tages over LiDAR since it is better suited for extensive areas and does not require

clear weather conditions although it has to be remarked that the high humidity (e.g.

in regions with sub-tropical monsoon climate) proved to be very challenging when

using InSAR (Muller et al., 2008). InSAR also showed insensitivity to differences

in forest biomass above 150 Mg/ha, which is significantly smaller than the values in

many tropical and temperate forests (Means et al., 1999; Muller et al., 2008).

2.3 LiDAR waveform simulation via Monte Carlo

ray tracing

LiDAR waveforms will be simulated in this project by applying a Monte Carlo ray

tracing model. Ray tracing involves firing photons into the canopy and quantifying

its behaviour at each intersection with a scene element. At each intersection diffuse

ray paths are generated as well which will contribute to multiple scattering within the

canopy scene. A diffuse path can be terminated either by escaping out of the scene

or after a certain threshold of scattering interactions is reached. The radiometric

properties (i.e. transmission, absorption and reflection) of the scene elements are

used to modulate the signal and to determine the attenuation along its trajectory

(Disney et al., 2000; Saich et al., 2003). Each single ray samples only one of the

infinite possible photon trajectories and to reach a convergent solution, Monte Carlo

sampling is used. By firing more photons into the canopy, the number of samples (n)

9


will increase and a steady solution will be achieved at a rate of n−1/2 (Disney et al.,

2010).

Compared to models based on geometric optics or radiative transfer, MCRT re-

quires more computing resources. It has the advantage, however, that once a solution

for radiative transfer is found, simulations of canopy reflectance can be obtained at

any view angle (Disney et al., 2000).

10

Chapter 3

Data and Methodology

3.1 Introduction

Analysis was carried out in different phases. In the first step, different tree models

were created and virtual single tree scenes were constructed. The next step involved

simulating airborne LiDAR signals over these scenes using a Monte Carlo ray tracing

model. The main objective was to interpret the resulting LiDAR waveforms by

comparing them with the analytical radiative transfer solutions for different canopy

shapes. To get a full understanding of these waveforms, analysis was first carried out

on trees with simple generic shapes. In the last phase of the project, more realistic

representations of Birch trees were considered.

3.2 Generation of synthetic tree models

3.2.1 Simple tree shapes

Simple generic trees with different characteristics were created by adjusting the input

parameters of the code used to generate the tree objects. The code was written in C

and was provided by Disney (2010). The output of this phase were single tree object

scenes which were used as input for the LiDAR simulations.

11

3. Data and Methodology

Canopy structure and characteristics of the biophysical variables

Three canopy shapes were considered: cuboids, cones and prolate spheroids. The

prolate spheroid trees required the semi-major radius and semi-minor radius to be

specified and its shape is obtained by rotating an ellipse around its semi-major axis.

The spheroid is defined by equation 3.1 where a is the semi-minor radius and c the

semi-major radius (Weisstein, 2010).

1 =x2 + y2

a2+z2

c2(3.1)

The shape of the cone was determined by the height and base-radius of the canopy.

Cuboid trees were specified by height, width and length. In this case width and length

were equal so the cross section for cuboid trees was a square. It has to be mentioned

that cylindrical trees could be interpreted the same way as cuboid trees since their

only difference is the shape of the cross section (i.e. circular for cylindrical trees).

The trunk underneath the canopy was constructed by a single cylinder, parametrized

by its height and radius: the trunk radius was fixed at 0.2 m for all the trees and the

trunk height was set to a quarter of the canopy height.

Leaves were the only material present in the canopy (no branches etc.) and were

shaped as small disks with a radius of 0.01 m. The leaf angle distribution was chosen

to be uniform (spherical) throughout the canopy. Because overlapping leaves were

not preferred, the algorithm for generating the trees also checked on the presence of

overlapping leaves and made adjustment in case any were found. The final parameter

used to parameterise the canopy is the LAI. This parameter should be interpreted

as the one-sided area of leaf surfaces per unit ground surface area (Jensen, 2007).

One knows that the leaf area density function ul is dependant on the position in the

canopy. When Nv is the leaf number density (i.e. the numbers of leaves per unit

volume) and Al is the leaf area, the leaf area density function is:

ul(z) = Nv(z)Al (3.2)

In this project Al was a constant because all the leaves had the same dimensions

and the leaf angle distribution was uniform. The leaves were spread equally over

the canopy and therefore Nv was a constant as well. Based on equation 3.2, ul was

therefore constant throughout the canopy and its units were [m2 / m3]. In the cuboid

12


canopy case the leaf area density is constant for each horizontal layer and therefore

ul(z) = LAI/h where h is the canopy height.

Figure 3.1: Examples of different canopy shapes used in tree generation: left figurerepresents a cuboid canopy, conical canopy is displayed in the middle figure and aspheroid canopy in the right figure

An overview of the the parameters used to generate the synthetic tree data set

can be found in table 3.1. All trees were generated with varying LAI values; the

values used were 1, 2, 3, 4, 5, 6, 8, 10, 15 and 20 m2/m2.

Shape Dimensions [m]Cuboid Height 4m; Width and Length 1m

Height 6m; Width and Length 2mHeight 8m; Width and Length 14

Cone Height 4m; Base radius 1mHeight 6m; Base radius 2mHeight 8m; Base radius 4m

Prolate spheroid Semi-major 2m; Semi-minor 1mSemi-major 3m; Semi-minor 2mSemi-major 4m; Semi-minor 4m

Table 3.1: Parameter ranges of the synthetic tree data set

Radiometric properties

Three main building blocks could be considered in the final tree models: leaves, trunk

and soil. Each of these elements consisted of their own typical material which had

13


a specific spectral reflectance function. The functions used for these trees can be

found in figure 3.2 and were similar to the reflectance properties used in previous

studies (Disney et al., 2006, 2010). The bark reflectance used in the trunk element

was a laboratory-measured spectrum of bark material, measured in the European

Goniometric Observatory (EGO) facility at the Joint Research Centre (JRC) in Is-

pra (Italy) and was part of the LOPEX experiment (Hosgood et al., 1995). The soil

was considered to be a simple Lambertian soil surface and the its reflectance prop-

erties were constructed from the soil basis functions of Price (1990) which proved

to rebuild 99.6% of observed variance of the measured spectra (Disney et al., 2006).

The radiometric properties of the leaves were based on field measurements on Birch

leaves using integrating sphere (Analytical Spectral Devices Inc., Boulder, CO).

Figure 3.2: Standard reflectance of material used in the generated tree models

3.2.2 Realistic tree models

The Birch (Betula pubescens) tree models which were utilised in Disney et al. (2010)

were used as realistic tree models in this project. They were derived from the

14


OnyxTREE c© software and were parameterised with field data obtained in a Birch

forest site in Sweden (Disney et al., 2009). Radiometric properties were the same as

for the simple tree models.

LAI information about the Birch trees was available for the different elements of

the tree (see table 3.2). A weighted summation (based on the radiometric properties)

of these values will give the true LAI value of each Birch tree whereas a simple

summation of these individual values gives a rough approximation of the true LAI.

Leaf Trunk Bough Branch1 Branch2 Branch3 Total (approximated LAI)Birch 1 3.58 0.42 0.07 0.05 0.07 0.06 4.19Birch 2 1.38 0.19 0.05 0.03 0.04 0.04 1.69Birch 3 2.74 0.29 0.08 0.06 0.07 0.07 3.23Birch 4 3.55 0.54 0.13 0.05 0.06 0.06 4.34Birch 5 4.26 0.38 0.62 0.21 0.11 0.08 5.58Birch 6 2.94 0.38 0.23 0.06 0.06 0.05 3.66Birch 7 1.23 0.18 0.06 0.02 0.03 0.03 1.51

Table 3.2: LAI values for realistic birch trees

3.3 LiDAR simulations

LiDAR measurements were simulated by using the Monte Carlo ray tracing (MCRT)

software that was developed and made available by Lewis (2010b). This librat model

was based on the ararat MCRT model (Lewis, 1999). This code has been tested

extensively in previous studies, as well as against numerous other models (Pinty

et al., 2004; Widlowski et al., 2007) as observations (Disney et al., 2006, 2010, 2009).

To improve the speed of the Monte Carlo ray tracing LiDAR simulations, bound-

ing boxes were implemented within the tree object files. Bounding boxes allow for

efficient intersection determination in the ray tracing and minimise the testing nec-

essary to isolate scattering elements and are therefore one of the most important

efficiency algorithms. If a photon is fired into the canopy and the ray does not in-

tersect with a bounding box at a certain hierarchy, it is not necessary to test for

interactions with scattering elements at lower levels of hierarchy from that bound-

ing box (Disney et al., 2000). The number of hierarchy of the bounding boxes was

increased until the average number of elements within a bounding box was smaller

than 16.

15


Optimizing the parameters for LiDAR simulations

A default LiDAR camera and illumination source were created and parameters were

adjusted where necessary for each simulation. The scene reflectance was simulated as

observed from nadir and camera and light properties geometries were identical and

therefore one was looking in a ’hot spot’ scenario during these simulations.

For all the simulations, the boomlength (i.e. the flying height) was set to 5500 m.

This relatively high flying altitude was preferred because this would result in a very

small field of view for the sensor. A small field of view was preferred because then

the rays of the emitted LiDAR beam would be almost parallel when the canopy was

reached and this would make the interpretation of the LiDAR signal more objective.

Another advantage of the small field of view was that the there was only one solid

ground return.

A large footprint LiDAR was designed and simulations with different footprint

sizes were simulated. The effect of different footprint sizes was explored for a prolate

spheroid tree with LAI 6 and a canopy height of 6 m and semi-minor of 2 m. The

resulting waveforms were visualised figure 1 (see Appendix). It was observed that a

smaller footprint gave higher reflectance values for the returned signal than a larger

footprint. Because the scene had only one tree this can easily be explained by the

fact that the relative proportion of tree in the footprint increased when the footprint

size decreased. Finally the area viewed on the ground was selected to be a square

with dimensions 20 m x 20 m (i.e. the spatial resolution). Therefore the vertical

extent of the tree was always completely within the viewed area.

