Monte Carlo simulation of multiple photon scattering in sugar maple tree canopies

Monte Carlo simulation of multiple photon scattering insugar maple tree canopies

Michael A. Greiner,1,* Bradley D. Duncan,2,3 and Matthew P. Dierking2,4

1Electro-Optics Program, University of Dayton, 300 College Park,Dayton, Ohio 45469-1245, USA

2Sensors Directorate, Air Force Research Laboratory (AFRL/RYJM), 3109 P. Street, Building 622,Wright Patterson Air Force Base, Ohio 45433-7700, USA

[email protected]@wpafb.af.mil

*Corresponding author: [email protected]

Received 7 August 2009; revised 13 October 2009; accepted 14 October 2009;posted 14 October 2009 (Doc. ID 111750); published 2 November 2009

Detecting objects hidden beneath forest canopies is a difficult task for optical remote sensing systems.Rather than relying upon the existence of gaps between leaves, as other researchers have done, ourultimate goal is to use light scattered by leaves to image through dense foliage. Herein we describethe development of a Monte Carlo model for simulating the scattering of light as it propagates throughthe leaves of an extended tree canopy. We measured several parameters, including the gap fraction andmaximum leaf-area density, of a nearby sugar maple tree grove and applied them to our model. We reportthe results of our simulation in both the ground and the receiver planes for an assumed illuminationangle of 80°. To validate our model, we then illuminated the sugar maple tree grove at 80° and collecteddata both on the canopy floor and at our monostatic receiver aperture. Experimental results were foundto correlate well with our simulated expectations. © 2009 Optical Society of America

OCIS codes: 290.4210, 290.7050, 290.1483.

1. Introduction

Detecting objects beneath forest canopies is a diffi-cult task for optical remote sensing systems. Currentfoliage penetration methods depend heavily upon theability to look through gaps between the leaves andthe branches [1]. For example, as an airborne sensoris flown over a forested area, small pieces of a targetunder the canopy may be sequentially imaged andlater pieced together to form a composite image[2,3]. Unfortunately, as shown in Fig. 1, the gap frac-tion quickly drops to zero as the foliage density and/or zenith angle increases, leaving very few opportu-nities to exploit gaps in the canopy [4]. Developingthe capability to look directly through leaves while

not relying on the existence of gaps will greatlyaid in the imaging of objects beneath the canopy.

Imaging from scattered photons presents new andunique challenges. The scattering distributions of in-dividual leaves have been studied extensively, butunderstanding what happens to light as it is scat-tered through a random collection of leaves is a morecomplex problem [5,6]. In particular, after propaga-ting through a canopy, a light beam will experiencespatial, angular, and temporal dispersion. The dis-tribution of photons exiting the canopy will then bephysically wider, decollimated, and temporally dis-persed from the incident beam [7].

For our current work we have created a canopypropagation model that allows us to simulate thepropagation and scattering of photons through aforest canopy with 100% foliage obscuration. TheMonte Carlo algorithm we use in this simulation is

0003-6935/09/326159-13$15.00/0© 2009 Optical Society of America

10 November 2009 / Vol. 48, No. 32 / APPLIED OPTICS 6159

described in detail in Section 2. In Section 3 we dis-cuss the measurement of several parameters from anearby sugar maple tree grove, which we then ap-plied to our model. By using real forest parameters,we intend to achieve the most accurate and reprodu-cible results. In Section 4 we report the results of oursimulations. There we explore several phenomena,including the beam footprint and temporal pulsewidths expected on the ground, as well as the spatial,temporal, and angular returns measured by a virtualdetector looking down on the canopy from an ele-vated position several hundred meters away. Then,in Section 5, we discuss the experimental validationof our model by comparing several predicted pheno-mena to those we measured after propagating a realbeam through the canopy of our sugar maple treegrove. Our conclusions are presented in Section 6.