The LiDAR simulations were initially explored for two different bin size parame-

ters: 0.05 m and 0.1 m (i.e. the vertical resolution). There will be double the amount

of bins (i.e. the number of samples) in the resulting LiDAR waveform of the simula-

tion with bin size 0.05 m than there will be when bin size 0.1 m is used. Additionally,

the amount of energy returned within a bin of size 0.05 m is smaller then for bin size

0.1 m because returned energy has been sampled over a shorter length through the

canopy. This has been illustrated for a prolate spheroid tree with LAI 1 and 6 and

a canopy height of 6 m and semi-minor of 2 m in figure 2 and 3 (see Appendix). In

the end a bin size of 0.1 m was chosen since this was analogous with the bin size of

LVIS.

Sampling characteristics were defined for the camera: in this case 1048576 x 100

16


primary rays will be used in a square pattern of 1024 by 1024 pixels. The number

of primary rays describes the number of photons fired at one pixel. The returned

signal is then averaged over the amount of photons fired. In general this means that

the higher the number of primary rays per pixel (also called RPP), the smoother the

LiDAR waveform. This is illustrated in figure 4 and 5 (see Appendix). Using only 1

primary ray per pixel made the resulting waveform relatively noisy. This noise could

be reduced by increasing the rays per pixel, which will increase the signal to noise

ratio and the solution will converge towards a more stable solution. However, the

cost of using more RPPs was an increase in computing speed: e.g. using 32 rays

per pixel would increase the speed of computing by a factor 32. In the end it was

decided to use a value of 100 for the rays per pixel as this provided the best results

considering the trade-off between noise reduction and increase in computing time. It

needs to be remarked that even at 300 RPP the LiDAR waveform was not completely

smooth due to the chosen leaf size of the generated trees which was not completely

infinitesimal (disk radius of 0.01 m) and therefore the tree was not completely a

turbid medium (see Appendix, figure 6). Generating trees with smaller disk radius

required more computing time and considering the trade off between computing time

and smaller leaves, 0.01 m was the best compromise within the time frame of the

project.

The diffuse illumination field (’black’ sky) was switched off and only first order

scattering (i.e. only one interaction with soil or canopy elements) was taken into

account. This was a good approximation of reality because the LiDAR signal was

dominated by first order scattering as can be seen from figure 3.3. The wavelength

used for the LiDAR simulation was 1064 nm. This wavelength is situated in the

near-infrared (NIR) band and is frequently used in vegetation monitoring since veg-

etation reflectance is much higher in the NIR than in the visual spectrum. A better

reflectance signal will therefore be received for the NIR band because the signal-to-

noise ratio (S/N) is higher than for the visual bands (Sabins, 1997).

17


Figure 3.3: LiDAR simulations are first order scattering dominated: illustrated for aspheroid canopy with LAI 8 and canopy height 4 m and semi minor axis 1 m

3.4 Analysis and interpretation of the LiDAR wave-

form

3.4.1 Solution to the radiative transfer

The solution for first order scattering in the optical case is expected to reconstruct

the expected LiDAR waveform theoretically. If Ωs is the direction of scattering and

Ω0 the direction of the incident LiDAR pulse then the received backscattered energy

by the sensor I(Ωs, z) is:

I(Ωs, z) =e

−κe(Ωs)(z − (−H))

µs ρsoil(Ωs,Ω0)e

−κe(Ω0)(−H)

µ0 I0δ(Ωs − Ω0)+

I0

µs

∫ z′=z

z′=−He

−κe(Ωs)(z − z′)µs e

κe(Ω0)z′

µ0 P (Ω0 → Ωs) dz′

(3.3)

18


Parameter ValueFlying height 5500 mFootprint size 20 m x 20 mBin spacing 0.1 mRPP 100Wavelength 1064 nm

Table 3.3: Summary of LiDAR camera parameters used for MCRT simulations

Figure 3.4: Overview of plane parallel medium geometry: z = 0 is the top of thecanopy and H is the tree height. (from Lewis (2010c))

The purpose of these formula was to measure the backscattered energy of the

canopy with a sensor which was located above the canopy and therefore the value of

z was set to zero (i.e. the top of the canopy: see figure 3.4). Also the volume scattering

phase function was defined as P(Ω0→ Ωs) =ulµs

Γ(Ω0→ Ωs) where Γ(Ω0→ Ωs) is the

area scattering phase function (Lewis, 2010c).

I(Ωs, 0) =e

−κe(Ωs)H

µs e

−κe(Ω0)(−H)

−µ0 ρsoil(Ωs,Ω0)I0δ(Ωs − Ω0)+

I0ulΓ(Ω0 → Ωs)

µsµ0

∫ z′=0

z′=−He

−κe(Ωs)(−z′)µs e

κe(Ω0)z′

µ0 dz′

(3.4)

19


Because the sensor emits and receives LiDAR pulses at nadir, µs and µ0 were equal

to cos(0) = 1 according to see figure 3.4. In the specific case of a LiDAR sensor, Ωs

and Ω0 were the same (i.e. Ωs = -Ω0) and therefore this direction will now be called

Ω and hence equation 3.4 becomes:

I(Ω, 0) =e−2Hκe(Ω)ρsoil(Ω)I0δ(Ω) + I0ulΓ(Ω→ −Ω)

∫ z′=0

z′=−He2z′κe(Ω) dz′ (3.5)

Working out the integral leads to:

I(Ω, 0) =e−2Hκe(Ω)ρsoil(Ω)I0δ(Ω) +I0ulΓ(Ω→ −Ω)

2κe(Ω)

[1− e−2Hκe(Ω)

](3.6)

According to Lewis (2010c), the optical extinction coefficient κe can be replaced by

ulG(Ω):

I(Ω, 0) =e−2G(Ω)ulHρsoil(Ω)I0δ(Ω) +I0Γ(Ω→ −Ω)

2G(Ω)

[1− e−2G(Ω)ulH

](3.7)

If a photon is able to follow a specific path down through the canopy it should also be

able to leave the canopy via the same path. This is the case for LiDAR measurements

because the sensor operates in the hotspot and therefore a correction factor needs to

applied for the joint gap probability term:

I(Ω, 0) =e−G(Ω)ulHρsoil(Ω)I0δ(Ω) +I0Γ(Ω→ −Ω)

2G(Ω)

[1− e−G(Ω)ulH

](3.8)

One is not only interested in the final returned energy, but also in the intermediate

interactions at each level of z and therefore equation 3.5 was integrated the same way

but with different borders [0,-z], where z is any value between 0 and -H. The result

is analogue to equation 3.8:

I(Ω, 0) =e−G(Ω)ulHρsoil(Ω)I0δ(Ω) +I0Γ(Ω→ −Ω)

2G(Ω)

[1− e−G(Ω)ulz

](3.9)

The equation derived above assumed that the whole illuminated pixel was covered

by the canopy. This is often not the case in reality and also the simulations in this

project do not fulfil this condition since only one tree at a time was viewed. Therefore

the returned backscatter above needs to be adjusted by a factor which contributes

20


for canopy cover. Hence equation 3.10 becomes:

I(Ω, 0) = C

[e−G(Ω)ulHρsoil(Ω)I0δ(Ω) +

I0Γ(Ω→ −Ω)

2G(Ω)

[1− e−G(Ω)ulz

] ]+ (1− C)ρsoil

(3.10)

where C is the canopy cover, which can range from one to zero. The focus in this

project was on describing the canopy shape and for ease of reference, only the canopy

contribution of equation 3.10 will be considered in the derivation of the analytical

approach. The reflectance contribution of the soil can easily be added again after-

wards.

I(Ω, 0, canopy, z) = CI0Γ(Ω→ −Ω)

2G(Ω)

[1− e−G(Ω)ulz

](3.11)

It needs to be remarked that equations derived in this section are for the general

(cuboid) case where the value z (the depth with respect to the top of the canopy)

is equal to the value z’ (the effective penetration depth within the canopy). This

is not the case for irregular crown shapes (e.g. cone and spheroid) and this will be

discussed later in the appropriate section.

3.4.2 Analytical solutions to the inversion problem

An analytical solution to the inversion problem is preferred over a numerical ap-

proach, because such a solution will be more practical for the end user. For example

it can be easily used to generate a look up table for a specific case study by adjusting

the parameters in the analytical equation.

The underlying concept is to treat the canopy shapes as a summation of homoge-

neous annuli with equal effective penetration depth z’ centred around the z axis (i.e.

elevation axis) . Unlike a numerical approach, the bin step and annulus width will be

infinitesimal. Therefore the analytical solution can be seen as an integral, whereas

the summation in a numerical solution can be interpreted as a Riemann sum.

Trees with cuboid canopy

Because the top of the canopy was a horizontal plane, the value of z’ was constant

throughout the extent of any horizontal layer. In other words, the value of z was

equal to the effective penetration depth z’ into the canopy at any position in a

21


horizontal layer. No adjustments needed to be made since the general equation 3.11

was derived for a horizontally infinite homogeneous canopy and it could therefore be

used to describe the behaviour of photons passing through the canopy of a cuboid

tree.

To reconstruct the LiDAR waveform, the contribution of each individual infinites-

imal horizontal canopy layer is needed. This can be achieved by taking the derivative

of equation 3.11 with respect to z:

δI(Ω, 0, canopy, z)

δz= C

I0Γ(Ω→ −Ω)

2G(Ω)(−G(Ω)ul)

(−e−G(Ω)ulz

)= C

I0Γ(Ω→ −Ω)ul2

e−G(Ω)ulz

(3.12)

A normalised expression of this equation was preferred for further analysis and there-

fore the equation was normalised by dividing it by the total returned canopy LiDAR

reflectance. This value was derived from equation 3.11 with z = h where h is the

canopy height. Therefore the normalised expression of equation 3.12 becomes:


δz=

CI0Γ(Ω→ −Ω)ul

2e−G(Ω)ulz

CI0Γ(Ω→ −Ω)

2G(Ω)[1− e−G(Ω)ulh]

=G(Ω)ule

−G(Ω)ulz

[1− e−G(Ω)ulh]

(3.13)

This solution can also be written as a function of LAI instead of ul. One knows that

ul =AcoverLAI

Vtot(3.14)

where Acover is the area of the tree cover projection and Vtot the total tree volume.

Hence for a cuboid trees ul =LAI

hand therefore the normalised contribution of each

individual infinitesimal horizontal canopy layer is:


δz=G(Ω)LAIe−

G(Ω)LAIzh

h [1− e−G(Ω)LAI ](3.15)

Finally, the expression for the normalised cumulative I function was derived. This is

the integral of equation 3.15 with respect to z for the interval [0,z]. The normalised

22


cumulative I function is dependent on z and will range from zero to one:

∫δI(Ω, 0, canopy, z) =

∫ z

0

G(Ω)LAIe−G(Ω)LAIz

h

h [1− e−G(Ω)LAI ]

I(Ω, 0, canopy, z) =1− e

−G(Ω)LAIzh

1− e−G(Ω)LAI

(3.16)

Trees with conical canopy

The difference with the cuboid trees discussed in the paragraph above is that within

one horizontal layer, different parts of the canopy will be subject to different within

canopy path lenghts z’. Therefore the solution for the cuboid case (see equation

Figure 3.5: Vertical cross section of conical canopy: overview of the symbology used

3.12) needs to be adjusted for shapes which have different z’ values within a single

horizontal layer (this is the case for conical and spheroid trees).


δz= C

I0Γ(Ω→ −Ω)ul2

e−G(Ω)ulz′

(3.17)

Each horizontal layer can be seen as the sum of infinitesimal annuli with constant

z’. The reflectance of each annulus can be seen as the product of equation 3.17 with

its area. Hence the total reflectance of the horizontal a horizontal layer is the integral

(i.e. the summation) over all the annuli. It is known that for a layer at depth z, r

=Rz

hand therefore the integral needs to be calculated for the r’ interval

[0,Rz

h

],

23


where r’ can be any intermediate position in the interval. Normalising was done by

dividing by πR2.


δz= C

∫ r′=Rzh

r′=0

I0Γ(Ω→ −Ω)ul

(e−G(Ω)ulz

′)

2πr′dr′

2πR2(3.18)

As mentioned before z’ has a different value for each annulus. From figure 3.5 it

can be seen that z’ is a function of z and r’: z’ = z-hr′

Rand therefore equation 3.18

can be written as:


δz= C

∫ r′=Rzh

r′=0

I0Γ(Ω→ −Ω)ul

(e−G(Ω)ul(z−hr′

R))r′dr′

R2

= CI0Γ(Ω→ −Ω)ul

(e−G(Ω)ulz

)R2

∫ r′=Rzh

r′=0

(e

G(Ω)ulhr′

R

)r′dr′

(3.19)

The integral was solved by using the integration by parts method: if u = f(x) and

g = f(x) then∫

udv = uv -∫vdu. When this method was applied to solve equation

3.19 this became u = r’, du = dr’, dv = eG(Ω)ulhr

′

R dr’ and v = RG(Ω)ulh

(e

G(Ω)ulhr′

R

).

HenceδI(Ω, 0, canopy, z)

δzbecomes:


(e−G(Ω)ulz

)R2

[Rr′e

G(Ω)ulhr′

R

G(Ω)ulh− R2e

G(Ω)ulhr′

R

G(Ω)2u2l h

2

]r′=Rzh

r′=0

= CI0Γ(Ω→ −Ω)

hRG(Ω)

(e−G(Ω)ulz

)[r′e

G(Ω)ulhr′

R − ReG(Ω)ulhr

′

R

G(Ω)ulh

]r′=Rzh

r′=0

= CI0Γ(Ω→ −Ω)

hRG(Ω)

(e−G(Ω)ulz

)(RzheG(Ω)ulz − ReG(Ω)ulz

G(Ω)ulh+

R

G(Ω)ulh

)

= CI0Γ(Ω→ −Ω)

h2G(Ω)

(z − 1

G(Ω)ul+e−G(Ω)ulz

G(Ω)ul

)(3.20)

To normalise this equation it was divided by its total returned canopy LiDAR

reflectance. This value can be derived from integrating equations 3.20 with z = h.

24


Therefore the normalised expression of 3.20 becomes:


δz=

CI0Γ(Ω→ −Ω)

h2G(Ω)

(z − 1


G(Ω)ul

)

CI0Γ(Ω→ −Ω)

h2G(Ω)

(h2

2− h

G(Ω)ul+

1− e−G(Ω)ulh

G(Ω)2u2l

)

=

z − 1


G(Ω)ulh2

2− h

G(Ω)ul+

1− e−G(Ω)ulh

G(Ω)2u2l

(3.21)

In the final step ul will be replace by leaf area index. It is known that for a conical

tree Acover=πR2 and Vtot =πR2h

3. Based on equation 3.14, ul is therefore equal to

3LAI

hand 3.22 becomes:


δz=

z − h

3G(Ω)LAI+h(e−3G(Ω)LAIz

h

)3G(Ω)LAI

h2

2− h2

3G(Ω)LAI+h2(

1− e−3G(Ω)LAI)

9G(Ω)2LAI2

=

2

(z

h−

(1− e

−3G(Ω)LAIzh

)3G(Ω)LAI

)

h

(1− 2

3G(Ω)LAI+

2(

1− e−3G(Ω)LAI)

9G(Ω)2LAI2

)

(3.22)

Trees with spheroid canopy

Based on figure 3.6, the equation of the ellipse (i.e. the vertical cross section of the

prolate spheroid) can be written as:

1 =r′2

a2+z′′2

c2(3.23)

25


and therefore z” = c

√1− r′2

a2and r’ = a

√1− z′′2

c2where (r’, z”) are the locations

of the points that shape the ellipse. Because z” and z do not use the same reference

system and because the value of z was used an input for the analysis the former

equations needs to rewritten to r’ = a

√1− (c− z)2

c2and will serve as the upper

boundary of the integral over r’.

From the same figure it can be seen that for the upper hemisphere of the canopy

z’ can be expressed as z”- (c-z) = c

√1− r′2

a2- (c-z). It needs to be remarked again

that all z and r’ values are distances and will therefore be positive values. Equation

Figure 3.6: Spheroid vertical cross section (upper hemisphere): overview of the sym-bology used

3.18 can be used as a starting point to come to the analytical solution since spheroid

26


and cones both have a circular horizontal cross section (for the spheroid case R = a):


δz= C

∫ r′=a

√1− (c−z)2

c2

r′=0

2πI0Γ(Ω→ −Ω)ul

(e−G(Ω)ul(c

√1− r′2

a2 −(c−z)))r′dr′

2πa2


(eG(Ω)ul(c−z)

)a2

∫ r′=a

√1− (c−z)2

c2

r′=0

e−G(Ω)ulc√

1− r′2a2 r′dr′

(3.24)

In the first step to solve the integral in equation 3.24 substitution will be used with

t2 = (1-r′2

a2) and dt =

−r′dr′

a2

√1− r′2

a2

=−r′dr′

a2t. Therefore r’dr’= -a2tdt and the integral

can be rewritten as:

−a2

∫e−G(Ω)ulcttdt (3.25)

This can now be solved by using the integration by parts method with u = t, du =

dt, dv = e−G(Ω)ulctdt and v =−e−G(Ω)ulct

G(Ω)ulc. Hence the integral becomes:

− a2(−te−G(Ω)ulct

G(Ω)ulc− e−G(Ω)ulct

G(Ω)2u2l c

2

)= a2

(te−G(Ω)ulct

G(Ω)ulc+

e−G(Ω)ulct

G(Ω)2u2l c

2

)=a2e−G(Ω)ulct

G(Ω)ulc

(t+

1

G(Ω)ulc

) (3.26)

From the substitution step it is known that t =√

1− r′2

a2 and therefore this solution

can be rewritten as:

a2e−G(Ω)ulc√

1− r′2a2

G(Ω)ulc

(√1− r′2

a2+

1

G(Ω)ulc

)(3.27)

The final solution of the integral can be plugged back into equation 3.24 and the

definite integral be calculated which gives the final solution of δI/δz for the upper

27


hemisphere of the spheroid crown.


δz

= CI0Γ(Ω→ −Ω)ule

G(Ω)ul(c−z)

a2

[a2e−G(Ω)ulc

√1− r′2

a2

G(Ω)ulc

(√1− r′2

a2+

1

G(Ω)ulc

)]r′=a√1− (c−z)2

c2

r′=0

= CI0Γ(Ω→ −Ω)eG(Ω)ul(c−z)

G(Ω)c

(e−G(Ω)ul(c−z)

(c− zc

+1

G(Ω)ulc

)− e−G(Ω)ulc

(1 +

1

G(Ω)ulc

))

= CI0Γ(Ω→ −Ω)

G(Ω)c

(c− zc

+1

G(Ω)ulc− e−G(Ω)ulz

(1 +

1

G(Ω)ulc

))(3.28)

From figure 3.7 it can be seen that for the lower hemisphere of the canopy z’

can be rewritten as c

√1− r′2

a2+ (z-c). This can be rewritten as c

√1− r′2

a2- (c-z)

and therefore the expression for z’ is the same for the upper and lower hemisphere.

However, the limit for the upper border of the integration interval for r’ is different.

As mentioned earlier, z” uses a different coordinate system then z and hence for the

lower hemisphere r’ = a

√1− (z − c)2

c2.

Equation 3.24 can now be solved analogue to the upper hemisphere to give δI/δz

for the lower hemisphere of the spheroid canopy:


δz

= CI0Γ(Ω→ −Ω)ule

G(Ω)ul(c−z)

a2

[a2e−G(Ω)ulc

√1− r′2

a2

G(Ω)ulc

(√1− r′2

a2+

1

G(Ω)ulc

)]r′=a√1− (z−c)2

c2

r′=0

= CI0Γ(Ω→ −Ω)eG(Ω)ul(c−z)

G(Ω)c

(e−G(Ω)ul(z−c)

(z − cc

+1

G(Ω)ulc

)− e−G(Ω)ulc

(1 +

1

G(Ω)ulc

))

= CI0Γ(Ω→ −Ω)

G(Ω)c

(e−2G(Ω)ul(z−c)

(z − cc

+1

G(Ω)ulc

)− e−G(Ω)ulz

(1 +

1

G(Ω)ulc

))(3.29)

Equation 3.28 will be used when 0 ≤ z ≤ c and equation 3.29 is applied when

c ≤ z ≤ 2c. It can be seen that, as required, both equations give the same value

for δI/δz for z = c. Based on both equations the total returned LiDAR reflectance

28


Figure 3.7: Vertical cross section of spheroid canopy (lower hemisphere): overviewof the symbology used

of both canopy hemispheres can be calculated. This value can then be used to

normalise equation 3.28 and 3.29. The contribution of the upper hemisphere to the

total reflectance was obtained by integrating equation 3.28 for the z interval [0; c]

and the lower hemisphere contribution was derived from the integral of equation 3.29

for the z interval [c; 2c]. The leaf area density function ul will also be replaced by

leaf area index. It is known that for a spheroid tree Acover=πa2 and Vtot =4πa2c

3.