2. Monte Carlo Algorithm

The propagation of light through a forest canopy wasmodeled by means of an extensive Monte Carlo simu-lation, in whichmultiple scattering events within thecanopy are treated as a sequence of interactions be-tween a single photon and a discrete scatterer. Sim-plification lies in the fact that the end result does notcome through looking at the canopy as a whole, butrather by considering each photon individually andeach interaction sequentially. The advantage ofusing Monte Carlo methods is that multiple scatter-ing events can be calculated without complex anal-ysis, as only single variable scattering probabilitydensity functions (PDFs) are required.We use ourMonte Carlo algorithm to track the pro-

pagation of photons through a simulated canopy,which we have modeled as an elliptically cylindricalcollection of leaves that are randomly, but not uni-formly, distributed. As shown in Fig. 2, photonsare launched at an illumination angle of θINC froman elevated monostatic direct detection ladar systemand are aimed at an assumed primary target locatedon the ground at the geometric center of the canopy.We have also assumed a secondary ground target be-neath the location where photons first enter the ca-nopy, as preliminary simulations showed a large spoton the ground at this location due to initial down

scattering. Both virtual targets are 10m in diameterand are tilted so that their surface normals arepointed back toward the receiver.

The algorithm used is based on the cloud propaga-tion model presented by Bucher and is depicted inthe flow chart provided in Fig. 3 [7]. According toour model, a single incident photon enters the canopyand travels a random distance before it experiencesits first scattering interaction. On this initial propa-gation the photon cannot stray from its trajectoryand, therefore, may only encounter a leaf or the pri-mary target at which it is initially aimed. However,there are typically four possible options after randomphoton propagation: (i) leaf interaction, (ii) targetinteraction, (iii) ground interaction, or (iv) canopydeparture.

If the photon remains within the bounds of the can-opy after its initial propagation event, then it musthave struck a leaf, and either absorption or scatter-ing will occur. We determine whether the photon isabsorbed by selecting a value of a random variableuniformly distributed between 0 and 1. If the randomvalue is smaller than the leaf absorption coefficient,then the photon is absorbed and we launch a newphoton into the canopy. Otherwise, the photon isscattered from the leaf, in which case we select ran-dom leaf orientation angles (θL and ϕL) and randomleaf scattering angles (θS and ϕS).

As shown in Fig. 4(a), the leaf orientation angles(θL and ϕL) are defined in the canopy coordinate sys-tem, where θL is the zenith angle made between the zaxis and the leaf normal vector n, and ϕL is the azi-muth angle between the x axis and the projection ofthe leaf normal vector onto the x–y plane. The canopycoordinate system is, in turn, defined so that the ori-gin is located on the ground at the center of the pri-mary target, where the positive x axis is parallel tothe ground and points directly away from the trans-mitter, the z axis points straight up, and the y axis isdefined according to the right-hand rule. As shown inFig. 4(b), the leaf scattering angles (θS and ϕS) arethen defined in the leaf coordinate system, whereθS is the zenith angle made between the leaf surfacenormal and the scattered photon’s unit propagationvector s, and ϕS is the azimuth angle made between

Fig. 1. Example hemispheric image captured looking upwardwithin a grove of sugar maple trees near Dayton, Ohio.

Fig. 2. Canopy illuminated by a monostatic ladar system at anangle θINC. The primary target is on the ground at the geometriccenter of the canopy, while a secondary target is located directlybelow the point where the incident beam enters the canopy.