Based on equation 3.14, ul is therefore equal to3LAI

4c. Therefore the solution for

the integral of the upper hemisphere is

C4I0Γ(Ω→ −Ω)

G(Ω)

[1

8+

1

3G(Ω)LAI

(1+(e−3G(Ω)LAI

4 +1)(

1+4

3G(Ω)LAI

))](3.30)

and for the lower hemisphere

C4I0Γ(Ω→ −Ω)

G(Ω)

[−3

16+e−3G(Ω)LAI

2

3G(Ω)LAI

(−1

2− 1

3G(Ω)LAI

)+e−3G(Ω)LAI

4

3G(Ω)LAI

(1+

4

3G(Ω)LAI

)](3.31)

29


The sum of equation 3.30 and 3.31 is the the total returned LiDAR reflectance of

the whole canopy of a spheroid crown and is equal to:

C4I0Γ(Ω→ −Ω)

G(Ω)

[−1

16+

1

3G(Ω)LAI

(1 +

(2e−3G(Ω)LAI

4 + 1)(

1 +4

3G(Ω)LAI

)− e

−3G(Ω)LAI2

(1

2+

1

3G(Ω)LAI

))](3.32)

If ul is replaced in equation 3.28 and 3.29, the normalised equations for δI/δz for

a spheroid tree are:

c− zc

+4

3G(Ω)LAI− e

−3G(Ω)LAIz4c

(1 +

4

3G(Ω)LAI

)4c

[−1

16+

1

3G(Ω)LAI

(1 +

(2e−3G(Ω)LAI

4 + 1)(

1 +4

3G(Ω)LAI

)− e

−3G(Ω)LAI2

(1

2+

1

3G(Ω)LAI

))](3.33)

for the upper hemisphere and

e−6G(Ω)LAI(z−c)

4c

(z − cc

+4

3G(Ω)LAI

)− e

−3G(Ω)LAIz4c

(1 +

4

3G(Ω)LAI

)4c

[−1

16+

1

3G(Ω)LAI

(1 +

(2e−3G(Ω)LAI

4 + 1)(

1 +4

3G(Ω)LAI

)− e

−3G(Ω)LAI2

(1

2+

1

3G(Ω)LAI

))](3.34)

for the lower hemisphere.

3.5 Validation of the analytical solutions using sim-

ulated LiDAR waveforms of simple tree mod-

els

The analytical solutions were validated by comparing them against the MCRT simu-

lated LiDAR waveforms. The trees which were used for these simulations had specific

characteristcs and therefore the general analytical solutions could be specified more

30


precisely. The spherical leaf angle distribution was used in the generated trees and

from figure 3.8 it could be seen that G(Ω) is equal to 0.5 (Lewis, 2010c).

Figure 3.8: G-functions (leaf projection functions) for different leaf angle distribu-tions. (from Lewis (2010c))

3.5.1 Tree height is exactly known

For as well simulated as analytical waveforms, the returned energy was quantified.

The plot of the normalised cumulative LiDAR backscatter against the normalised

canopy height was a useful tool to determine the fraction of normalised cumulative

LiDAR backscatter at which the possibility to distinguish between the different LAI

scenarios was most favourable. The normalised canopy height position was defined

byz

hand ranged from from 0 (top) to 1 (bottom). The exact normalised canopy

height for a specific fraction was obtained via linear interpolation between the known

31


values. To achieve good accuracy and precision it was important to keep the sampling

interval minimal.

These fractions of normalised cumulative LiDAR backscatter and corresponding

normalised canopy height derived from the analytical waveforms were then used to

generate a look up table. The LUT inversion technique is widely used since it is

very simple and efficient. A LUT inversion generally consists of two parts. First the

LUT is generated (in this case based on information from the analytical solutions) by

sampling the complete parameter space and then the solution of the LUT is chosen.

In other words, one determines which LUT entry is closed related to the input entry

(in this case information from the simulated waveform). The root mean squared

error (RMSE) was used to determine the LUT solution. RMSE is a frequently used

measure for the goodness of fit. RMSE has the advantage of having the same units

of the variable being estimated. According to Dennison and Roberts (2003), RMSE

is calculated as:

RMSE =

√(∑ni=1(Yi − Yi)2

n

)(3.35)

Finally, the RMSE values for all LAI combinations between simulated and an-

alytical data were put together in a single cross table per tree shape to give an

indication of the agreement between the LAI predicted via the analytical solution

and the true LAI value of the tree. All validation analysis was carried out in the

statistical computing environment R (R Development Core Team, 2009).

3.5.2 Tree height is not exactly known: robust method

Knowing the exact tree height is not always possible and therefore a simple more

robust method was developed as well. The simulated MCRT LiDAR waveforms were

taken and δI/δz was normalised. All values smaller than 0.001 were left out and this

subset of the data was normalised again for both cumulative backscatter and canopy

height. For small LAI values there was a small cut off at the top of the canopy, for

larger LAI values there was a significant loss of data at the bottom of the canopy.

Hence the normalised cumulative I functions were shifted away from the original

functions and thus also from the reference analytical waveforms which were used in

the look up table. Therefore a link needed to be established between the original and

the thresholded I functions.

32


It is assumed that by thresholding the original data (and therefore altering the

true canopy height), the loss of received backscatter by the sensor is minimal, al-

though the loss in canopy height can be significant. Hence re-normalising the cu-

mulative I function will be dominated by a linear transformation of the normalised

canopy height. This means that the ratio of the normalised canopy height of two

different fractions of normalised cumulative backscatter will neutralise this linear

transformation.

3.6 Applying the analytical solution to simulated

LiDAR waveforms of realistic Birch trees

In this analysis, the normalised I functions of the Birch LiDAR waveforms will be

compared with the analytical solution developed for the generic shapes. The simple

robust method (ratio normalised canopy height of 50% and 75% fraction of normalised

cumulative I) was applied. Additionally, normalised canopy height was determined

for fractions of cumulative backscatter spread across the whole function interval and

RMSE measures were calculated based on those values. Because the tree height of

the Birch trees was exactly known, no ratio was used in this approach.

33

Chapter 4

Results



els


Trees with cuboid canopy

Based on equation 3.12, the LiDAR signal was generated via the analytical solution.

Since the term CI0Γ(Ω→ −Ω)

2was the same constant for all LAI scenes it will have

no influence on the shape of the waveform since it was only a scaling factor and will

therefore not be of any importance in the validation methodology applied here. For

a cuboid canopy with canopy height 8 m and width and length 4 m, the resulting

LiDAR signal can be found in figure 4.1 for the Monte Carlo ray tracing LiDAR

simulation and in figure 4.2 for the analytical build-up approach. LiDAR waveforms

for other canopy extents showed similar results (see Appendix figure 7 and figure 8).

Analogue to equation 3.15, these LiDAR waveforms were normalised by dividing

each δI by the total sum of δI’s (see figure 9 in the Appendix for an illustration of the

analytical solution normalised LiDAR waveform). This was then used to derive the

normalised cumulative I functions analogous to equation 3.16. Since the analytical

normalised cumulative I functions were the same for each combination of different

34

4. Results

Figure 4.1: MCRT LiDAR simulations for a tree with a cuboid canopy and a canopyheight of 8 m and width and length of 4 m

35

4. Results

Figure 4.2: Analytical LiDAR waveforms for a tree with a cuboid canopy and acanopy height of 8 m and width and length of 4 m

36

4. Results

Figure 4.3: Comparison of the normalised cumulative I functions for a tree with acuboid canopy

37

4. Results

canopy height, width and length, they only needed to be constructed once. This was

done by using a tree of height 8 m with a sampling interval of 0.05 m and hence

the total number of samples was 161. The normalised cumulative I functions for the

simulated LiDAR waveform were obtained by adding the individual bin reflectance

in function of height and dividing by the total sum. Comparison of the normalised

cumulative I functions derived from the analytical and the MCRT method can be

found in figure 4.3.

Based on the normalised I functions in figure 4.3, the 75% fraction of normalised

backscatter received by the sensor was chosen to distinguish well between the different

LAI scenes. Normalised canopy height position for both the analytical solution and

the MCRT simulation LiDAR waveforms can be found in the Appendix table 1.

This information was used to calculate the root mean squared error for all LAI

combinations between the simulated and the analytical waveforms. It needs to be

remarked that RMSE was calculated with only one pair of numbers. The resulting

RMSE table for for the cuboid canopy case with a canopy height of 8 m and width

and length of 4 m can be found in table 4.1. The other two cuboid tree extents

showed similar results and their RMSE table can be found in the Appendix (see

table 2 and table 3).