6160 APPLIED OPTICS / Vol. 48, No. 32 / 10 November 2009

the projections of s and the photon’s unit propagationvector p as it is incident upon the leaf surface.The photon propagation angles (θ and ϕ), defined

in the canopy coordinate system as shown in Fig. 4(a), are then determined using a simple matrix trans-formation. In particular, after scattering, the scalarcomponents of the unit vector describing the propa-gation direction s can be expressed in terms of theleaf coordinate system as

" xSySzS

#¼

" sin θS cosϕS

sin θS sinϕS

cos θS

#: ð1Þ

This vector must then be transformed into the can-opy coordinate system by applying rotation anglesϕL and θL about the z and y axes, respectively, accord-ing to264x

y

z

375 ¼

264cos θL cosϕL − sinϕL sin θL cosϕL

cos θL sinϕL cosϕL sin θL sinϕL

− sin θL 0 cos θL

375

×

264sin θS cosϕS

sin θS sinϕS

cos θS

375: ð2Þ

From Eq. (2), the desired photon propagation anglescan then be determined:

θ ¼ tan−1

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pz

�; ϕ ¼ tan−1

�yx

�: ð3Þ

Next, if the photon reaches the ground plane, itslocation is compared to the locations of the two tar-gets. If the photon strikes either of the targets, bothof which are assumed to have unit reflectance, wesave the spatial, temporal, and angular ground planedata and select random target scattering angles (θTand ϕT). The target scattering angles are then de-fined in a target coordinate system analogous tothe leaf coordinate system, in the sense that the ze-nith and azimuth angles are determined by the nor-mal vector of the target. Also, by substituting (θT andϕT) for (θS and ϕS), the same matrix transformationdescribed by Eqs. (1)–(3) can be applied to determinethe new photon propagation angles in the canopycoordinate system.

If, however, the photon reaches the ground planebutdoesnot interactwitha target, itmusthave struckthe ground, which we also assume to have unit reflec-tance. In this case, we save the spatial, temporal, andangular ground plane data and select random groundscattering angles (θG and ϕG). Note that the groundscattering angles are defined in the canopy coordinatesystemand, therefore, describe thenewphotonpropa-gation angles without further transformation.

Finally, if the random propagation distance placesthe photon outside the bounds of the canopy, we

Fig. 3. Monte Carlo algorithm flow chart describing photon propagation through a tree canopy.


propagate the photon back to the receiver plane. Tomatch our experimental conditions, the receiverplane in our model was assumed to be located45m above the ground at a range of 255m (for 80°illumination) and tilted such that its surface normalis directed toward the primary target at the center ofthe canopy. If a photon hits the receiver plane withina 25 cm diameter pupil centered on the monostatictransmit/receive ladar axis and within a 45° fieldof view, then we save the spatial, temporal, and an-gular receiver plane data and launch a new photon.Otherwise, the photon is considered undetected, wedo not save any data, and then we launch a new pho-ton. In our model, photons that exit the canopy tra-veling nominally away from the receiver plane willarrive at the detector in the negative time domain.These photons are also discarded as undetected.Each photon can travel any distance in any direc-

tion according to a number of random variables thatmust be examined at every step. In particular, thecanopy is described completely by the probabilitydensity functions describing the following randomvariables: the propagation distance d; the probabilitythat the photon is reflected R, transmitted T, orabsorbed A by the leaf; the leaf angular orientationangles θL and ϕL; the leaf scattering angles θS andϕS; the ground scattering angles θG and ϕG; andthe target scattering angles θT and ϕT . The probabil-ity density functions for each of the random variableswill be described in the following subsections. Forconvenience, a simple method for generating specificvalues of a random variable, given its probabilitydensity function, is discussed in Appendix A.

A. Random Propagation Distance

For a homogeneous medium of constant leaf-numberdensity, the distance d that the photon travels be-tween scattering events is a random variable de-scribed by the following exponential probabilitydensity function [7]:

pdðdÞ ¼1Dexp

�−dD

�; ð4Þ

where the parameter D is the mean free path, oraverage distance traveled by the photon between

interactions. The mean free path Dðz; θÞ is, in turn,inversely proportional to the product of the meanprojected leaf area ApðθÞ in the direction of photonpropagation and the leaf-number density NðzÞ ofthe canopy in the region surrounding the photon.In particular,

Dðz; θÞ ¼ 1�ApðθÞ ·NðzÞ ; ð5Þ

where θ is defined in Eq. (3) and z is the verticalheight of the photon within the canopy [8].

1. Mean Projected Leaf Area

Because leaves mostly face upward, toward the Sun,photons traveling at zenith angles θ far off the ver-tical will see less leaf surface area than will photonstraveling at smaller angles. Therefore, photons pro-pagating near the vertical will, in general, experi-ence a greater number of leaf interactions and havea smaller mean free path than those traveling atgreater angles.

To derive an expression for the mean projectedarea, first suppose that the photon is traveling alongthe direction s, which makes an angle of θ with re-spect to the z axis, as illustrated in Fig. 4(a). Becausewe will assume that the leaf angular distribution isuniform in the azimuth angle ϕ, we can assume thatp lies in the x–z plane and still arrive at a generalresult. Therefore, the photon propagation vectorcan be expressed as

p ¼ x sin θ þ z cos θ: ð6ÞThe leaf angle orientation is similarly described bythe normal vector n, which can be written as

n ¼ x sin θL cosϕL þ y sin θL sinϕL þ z cos θL; ð7Þwhere x, y, and z are Cartesian coordinate systemunit vectors. The projected leaf area is then foundby means of a vector dot product. In particular, theprojected area in the direction of photon propagationcan be expressed as

Ap ¼ A0n⇀• p

⇀; ð8Þ

where A0 ¼ πd2L=4 is the actual leaf area and where

we have tacitly assumed circular leaves with a meaneffective diameter dL of 7:5 cm [9]. Inserting theexpressions from Eqs. (6) and (7) into Eq. (8), andsimplifying, yields the following result:

Ap ¼ A0½sin θL cosϕL sin θ þ cosϕL cos θ�: ð9Þ

The last expression is the projected area of a singlegeneralized leaf described by a specific set of leaf or-ientation angles. To find the mean projected areaApðθÞ, we must then average this expression overall possible orientation angles. Note, though, that

Fig. 4. (a) Leaf orientation angles are defined in the ðx; y; zÞcanopy coordinate system, while (b) leaf scattering angles aredefined in the leaf coordinate system.


the angles θL and ϕL are governed by their own sta-tistically independent PDFs pθLðθLÞ and pΦL

ðϕLÞ,respectively, and so are included in the integrationas follows [10]:

ApðθÞ ¼Z∞−∞

Z∞−∞

ApðθL;ϕL : θÞpθLðθLÞpϕLðϕLÞdϕLdθL:

ð10ÞMost often the azimuth angle distribution is as-sumed to be uniform on the range ½0; 2πÞ, with PDFpΦL

then written as [11,12]

pΦLðϕLÞ ¼

12π ½uðϕLÞ − uðϕL − 2πÞ�; ð11Þ

where uðx − x0Þ ¼ ½ð1 if x ≥ x0Þorð0 if x < x0Þ� is theunit step function [13]. The zenith angle PDF pΘL

is, however, commonly considered to be cosinusoidal,being expressed as

pΘLðθLÞ ¼

�uðθLÞ − u

�θL −

π2

��cos θL: ð12Þ

Note that this PDF corresponds to leaf normalvectors that are primarily vertical and only rarelyhorizontal [9].The expression for the mean projected area can

then be found by inserting Eqs. (9), (11), and (12) intoEq. (10). Solving the integral and simplifying yieldsthe following final expression for the mean projectedarea:

ApðθÞ ¼d2

4

�sin θ þ π2

4cos θ

�: ð13Þ

2. Leaf-Number Density

It is commonly accepted that the volume distributionof leaves varies as a function of vertical height zwith-in the canopy [14,15]. Often the vertical profile of acanopy is described in terms of leaf-area density LðzÞ,which is a measure of the total leaf area containedwithin a volume of the canopy. This is a convenientway to characterize the distribution, as the leaf-areadensity is readily converted to number density bydividing by the actual mean leaf area.The expression we used to define the leaf-area

density function was empirically derived by otherresearchers and is based on several previously ar-chived forest parameters, including total canopyheight h, maximum leaf-area density Lm, and thecorresponding canopy height zm, at which the leaf-area density takes on its maximum value [14]. Inparticular,