Analytical solutionMCRT simulation LAI 1 LAI 2 LAI 3 LAI 4 LAI 5 LAI 6 LAI 8 LAI 10 LAI 15 LAI 20LAI 1 0.004 0.054 0.115 0.175 0.232 0.283 0.366 0.426 0.515 0.561LAI 2 0.063 0.005 0.055 0.116 0.173 0.224 0.307 0.367 0.456 0.502LAI 3 0.121 0.063 0.002 0.058 0.115 0.166 0.249 0.309 0.398 0.444LAI 4 0.181 0.124 0.063 0.002 0.054 0.106 0.189 0.249 0.337 0.383LAI 5 0.233 0.176 0.115 0.055 0.002 0.053 0.136 0.197 0.285 0.331LAI 6 0.281 0.224 0.163 0.102 0.045 0.006 0.089 0.149 0.238 0.284LAI 8 0.365 0.308 0.247 0.186 0.130 0.078 0.005 0.065 0.153 0.199LAI 10 0.424 0.366 0.305 0.245 0.188 0.137 0.054 0.006 0.095 0.141LAI 15 0.511 0.453 0.392 0.332 0.275 0.224 0.141 0.081 0.008 0.054LAI 20 0.558 0.500 0.439 0.379 0.322 0.271 0.188 0.128 0.039 0.007

Table 4.1: Resulting RMSE values for all LAI combinations between the simulatedand the analytical waveforms derived from comparing normalised canopy height ofthe 75% fraction of the normalised cumulative I functions. Cuboid canopy case witha canopy height of 8 m and width and length of 4 m

38

4. Results

Trees with conical canopy

The LiDAR signal was generated analytically by applying equation 3.20. Since the

term CI0Γ(Ω→ −Ω)

h2G(Ω)remained constant for all the LAI scenes for a specific canopy

extent, it had no influence on the shape of the waveform for the same reason men-

tioned in the paragraph above. For a conical canopy with canopy height 6 m and

base radius 2 m the LiDAR waveform can be found in figure 4.4 for the MCRT

simulation and in figure 4.5 for the analytical method. MCRT simulations for other

conical canopy extents showed similar results (see Appendix figure 10 and figure 11).

It must be noted that there was no MCRT LiDAR waveform data available for LAI

10, 15 and 20 for trees with a canopy height of 8 m and 4 m base radius. They could

not be generated within the time frame of the project due to the heavy computing

load.

Figure 4.4: MCRT LiDAR simulations for a tree with a conical canopy and a canopyheight of 6 m and base radius of 2 m

39

4. Results

Figure 4.5: Analytical LiDAR waveforms for a tree with a conical canopy and acanopy height of 6 m and base radius of 2 m

40

4. Results

The normalised LiDAR waveform was calculated analogue to equation 3.22 by

dividing each δI by the total sum of δI’s (figure 12 in the Appendix illustrates the

normalised LiDAR waveform for the analytical approach). Based on this normalised

waveforms, the normalised cumulative I functions were obtained. A comparison of

the normalised cumulative I functions can be found in figure 4.6. These normalised

I functions did not reveal any region with a significant range to distinguish between

LAI. Most favourable regions were in the lower fractions and therefore normalised

canopy height position for four fractions (10%, 20%, 30% and 40%) was determined

(see Appendix table 4) These positions were used to calculate the RMSE table which

holds information about all possible LAI combinations between MCRT and analytical

LiDAR waveforms. Results showed good agreement for conical canopies with canopy

height 8 m (see table 4.2), but LAI tended to be overestimated when canopy height

was 4 m (see table 4.3). Intermediate results were obtained for a canopy height 6 m

(see Appendix table 5).

Figure 4.6: Comparison of the normalised cumulative I functions for a tree with aconical canopy

41

4. Results

Analytical solutionMCRT simulation LAI 1 LAI 2 LAI 3 LAI 4 LAI 5 LAI 6 LAI 8 LAI 10 LAI 15 LAI 20LAI 1 0.004 0.015 0.029 0.039 0.048 0.054 0.063 0.069 0.079 0.084LAI 2 0.020 0.002 0.013 0.024 0.032 0.038 0.047 0.054 0.063 0.068LAI 3 0.034 0.015 0.001 0.010 0.018 0.024 0.034 0.040 0.049 0.054LAI 4 0.042 0.023 0.009 0.002 0.010 0.016 0.025 0.031 0.041 0.046LAI 5 0.050 0.031 0.017 0.006 0.002 0.008 0.017 0.024 0.033 0.038LAI 6 0.056 0.037 0.023 0.012 0.004 0.002 0.011 0.018 0.027 0.032LAI 8 0.064 0.045 0.031 0.021 0.012 0.006 0.003 0.010 0.019 0.024LAI 10 NA NA NA NA NA NA NA NA NA NALAI 15 NA NA NA NA NA NA NA NA NA NALAI 20 NA NA NA NA NA NA NA NA NA NA

Table 4.2: Resulting RMSE values for all LAI combinations between the simulatedand the analytical waveforms derived from comparing normalised canopy height offour different fractions of the normalised cumulative I functions. Conical canopy casewith a canopy height of 8 m and base radius 4 m


Table 4.3: Resulting RMSE values for all LAI combinations between the simulatedand the analytical waveforms derived from comparing normalised canopy height offour different fractions of the normalised cumulative I functions. Conical canopy casewith a canopy height of 4 m and base radius 1 m

42

4. Results


The analytical solution to the first order scattering in a spheroid canopy was used to

construct the LiDAR waveforms. One used equation 3.28 for the upper hemisphere

and equation 3.29 for the lower hemisphere of the canopy. CI0Γ(Ω→ −Ω)

G(Ω)cwas only

a scaling factor and had no influence in the validation methodology. For a spheroid

canopy with canopy height 6 m and semi minor axis 2 m the analytical waveform can

be found in figure 4.8 and the MCRT LiDAR simulation can be found in figure 4.7.

MCRT simulations for other canopy extents can be found in the Appendix (see figure

13 and figure 14). Four LAI versions (LAI 5, 10, 15, 20) of the spheroid canopy with

canopy height 8 m and 4 m had no MCRT LiDAR data available due to computing

issues (LAI 5) or the heavy computing load for generating the trees. Analogue to

Figure 4.7: MCRT LiDAR simulations for a tree with a spheroid canopy and a canopyheight of 6 m and semi minor axis of 2 m

equation 3.33 and equation 3.34, these waveforms were normalised by dividing each

43

4. Results

Figure 4.8: Analytical LiDAR waveforms for a tree with a spheroid canopy and acanopy height of 8 m and semi minor axis of 4 m

44

4. Results

δI by the total sum of δI’s (see Appendix figure 15 for an illustration of the normalised

analytical waveform). The same approach was used for the MCRT simulations and

based on that information the normalised cumulative I functions could be constructed

which were compared in figure 4.9. Based on these functions and and for the same

reasoning discussed in the cuboid canopy section,the 75% fraction of normalised I was

selected as region which distinguished well between between the different LAI scenes.

Normalised canopy height positions for this fraction can be found in Appendix table 6

and based on this data RMSE could be calculated. Results showed perfect agreement

for spheroid canopies with canopy height 4 m and 6 m (see Appendix table 7 and

table 8). Good results were achieved for canopy height 8 m as well, however one

outlier was present (see table 4.4).

Figure 4.9: Comparison of the normalised cumulative I functions for a tree with aspheroid canopy

45

4. Results

Analytical solutionMCRT simulation LAI 1 LAI 2 LAI 3 LAI 4 LAI 5 LAI 6 LAI 8 LAI 10 LAI 15 LAI 20LAI 1 0.004 0.041 0.081 0.116 0.146 0.171 0.209 0.237 0.280 0.303LAI 2 0.013 0.032 0.072 0.107 0.137 0.162 0.200 0.228 0.271 0.294LAI 3 0.086 0.041 0.001 0.034 0.064 0.089 0.127 0.155 0.198 0.221LAI 4 0.120 0.075 0.035 0.000 0.030 0.055 0.093 0.121 0.164 0.187LAI 5 NA NA NA NA NA NA NA NA NA NALAI 6 0.174 0.129 0.089 0.054 0.024 0.001 0.039 0.067 0.110 0.133LAI 8 0.212 0.167 0.127 0.092 0.062 0.037 0.001 0.029 0.072 0.095LAI 10 NA NA NA NA NA NA NA NA NA NALAI 15 NA NA NA NA NA NA NA NA NA NALAI 20 NA NA NA NA NA NA NA NA NA NA

Table 4.4: Resulting RMSE values for all LAI combinations between the simulatedand the analytical waveforms derived from comparing normalised canopy height ofthe 75% fraction of the normalised cumulative I functions. Spheroid canopy casewith a canopy height of 8 m and semi minor axis 4 m



A simple but more robust method was tested on trees with a spheroid canopy and

unknown canopy height. Since previous analysis suggested that canopy extent for a

specific generic canopy shape had no influence on the MCRT simulation, only trees

with canopy height 6 m and semi minor axis 2 m were considered. After threshold-

ing δI/δz values smaller than 0.001, the normalised I functions were generated again.

Figure 4.10 compares the re-normalised I functions with the original analytical ones

which serve as the basis for this look up table inversion. Table 4.5 shows the per-

centage of true canopy height and reflectance detected after thresholding the small

values.

LAI 1 LAI 2 LAI 3 LAI 4 LAI 5 LAI 6 LAI 8 LAI 10 LAI 15 LAI 20% true h detected 93.3 93.3 91.7 86.7 84.2 80.0 74.2 68.3 60.8 55.8% true I detected 99.7 99.6 99.7 99.3 99.3 99.1 99.2 99.1 99.4 99.4

Table 4.5: Percentage of true canopy height and backscatter detected after thresh-olding δI/δz values smaller than 0.001 for a spheroid canopy with 6 m height andsemi minor of 2 m

The normalised canopy height position of the 50% and 75% fraction of normalised

cumulative backscatter and their ratio can be found in table 4.6. Based on this ratio,

RMSE values were calculated and summarised in table 4.7.

46

4. Results

Figure 4.10: Overview of the normalised cumulative I functions for a tree with aspheroid canopy. MCRT LiDAR waveform after thresholding normalised δI/δz valuessmaller than 0.001 vs. original analytical normalised I functions

LAI 1 LAI 2 LAI 3 LAI 4 LAI 5 LAI 6 LAI 8 LAI 10 LAI 15 LAI 20Analytical 50% 0.453 0.414 0.382 0.355 0.332 0.314 0.285 0.263 0.230 0.210

75% 0.625 0.580 0.540 0.505 0.475 0.450 0.412 0.384 0.341 0.31850%/75% 0.725 0.714 0.707 0.703 0.699 0.697 0.691 0.686 0.673 0.662

MCRT threshold 50% 0.461 0.422 0.403 0.398 0.381 0.373 0.379 0.369 0.373 0.37275% 0.642 0.597 0.573 0.568 0.548 0.543 0.546 0.543 0.553 0.560

50%/75% 0.719 0.706 0.702 0.700 0.696 0.688 0.694 0.681 0.675 0.663

Table 4.6: Normalised canopy height of the 50%, 75% and their ratio fraction ofnormalised I for both the analytical and the MCRT simulated LiDAR waveform fora spheroid canopy case with a canopy height of 6 m and semi minor axis 2 m afterthresholding δI/δz values smaller than 0.001

47

4. Results

Analytical solutionMCRT simulation (threshold) LAI 1 LAI 2 LAI 3 LAI 4 LAI 5 LAI 6 LAI 8 LAI 10 LAI 15 LAI 20LAI 1 0.006 0.005 0.012 0.016 0.020 0.023 0.028 0.033 0.046 0.057LAI 2 0.019 0.008 0.001 0.003 0.007 0.009 0.015 0.020 0.033 0.044LAI 3 0.023 0.012 0.005 0.001 0.003 0.006 0.011 0.016 0.029 0.040LAI 4 0.025 0.014 0.007 0.003 0.000 0.003 0.008 0.014 0.026 0.038LAI 5 0.029 0.018 0.011 0.007 0.004 0.001 0.004 0.009 0.022 0.034LAI 6 0.037 0.026 0.019 0.015 0.012 0.009 0.004 0.002 0.014 0.026LAI 8 0.031 0.020 0.013 0.009 0.006 0.003 0.002 0.008 0.020 0.032LAI 10 0.044 0.033 0.026 0.022 0.019 0.016 0.011 0.005 0.007 0.019LAI 15 0.050 0.039 0.032 0.028 0.025 0.022 0.017 0.011 0.001 0.013LAI 20 0.061 0.050 0.044 0.039 0.036 0.033 0.028 0.023 0.010 0.002

Table 4.7: Resulting RMSE values for all LAI combinations between the simulatedand the analytical waveforms derived from comparing normalised canopy height ofthe ratio fraction of the normalised cumulative I functions. Spheroid canopy casewith unknown canopy height



Figure 4.11 shows the LiDAR waveforms for the realistic Birch trees obtained via

MCRT simulation. The normalised I functions were calculated the same way as

described in the paragraphs above and are displayed in figure 4.12. Figure 4.13

compares the normalised I functions of the Birch trees with the analytical waveform

for a spheroid canopy and comparison with the analytical waveform for a cuboid

canopy can be found in figure 4.14.

The simple robust method used in the previous paragraph was applied (canopy

height positions for selected fractions and their ratio can be found in the Appendix,

table 9 and table 10). Resulting RMSE values can be found in table 4.8 (compar-

ison with analytical solution for spheroid canopy) and table 4.9 (comparison with

analytical solution for cuboid canopy).

Additionally, an RMSE was calculated based on the normalised canopy height

positions of the 20%, 50%, 75% and 80% fraction of normalised cumulative backscat-

ter (see Appendix table 11 and table 12). Since visual interpretation indicated that

the normalised cumulative backscatter functions of the Birch trees were most similar

to the spheroid canopy case, only these analytical waveforms were used in this part

of the analysis. Results are displayed in table 4.10.

48

4. Results

Birch 1

Birch 2

Birch 3

Birch 4

Birch 5

Birch 6

Birch 7

Figure 4.11: Overview of MCRT Lidar simulations on realistic Birch trees: x-z (left)and y-z (middle) visualisation of the birch tree model and LiDAR waveform (right)

49

4. Results

Figure 4.12: Normalised cumulative I functions for MCRT simulated Birch tree data

Figure 4.13: Normalised cumulative I functions comparison between MCRT simu-lated Birch tree data and spheroid canopy analytical waveforms

50

4. Results

Figure 4.14: Normalised cumulative I functions comparison between MCRT simu-lated Birch tree data and cuboid canopy analytical waveforms

Analytical solutionMCRT simulation LAI 1 LAI 2 LAI 3 LAI 4 LAI 5 LAI 6 LAI 8 LAI 10 LAI 15 LAI 20Birch 1 0.1268 0.1160 0.1093 0.1049 0.1016 0.0988 0.0935 0.0883 0.0755 0.0640Birch 2 0.1394 0.1286 0.1218 0.1175 0.1142 0.1113 0.1061 0.1009 0.0881 0.0766Birch 3 0.1407 0.1298 0.1231 0.1187 0.1154 0.1126 0.1073 0.1021 0.0894 0.0778Birch 4 0.1573 0.1465 0.1397 0.1354 0.1321 0.1292 0.1240 0.1188 0.1060 0.0945Birch 5 0.0069 0.0177 0.0244 0.0288 0.0321 0.0349 0.0402 0.0454 0.0582 0.0697Birch 6 0.1272 0.1164 0.1096 0.1052 0.1020 0.0991 0.0939 0.0887 0.0759 0.0644Birch 7 0.0110 0.0218 0.0286 0.0330 0.0362 0.0391 0.0443 0.0495 0.0623 0.0738

Table 4.8: Resulting RMSE values derived from comparing Birch MCRT wave-form simulations with the analytical solutions for a spheroid canopy (simple robustmethod)


Table 4.9: Resulting RMSE values derived from comparing Birch MCRT waveformsimulations with the analytical solutions for a cuboid canopy (simple robust method)

51

4. Results


Table 4.10: Resulting RMSE values derived from comparing Birch MCRT waveformsimulations with the analytical solutions for a spheroid canopy (based on 4 positionsof relative canopy height)

52

Chapter 5

Discussion



els


The shape specific analytical solutions for first order scattering proved to reconstruct

the simulated LiDAR waveforms well. Similar to findings in Ni-Meister et al. (2001),

the key concept in these analytical solutions (equation 3.12, 3.20, 3.28 and 3.29) was

the gap probability and its dependency on within crown path length. These formu-

lae all showed dependency on the area scattering phase function Γ(Ω → -Ω). In this

specific case of a spherical leaf angle distribution, nadir viewing and illumination

angles and assumption that leaf transmittance is linearly related to leaf reflectance,

Γ(Ω → -Ω) is equal toρl3

where ρl is the leaf reflectance (Lewis, 2010c). Therefore

the area phase scattering function will be a constant for trees with the same leaf

properties, which is the case in the tree models used in this project. Hence it would

only have a scaling influence on the LiDAR waveform. In the validation method ap-

plied, the LiDAR waveform was normalised by dividing by its integral over the whole

canopy (see equations 3.15, 3.22, 3.33 and 3.34). This implies that the normalised

LiDAR waveform was independent of this scaling factor since it appeared in the nu-

merator as well as in the denominator of the normalisation ratio. As a consequence

53

5. Discussion

it can be concluded that the specific shape of the normalised cumulative backscatter

functions (see figure 4.3, 4.6 and 4.9) will be independent of the wavelength used in

the LiDAR sensor. Additionally this means that one can solve, theoretically, for LAI

without knowing the the leaf reflectance.

The developed analytical solutions also revealed that for a specific canopy shape,

the extent had no influence on the LiDAR waveform and was also considered to be

a scaling factor. As discussed earlier, normalising the LiDAR waveform removes the

scaling effect and hence each canopy shape had only one set of analytical solutions,

independent of the canopy extent parameters.

Results of the look up table inversion in table 4.1 showed good agreement for the

cuboid canopy as the LAI values for the simulated trees could be predicted well by the

analytical solution. Similar results were obtained for canopies with a prolate spheroid

shape (see table 4.4), except for one small outlier in the results for a spheroid canopy

with canopy height 8. The results for the other spheroid canopy extents did not

show this outlier and thus a possible explanation for this is the noise in the MCRT

simulations. As described earlier this was due to the fact that the tree models were

only an approximation of a turbid medium because leaf size was not small enough

(however optimised regarding the trade off with computing time in tree generation).

One can see in the MCRT simulations in figure 4.1, 4.4 and 4.7 that the noise was

random and therefore had no significant influence on the analysis.

Analytical solutions for the conical canopy tended to overestimate with one unit

for trees with a small canopy height (see table 4.3), but showed good agreement

for trees with increased canopy height (see table 4.2). Because the LiDAR sensor

settings (and more specifically bin spacing) were the same for all simulations, less

within-canopy LiDAR returns were registered for smaller canopies compared to larger

canopies. Therefore the normalised cumulative backscatter functions had to be con-

structed from less data points and interpolation had to be applied over wider intervals

which decreased the accuracy of determining the exact position of normalised canopy

height for a specific fraction of normalised cumulative backscatter. This error, which

was introduced by interpolating, decreased as the sample size increased and therefore

better results were obtained for larger canopies. The reason why cuboid and spheroid

canopies do not show this poorer performance for small canopies can be explained by

the distribution of the normalised cumulative I functions for different LAI values. As

can be seen in figure 4.3 and 4.9, cuboid and spheroid canopy normalised cumulative

54

5. Discussion

backscatter functions showed the possibility to distinguish well between the different

LAI scenes. Figure 4.6 does not reveal such a region and therefore the interpolation

accuracy becomes more important. Another consequence of this difference in ease

to find a distinguishable region is that the LUT of the spheroid and cuboid canopy

only needed to contain one value (75% fraction) to perform well, whereas the conical

canopy LUT needed four values to function reasonable. The RMSE value which se-

lected the final LAI solution was also less defined (i.e. the difference with the RMSE

values for other LAI scenes was small) for conical canopies than for the cuboid or

spheroid canopies.


It is not always the case that one knows or can easily extract the true canopy height.