LðzÞ ¼ Lm

�h − zmh − z

�nexp

�n

�1 −

h − zmh − z

��; ð14Þ

where

n ¼�

6 0 ≤ z ≤ zm1=2 zm ≤ z ≤ h

:

As an example, the leaf-area density for the canopywe used in our simulation is plotted in Fig. 5, whereh ¼ 18m, Lm ¼ 0:6132 m−1, and zm=h ¼ 0:85. Notethat the value for Lm corresponds to our own experi-mental results, as will be described in Section 3,while the other values were chosen in accordancewith maple tree forest parameters measured byothers [14]. The leaf-number density function NðzÞis then found by dividing the leaf-area density bythe mean leaf area according to

NðzÞ ¼ LðzÞA0

¼ 4

πd2L

Lm

�h − zmh − z

�nexp

�n

�1 −

h − zmh − z

��: ð15Þ

Finally, the expression for the mean free path be-tween scatterers can be determined by insertingEqs. (13) and (15) into Eq. (5), and simplifying, toyield

Dðz;θÞ¼ π

½sinθþ π24 cosθ� ·Lm

�h−zmh−z

�nexp

�n

�1− h−zm

h−z

�� :ð16Þ

3. Selecting a Random Propagation Distance inan Inhomogeneous Medium

Recall that Eq. (4) gives the PDF for the randompropagation distance in a homogeneous medium.However, in actuality, tree canopies are inhomoge-neous, as evidenced by the leaf-number density chan-ging as a function of canopy height. Our approach toaccount for this fact was to break the canopy into50 vertically stacked horizontal slices, so that thehomogeneity of the leaf-number density was approxi-mately preserved within each slice. In order to mini-mize the total number of slices, we divided thecanopy into nonuniformly spaced regions, wherethe absolute value of the edge-to-edge change inleaf-area density was common to each region. As a

Fig. 5. Representative maple leaf-area density as a function ofcanopy height.


result, the canopy slices in regions where the leaf-area density changes rapidly are thinner than inthose regions where it varies slowly. The slices werethen treated as separate canopies stacked one on topof the other, and propagation through each slice wastreated on a region-by-region basis.In any given homogeneous slice of the canopy, a

random propagation distance d is selected by usingthe mean free path of that region. If the propagationdistance is less than the distance necessary for thephoton to leave the current region, then we assumethat there has been a leaf interaction. Otherwise, wetruncate the propagation at the edge of the currentregion and begin propagation again in the next re-gion. Note that, if the random propagation distancewould cause an upward traveling photon to leave thetopmost region of the canopy, we propagate thephoton back to the receiver plane and proceed as dis-cussed earlier. Similarly, if the random propagationdistance would cause a downward traveling photonto exit the bottommost region, we then terminatepropagation at the ground plane and determinewhether the photon has struck either the groundor one of the two targets.For example, consider a photon propagating at an

angle θ whose last leaf interaction occurred some-where in the kth region, as shown in Fig. 6(a). Thevertical distance between the current location ofthe photon and the boundary of the next region isdf and the total thickness of the kth region is givenby tk. Also, suppose a random value for dk is chosenthat places the photon outside the kth region; thatis, dk · cos θ > df . We then set the total distancetraveled in this jump to jdf = cos θj, and begin propa-gating again at the edge of the ðkþ 1Þth region. Nextwe use the mean free path for the ðkþ 1Þth region toselect a new propagation distance dkþ1, which is thencompared to the thickness of the ðkþ 1Þth region.If dkþ1 · cos θ > tkþ1, as is illustrated in Fig. 6(b),

then the cumulative distance traveled is set toj df = cos θ j þ j tkþ1= cos θ j and the process is re-started at the edge of the ðkþ 2Þth region. However,if dkþ1 · cos θ ≤ tkþ1, as shown in Fig. 6(c), then thephoton is assumed to have experienced a leaf inter-action in the ðkþ 1Þth region, the total distance tra-veled is set to jdf = cos θj þ dkþ1, and the propagationsequence is terminated.