In the case of a spheroid canopy, a threshold of 0.001 was applied at normalised data

of the simulated MCRT LiDAR waveform. This had a significant influence on the

estimated canopy height (see table 4.5). For low LAI values the impact was small and

most of the thresholded data were mainly located at the top of the canopy. These

findings are similar to the conclusions drawn in earlier studies. Disney et al. (2010),

Hopkinson (2007) and Chasmer et al. (2006) suggested that canopy height tend to be

underestimated since the laser pulse might need to penetrate into the foliage first to

return sufficient energy to trigger a significant return within the timing electronics of

the LiDAR sensor. This effect was more apparent for lower LAI values. Thresholding

had little to no effect for values at the top of the canopy for higher LAI values but

a significant part of the lower canopy was not detected. For leaf area index 20,

44.2% of the canopy was not detected whereas this was only 0.7% for LAI 1. These

observations of loss in canopy height were in contrast with the conservation of almost

all the returned backscatter which was between 99.1% and 99.7%. This is logical since

the applied threshold is a relatively small value and will therefore not remove any data

that contributes significantly to the return signal. Therefore it was justified to use a

ratio technique and to assume that re-normalising the thresholded MCRT simulation

was dominated by a linear transformation of the normalised canopy height although

it needs to be mentioned that results (see table 4.7) indicate that this method tends

to slightly overestimate the predicted LAI value. This small mismatch is probably

due to the small loss in returned backscatter, which will have a small effect on the

re-normalisation as well.

55

5. Discussion



The canopy properties of the simple trees were an extreme case and the realistic Birch

trees tend to behave differently. This could be seen from figure 4.11. Leaf material

was not spread homogeneous throughout the canopy and hence the leaf area density

function was not a constant. This could be noticed in the fluctuating behaviour of

the normalised cumulative backscatter functions in figure 4.12 where Birch trees with

more gaps showed more fluctuations. Another major difference between the Birch

trees and the simple trees is the presence of not only leaf material in the canopy, but

also branches. This will have a strong influence on the LiDAR waveform since the

LiDAR pulse will give a more solid return when it hits a branch than when it would

hit a leaf.

Visual interpretation of the canopy shape (figure 4.11) suggests that most of the

canopy shapes could be approximated with a generic spheroid shape. This was also

confirmed from the resulting MCRT LiDAR waveform. Birch 1, 2 and 3 showed a

typical spheroid canopy LiDAR waveform with a peak which is shifted more towards

the top of the canopy for higher LAI values. Birch 4 on the other hand can be better

approximated with a cuboid canopy, with a strong return on top of the canopy and the

return gradually decreasing with increased penetration in the canopy. The vertical

cross sections of birch 5, 6 and 7 suggested a spheroid canopy, but the presence of

significant gaps made canopy shape interpretation via the resulting LiDAR waveform

difficult.

A better way of obtaining information about the shape was to compare the cumu-

lative I functions of the Birch trees with the analytical solutions for a spheroid (figure

4.13) and cuboid (figure 4.14) canopy. This confirmed the initial visual interpretation

and it was therefore concluded that a spheroid canopy was a good approximation for

all Birch trees except for Birch 4 where a cuboid canopy approximation was the bet-

ter option. It has to be remarked that none of the Birch trees perfectly agreed with

normalised cumulative backscatter functions of the generic canopy shapes. The most

important difference between the cuboid and spheroid canopy shape is the strong

slope at the top of the canopy of the cuboid canopy. Together with the overall

curving behaviour this were the most important visual interpretation tools used to

address a generic shape to a realistic Birch tree.

56

5. Discussion

Comparing the RMSE results of the statistical analysis carried out with the simple

robust (ratio) method with the true LAI information of the Birch tree revealed that

only the LAI of Birch 7 was approximated reasonable (LAI 1 via LUT versus true

LAI 1.51) when the Birch waveforms were analysed as spheroid canopy waveforms

(see table 4.8). Birch 4, which was suggested to behave as a cuboid canopy, was

predicted an LAI of 3 (instead of true LAI 4.34).

These results were expected since visual interpretation of the normalised cumu-

lative I functions already suggested that the shape of the Birch curves did not match

the analytical solutions over the full extent of the normalised canopy. An analytical

method which uses only two data points to quantify the whole function will therefore

be very sensitive to the offset of the Birch functions. An improved method calculated

the RMSE based on four data points for the spheroid canopy case. Results in table

4.10 revealed an improvement, but still no satisfying results were obtained.

5.3 Further research

Potential for being a better and more robust method to interpret the Birch waveforms

could be curve fitting. Since the analytical equations for each canopy shape are

available, parameters (i.e. LAI) could be adjusted to obtain the best fit with the

Birch waveform.

The approach in this project assumed, like Ni-Meister et al. (2001), that the Li-

DAR signal was first order scattering dominated (see figure 3.3). Since vegetation

weakly absorbs laser pulses in the NIR, Kotchenova et al. (2003) suggested that

the inclusion of multiple scattering might results in more accurate descriptions of

LiDAR waveforms. In that study a time-dependent stochastic radiative transfer the-

ory was introduced to model the propagation of lasers pulses in the canopy and this

was solved numerically. The approach allowed for a more realistic description of

the canopy structure including clumping and gaps. A factor was used to describe

the correlation between foliage elements in different layers (vertical heterogeneity).

Although additional improvements need to be made towards more accurate charac-

terisation of the probability density function, this probability based approach is an

improvement compared with the averaging in the approach used in this project.

Analytical solutions derived in this project have the limitation that they can only

57

5. Discussion

be applied when the sensor is operated at nadir. In equation 3.4, µ was set to 1 for

this specific LiDAR operating conditions. It is suggested to work out the analytical

formulae analogous, but to re-introduce the µ factor. This will introduce a small

error in the solution, but it is expected to approximate reality well since off-nadir

viewing angles of LiDAR sensors are generally small.

As a final suggestion for further research, these formulae should be applied for

a scene with multiple trees. For a forest constructed with non-overlapping trees

with simple generic canopies, it is expected that the resulting waveform will be the

summation of the waveforms for the individual trees and this could be quantified

based on the analytical solutions derived for single trees.

58

Chapter 6

Conclusion

The research objective of this dissertation was the development of a set of ana-

lytical solutions to describe the LiDAR waveform for simple trees with a generic

canopy shape. Based on the radiative transfer solution for single order scattering in

the optical case, formulae for a cuboid, conical and spheroid canopy were derived.

These equations were validated against MCRT waveform simulations. Both the sim-

ulated and the analytical waveforms were used to construct normalised cumulative

backscatter functions. Based on these functions a LUT was generated and used for

inversion under the assumption that canopy height was known. Good agreement

was achieved for spheroid and cuboid canopies for all canopy extents. Agreement

for the conical case increased when canopy height was larger. This was because the

normalised cumulative backscatter functions for different LAI scenes did not show

a region which was suitable to distinguish well between the different functions. Be-

cause LiDAR sampling settings (i.e. bin size) were the same for all simulations, an

increase in canopy height therefore meant an increase in samples and thus a more

accurate representation of the normalised cumulative I functions.

Because it is not realistic to assume that canopy height is always known, a thresh-

old was applied on the simulated LiDAR waveforms of the spheroid canopies to filter

the small energy returns which might be missed in reality. It was noticed that only

little of the total canopy energy received by the sensor was lost by thresholding

(0.9%) whereas canopy height could be significantly reduced (up to 44.2% for high

LAI values). A simple robust ratio technique worked reasonable, although LAI values

tended to be slightly overestimated.

59

6. Conclusion

When the analytical solutions were applied on realistic Birch trees, no satisfying

LAI predications were achieved. This was mainly due to the fact that these analytical

formulae were developed for a specific case which assumes that the leaf area density

is a constant and only leaf material is present in the canopy. The Birch trees fail

both of these assumptions since gaps and branches were present and therefore simple

interpretation techniques proved not to work well for more realistic trees.

This project showed the great potential of these analytical equations to ex-

tract structural information from LiDAR waveforms. Since simple robust techniques

proved not to work for more realistic tree models, it is suggested that potential im-

provements can be made by applying a more robust technique such as curve fitting

to link the information of LiDAR waveforms with the analytical solutions. An im-

portant question to be asked is what such analytical solutions can really mean for

inversion problems and what practical use they can have. Earlier studies already

indicated that a combination of data from different sensors improved the assessment

of structure in forests significantly (Koetz et al., 2007). It is therefore suggested that

these analytical equations can serve as a useful tool to add a physical constraint to

this inversion problem.

60

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65

Appendix A:

Optimizing the parameters for

LiDAR simulations

Figure 1: LiDAR simulations for different footprint sizes for a prolate spheroid treewith LAI 6 m2/m2 and a canopy height of 6 m and semi-minor of 2 m

66

Appendix A: Optimizing the parameters for LiDAR simulations

Figure 2: LiDAR simulations with different bin size for a prolate spheroid tree withLAI 1 m2/m2 and a canopy height of 6 m and semi-minor of 2 m

67


Figure 3: LiDAR simulations with different bin size for a prolate spheroid tree withLAI 6 m2/m2 and a canopy height of 6 m and semi-minor of 2 m

68


Figure 4: LiDAR simulations with different rays per pixel (RPP) for a prolatespheroid tree with LAI 6 m2/m2 and a canopy height of 6 m and semi-minor of2 m

69


Figure 5: Zoom of LiDAR simulations with different rays per pixel (RPP) for aprolate spheroid tree with LAI 6 m2/m2 and a canopy height of 6 m and semi-minorof 2 m

70


Figure 6: The effect of leaf size on the MCRT LiDAR simulations for a spheroidcanopy with LAI 2 m2/m2 and a canopy height of 4 m and semi-minor of 1 m

71

Appendix B:

Validation of LiDAR waveforms:

cuboid canopy

Figure 7: MCRT LiDAR simulations for a tree with a cuboid canopy and a canopyheight of 4 m and width and length of 1m

72

Appendix B: Validation of LiDAR waveforms: cuboid canopy

Figure 8: MCRT LiDAR simulations for a tree with a cuboid canopy and a canopyheight of 6 m and width and length of 2m

LAI 1 LAI 2 LAI 3 LAI 4 LAI 5 LAI 6 LAI 8 LAI 10 LAI 15 LAI 20Analytical 0.697 0.640 0.579 0.518 0.461 0.410 0.327 0.267 0.178 0.132MCRT h4 0.691 0.639 0.575 0.519 0.455 0.410 0.334 0.270 0.176 0.134MCRT h6 0.684 0.635 0.571 0.517 0.459 0.410 0.329 0.273 0.180 0.135MCRT h8 0.693 0.634 0.576 0.516 0.464 0.416 0.332 0.273 0.186 0.140

Table 1: Normalised canopy height of the 75% fraction of normalised I for both theanalytical and the MCRT simulation LiDAR waveforms for a cuboid canopy