B. Leaf Scattering Angles

Recall that, when a photon interacts with a leaf,either absorption or scattering may occur. As pre-viously discussed , we determine whether the photonis absorbed by selecting a value of a random variableuniformly distributed between 0 and 1 and compar-ing it to the absorption coefficient A. In a similarfashion, when scattering occurs, we determine theform of scattering (i.e., either reflection or transmis-sion) by also selecting a value of a random variableuniformly distributed between 0 and 1. In this case,though, if the randomly selected value is between 0and R=ðRþ TÞ, where R and T are the reflection andtransmission coefficients of the leaf, respectively,then reflection is assumed to have occurred. Other-wise, the photon is assumed to be transmitted. Inour previous work, we measured R, T, and A forhealthy sugar maple leaves to be 0.4861, 0.4841,and 0.0298, respectively [16].

Regardless of the form of scattering, the leaf scat-tering angles θS and ϕS are generally assumed to bestatistically independent with separable PDFs [17].The scattering azimuth angle is most often takento be uniformly distributed on the interval ½0; 2πÞ,with its PDF pΦS

ðϕSÞ written in the same form asEq. (11), but with ϕS substituted for ϕL. We have alsopreviously shown that, in reflection, the zenith scat-tering angle can be modeled as the summation of twoprobability distributions: a Lambertian distributionaccurately models diffuse scattering from the leaf,while the specular reflection component can be mod-eled by a reversed Rayleigh distribution [16]. Thenormalized sum of the two components is thenexpressed as

pΘSðθSÞ ¼ ð1 − FsÞ cos θS þ FS

�π2 − θs

�ð2π − θIncÞ2

× exp�−

�π2 − θs

�2

2ð2π − θIncÞ2�; ð17Þ

where FS is the specular reflection fraction and θInc isthe leaf illumination angle [16]. In the event thattransmission takes place, the scattering distributionfunction is assumed to be perfectly Lambertian andthe specular reflection fraction is set to zero. Thus,the expression for pΘS

ðθSÞ, above, is valid for both re-flection and transmission and integrates to unit areafor each individually.

A sample bidirectional scattering distributionfunction (BSDF) for a sugar maple leaf (Acer saccar-um), fitted to data measured at an incident zenith

Fig. 6. Illustration of region-to-region propagation in a verticallysegmented tree canopy. If the random propagation distance placesa photon outside the current region, then we propagate the photonto the edge of the next region and select a new propagationdistance. If the random propagation distance places the photonwithin the current region, then the photon is assumed to haveexperienced a leaf interaction.


angle of 110° is shown in Fig. 7. Incident light, shownby the arrow, hits the leaf at 110° (or 70° above thehorizontal axis), while the leaf itself is assumed to beoriented vertically so that its surface normal vector nis parallel to the 0° axis. The BSDF lobe on the left isthen due to reflection and the lobe on the right is dueto transmission. In our previous work, we measuredBSDFs for both sugar maple and eastern cottonwood(Populus deltoides) trees in the local Dayton, Ohio,area and built extensive BSDF models, which wehave used in our Monte Carlo canopy scatteringsimulation [16].

C. Ground and Target Scattering Angles

When photons reach the ground plane, their locationis compared to the locations of the targets. Photonsthat do not hit either of the targets are assumed to bescattered from the ground. The azimuth and zenithground scattering angles are again commonly as-sumed to be statistically independent. In particular,the ground azimuth scattering angle distributionpΦG

ðϕGÞ is assumed to be uniform over the range½0; 2πÞ and can be expressed using Eq. (11) after sub-stituting ϕG for ϕL. The ground zenith scatteringangle distribution pΘG

ðθGÞ is then governed by a sim-ple Lambertian distribution, nominally describingupward scattering in the positive z direction, andcan be expressed using Eq. (12) after substitutingθG for θL [18].Photons that do hit a target are scattered accord-

ing to the PDFs governing the target zenith andazimuth scattering angles. Once again, the two ran-dom angles are considered to be statistically in-dependent. As with ground scattering, the targetazimuth scattering angle distribution pΦT

ðϕTÞ is as-sumed to be uniformly distributed over the interval½0; 2πÞ and the target zenith scattering angle distri-bution pΘT

ðθTÞ is taken to be Lambertian. The PDFspΦT

and pΘTcan then be found by substituting ϕT for

ϕL, and θT for θL in Eqs.(11) and (12), respectively.Recall, however, that both targets are assumed to

be tilted so that their surface normals are pointedback in the direction of the receiver. As a result,the target scattering angles must be transformed

into the canopy coordinate system as previously dis-cussed. We have done this for two reasons, the firstbeing that, in our simulation, we increase the prob-ability of receiving photons back from the targets ifwe nominally tilt them back toward the receiver. Andsecond, in real-world situations, targets will typicallyhave one or more facets that do, in fact, face backtoward the receiver.

3. Test Canopy Parameters

We wished to verify the results of our canopy propa-gation model with real data collected from within anearby grove of trees. Therefore, in order to achievethe most reproducible results, we used real param-eters measured for this tree grove in our simulation.These include the canopy dimensions, the illumina-tion angle geometry, and the leaf-area density.

We illuminated our tree grove using a rudimentarydirect detection ladar system located on the 11thfloor of a tower building at Wright–Patterson AirForce Base, Ohio [2]. Our ladar system employs aRaman scattering Nd:YAG coherent infinity pulsedlaser that operates at a wavelength of 1550nmand produces pulses of ∼1ns duration at a rate of10Hz. We frequency doubled the operating wave-length with a LiNbO3 crystal to 785nm and addedtelescoping optics to control the beam diameter downrange. The detector is a Princeton Instruments Pi-Max2 intensified CCD camera that is synchronizedwith the laser for range gating applications. Usinga differential GPS unit, we logged the locations ofthe primary target site and the ladar system locatedin the tower. We then determined the angle madebetween the two locations, as well as the distance se-parating them. We found that the illumination anglewas approximately 80° and the horizontal distancebetween the tower and the target was 255m.

We thenmeasured the dimensions of the tree groveby walking around the perimeter with a handheldGPS device and periodically logging waypointsaround the edge of the tree line. A best-fit ellipsewas then drawn around the perimeter, with majoraxis and minor axis dimensions found to be 358and 228m, respectively. Conveniently for us, boththe target site and the tower were found to lie alongthe major axis of this ellipse, allowing us to easilyincorporate the actual canopy dimensions into oursimulation.

Next we indirectly computed the maximum leaf-area density Lm by first measuring the gap fractionof our test forest at 80°, and by then finding the max-imum leaf-area density value that would yield thesame statistical probability of a photon passingthrough the canopy without interaction. To measurethe gap fraction, we began by capturing 60 hemi-spheric images, similar to the one shown in Fig. 1,from a 5 × 12 point grid of locations under the canopy.We then applied a threshold to the images, so thatpixels containing leaves and branches were given avalue of unity and those without were given a valueof zero. Each image was then broken into nine equal

Fig. 7. Example BSDF for a sugar maple leaf illuminated at anincidence zenith angle of 110°. The BSDF is uniform in azimuth.