73


Figure 9: Analytical normalised LiDAR waveforms for a tree with a cuboid canopyand a canopy height of 8 m and width and length of 4 m


Table 2: Resulting RMSE values for all LAI combinations between the simulatedand the analytical waveforms derived from comparing normalised canopy height ofthe 75% fraction of the normalised cumulative I functions. Cuboid canopy case witha canopy height of 4 m and width and length of 1 m

74



Table 3: Resulting RMSE values for all LAI combinations between the simulatedand the analytical waveforms derived from comparing normalised canopy height ofthe 75% fraction of the normalised cumulative I functions. Cuboid canopy case witha canopy height of 6 m and width and length of 2 m

75

Appendix C:


conical canopy

Figure 10: MCRT LiDAR simulations for a tree with a conical canopy and a canopyheight of 4 m and base radius of 1m

76

Appendix C: Validation of LiDAR waveforms: conical canopy

Figure 11: MCRT LiDAR simulations for a tree with a conical canopy and a canopyheight of 8 m and base radius of 4 m

LAI 1 LAI 2 LAI 3 LAI 4 LAI 5 LAI 6 LAI 8 LAI 10 LAI 15 LAI 20Analytical 10% 0.435 0.415 0.400 0.388 0.379 0.371 0.360 0.353 0.342 0.335

20% 0.557 0.537 0.522 0.510 0.502 0.495 0.485 0.478 0.468 0.46330% 0.644 0.626 0.612 0.602 0.594 0.588 0.579 0.573 0.565 0.56140% 0.714 0.698 0.686 0.677 0.671 0.666 0.658 0.654 0.647 0.643

MCRT h4 10% 0.419 0.404 0.392 0.380 0.370 0.359 0.350 0.344 0.334 0.32720% 0.547 0.527 0.509 0.498 0.490 0.484 0.475 0.470 0.462 0.45430% 0.638 0.615 0.597 0.592 0.584 0.577 0.571 0.564 0.559 0.55140% 0.707 0.686 0.669 0.669 0.663 0.655 0.649 0.644 0.637 0.632

MCRT h6 10% 0.426 0.408 0.397 0.384 0.374 0.370 0.360 0.350 0.341 0.33520% 0.552 0.531 0.516 0.507 0.498 0.493 0.481 0.475 0.469 0.46130% 0.638 0.621 0.606 0.597 0.588 0.585 0.576 0.571 0.564 0.55940% 0.708 0.693 0.680 0.672 0.667 0.661 0.654 0.650 0.645 0.641

MCRT h8 10% 0.432 0.416 0.399 0.391 0.380 0.374 0.364 NA NA NA20% 0.552 0.535 0.521 0.512 0.504 0.497 0.489 NA NA NA30% 0.639 0.624 0.611 0.602 0.595 0.590 0.582 NA NA NA40% 0.711 0.696 0.685 0.677 0.671 0.666 0.660 NA NA NA

Table 4: Normalised canopy height of selected fractions of normalised I for both theanalytical and the MCRT simulation LiDAR waveforms for a conical canopy

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Appendix C: Validation of LiDAR waveforms: conical canopy

Figure 12: Analytical normalised LiDAR waveforms for a tree with a conical canopyand a canopy height of 8 m and base radius of 4 m


Table 5: Resulting RMSE values for all LAI combinations between the simulatedand the analytical waveforms derived from comparing normalised canopy height offour different fractions of the normalised cumulative I functions. Conical canopy casewith a canopy height of 6 m and base radius 2 m

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Appendix D:


spheroid canopy

Figure 13: MCRT LiDAR simulations for a tree with a spheroid canopy and a canopyheight of 4 m and semi minor axis of 1m

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Appendix D: Validation of LiDAR waveforms: spheroid canopy

Figure 14: MCRT LiDAR simulations for a tree with a spheroid canopy and a canopyheight of 8 m and semi minor axis of 4 m

LAI 1 LAI 2 LAI 3 LAI 4 LAI 5 LAI 6 LAI 8 LAI 10 LAI 15 LAI 20Analytical 0.625 0.580 0.540 0.505 0.475 0.450 0.412 0.384 0.341 0.318MCRT CH4 0.621 0.572 0.532 0.494 0.466 0.442 0.405 0.377 0.334 0.310MCRT CH6 0.617 0.575 0.536 0.504 0.473 0.446 0.409 0.382 0.339 0.315MCRT CH8 0.621 0.612 0.539 0.505 NA 0.451 0.413 NA NA NA

Table 6: Normalised canopy height of the 75% fraction of normalised I for both theanalytical and the MCRT simulation LiDAR waveforms for a spheroid canopy

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Figure 15: Analytical normalised LiDAR waveforms for a tree with a spheroid canopyand a canopy height of 8 m and semi minor axis of 4 m


Table 7: Resulting RMSE values for all LAI combinations between the simulated andthe analytical waveforms derived from comparing normalised canopy height of the75% fraction of the normalised cumulative I functions. Spheroid canopy case with acanopy height of 4 m and semi minor axis 1 m

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Table 8: Resulting RMSE values for all LAI combinations between the simulated andthe analytical waveforms derived from comparing normalised canopy height of the75% fraction of the normalised cumulative I functions. Spheroid canopy case with acanopy height of 6 m and semi minor axis 2 m

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Appendix E:

Applying the analytical solution to

simulated LiDAR waveforms of

Birch trees

Birch 1 Birch 2 Birch 3 Birch 4 Birch 5 Birch 6 Birch 750% 0.214 0.310 0.310 0.295 0.503 0.385 0.48075% 0.358 0.530 0.531 0.520 0.688 0.645 0.653

50% - 75% ratio 0.598 0.585 0.584 0.567 0.731 0.597 0.736

Table 9: Normalised canopy height of the 50%, 75% and ratio fraction of normalisedI for MCRT simulations on the Birch trees

LAI 1 LAI 2 LAI 3 LAI 4 LAI 5 LAI 6 LAI 8 LAI 10 LAI 15 LAI 20Spheroid 50% 0.453 0.414 0.382 0.355 0.332 0.314 0.285 0.263 0.230 0.210

75% 0.625 0.580 0.540 0.505 0.475 0.450 0.412 0.384 0.341 0.31850% - 75% ratio 0.725 0.714 0.707 0.703 0.699 0.697 0.691 0.686 0.673 0.662

Cuboid 50% 0.434 0.375 0.323 0.278 0.240 0.209 0.163 0.131 0.086 0.06375% 0.697 0.640 0.579 0.518 0.461 0.410 0.327 0.267 0.178 0.132

50% - 75% ratio 0.623 0.587 0.558 0.536 0.520 0.509 0.497 0.491 0.483 0.476

Table 10: Normalised canopy height of the 50%, 75% and ratio fraction of normalisedI for spheroid and cuboid analytical solutions

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Appendix E: Applying the analytical solution to simulated LiDAR waveforms of Birch trees

Birch 1 Birch 2 Birch 3 Birch 4 Birch 5 Birch 6 Birch 720% 0.121 0.169 0.146 0.103 0.221 0.157 0.19750% 0.214 0.310 0.310 0.295 0.503 0.385 0.48075% 0.358 0.530 0.531 0.520 0.688 0.645 0.65380% 0.378 0.569 0.579 0.581 0.734 0.686 0.701

Table 11: Normalised canopy height of the 20%, 50%, 75% and 80% fraction ofnormalised I for MCRT simulations on the Birch trees

LAI 1 LAI 2 LAI 3 LAI 4 LAI 5 LAI 6 LAI 8 LAI 10 LAI 15 LAI 20Spheroid 20% 0.254 0.229 0.209 0.192 0.179 0.168 0.150 0.137 0.117 0.104

50% 0.453 0.414 0.382 0.355 0.332 0.314 0.285 0.263 0.230 0.21075% 0.625 0.580 0.540 0.505 0.475 0.450 0.412 0.384 0.341 0.31880% 0.666 0.621 0.580 0.543 0.512 0.485 0.444 0.414 0.369 0.345

Table 12: Normalised canopy height of the 20%, 50%, 75% and 80% fraction ofnormalised I for spheroid analytical solution

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Appendix F:

Overview of data folder

All analysis was carried out in the folder /data/raid7/2009/uceskca/dissertation.

This appendix will give an overview of the contents of the subfolders of this directory.

subfolder tree generation

Generation of simple tree models

genTree * lai generates the simple tree models (cub = cuboid, cone = conical, ell =

prolate spheroid). Generated objects can be found in folder gen objects.

Placing bounding boxes

BB * puts bounding boxes around the generated objects created in the previous step

and saves them in gen objectsBB. It uses the program filtMe to achieve this.

subfolder mcrt

MCRT simulations for simple tree models

RT lidar * is used to run the Monte Carlo ray tracing simulations. Input files are

located in gen objectsBB and output is written to results lidar.

The specific camera and light files for each simulation can be found in folder camera lidar

and light lidar. They are based on the default camera (camera.lidar.default.dat) and

light (light.lidar.default.dat) file.

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Appendix F: Overview of data folder

subfolder birch

Birch tree MCRT simulations

RT lidar birch is used to run the Monte Carlo ray tracing simulations. Input files

are located in gen object and output is written to results lidar.

The specific camera and light files for each simulation can be found in folder camera lidar

and light lidar. They are based on the default camera (camera.lidar.default.dat) and

light (light.lidar.default.dat) file.

subfolder analysis

Statistical analysis in R

The cuboid folder contains the analysis for the cuboid canopies, folder cone for con-

ical canopies, folder spheroid for spheroid canopies, folder spheroid unknownCH for

spheroid canopies with unknown canopy height, folder birch ratio for the ratio tech-

nique analysis on the Birch trees and birch20507580 for the analysis based on multiple

values of normalised canopy height.

Each of these folders contain:

(i) *.R in the main folder are the codes to determine the normalised canopy height.

Input csv-files are created in Microsoft Excel and are based on the MCRT

simulation output data. Output frac *.csv files are generated in R after running

the script.

(ii) The rmse folder contains *.R scripts to generate RMSE tables based on the

values of normalised canopy height. RMSE *.csv are the output files.

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Documents

Modelling lidar waveforms to solve for canopy properties