area annuli, whose bin centers have been mapped tononuniformly spaced zenith angles [19]. The numberof zero valued pixels was counted and then divided bythe total number of pixels for each annulus in orderto determine the gap fraction as a function of zenithangle. From the 60 individual images, mean gap frac-tion values were then calculated and plotted at thecenter of each angular bin, as shown in Fig. 8. Theinterpolated gap fraction value corresponding to80° was found to be 11.7e-3.Gap fraction, typically defined as the percentage of

canopy area not covered by leaves, also describes theprobability of encountering a gap at a specific zenithangle. That is, if a photon is launched into the simu-lation at a certain zenith angle, the probability that itwill reach the ground without interaction is equal tothe gap fraction of the canopy at that angle. Becausewemodeled propagation through the canopy as a sta-tistical process, we can, therefore, also calculate theprobability of a photon passing unscattered throughthe simulated canopy as a function of leaf-areadensity.Recall that the leaf-number density varies as a

function of canopy height and that, for our model,we have broken the canopy into 50 regions of con-stant density. Also, recall from Eq. (4) that we modelthe photon propagation distance between scatteringevents using a negative exponential PDF. Therefore,the probability that a photon traveling at zenithangle θ experiences a leaf interaction within thekth region [which has mean free path Dk and upperand lower bounds zk−1 and zk, respectively, as shownin Fig. 6(c)] can be found by first integrating this PDFover the region boundaries and then dividing by anormalization factor according to

Pfzk−1 < z < zkg ¼Rakak−1

1Dk

exp�−

xDk

�dxR

∞

ak−1

1Dk

exp�−

xDk

�dx

; ð18Þ

where ak ¼ zk= cos θ. We normalize the above slantpath integral in order to account for the fact that,if the photon experiences its first leaf interactionwithin the kth region, then it must have alreadypassed through the previous k − 1 regions withoutscattering.The probability that there will be no leaf inter-

action within the kth region is then found by sub-tracting Eq. (18) from unity. Correspondingly, the

probability of not having an interaction within theentire canopy can be found by evaluating the follow-ing product relationship:

Pn0 < z < h

o¼

YKk¼1

2641 −

Rakak−1

1Dk

exp�−

xDk

�dxR

∞

ak−1

1Dk

exp�−

xDk

�dx

375: ð19Þ

We evaluated Eq. (19) for an illumination angle of80° as we varied the maximum leaf-area density. [Re-call that the mean free path is a function of Lm, asgiven by Eq. (16).] The result is shown in Fig. 9,and the equation of the best-fit curve was found to be

Pf 0 < x < h : Lmgj80° ¼ expð−9:0490 · LmÞ: ð20ÞFrom this result, we found that a maximum leaf-areadensity of Lm ¼ 0:6132m2=m3 would yield the sameprobability of an unscattered photon reaching theground as was determined from the gap fractionmeasurements. This value is shown by the arrowin Fig. 9.

4. Simulation Results

Once the canopy parameters from the test forestwere incorporated into our model, we ran the MonteCarlo simulation and collected spatial, temporal, andangular data in both the ground and receiver planes.Over the period of several weeks, we simulated thelaunching of 10 billion photons into the canopy atan illumination angle of 80°. This ensured that a suf-ficient number of photons returned back to the detec-tor plane, allowing us to create smooth PDFs. Datafor all photons that struck the ground plane wereconsidered for the ground plane analysis, while onlythose photons that returned to the detector, whetherthey hit the ground or not, were considered in thereceiver plane data analysis.

We first divided the virtual ground plane into agrid of 100 × 100 rectangular bins (i.e., the rangeand cross-range bin dimensions were 3.58 and2:28m, respectively) and created temporal wave-forms, such as the one shown in Fig. 10, from the si-mulated photons arriving within a 45° field of view.This field of view was selected to match that of thedetector used in our experiment, as we will discuss

Fig. 8. Mean gap fraction as a function of zenith angle for ournearby grove of sugar maple trees.

Fig. 9. Probability of an unscattered photon reaching the groundas a function of Lm for a zenith illumination angle of 80°. The arrowdepicts the location where the probability of an unscattered photonis equal to the measured gap fraction of the canopy.


in Section 5. These waveforms are generally charac-terized by a sharp peak due to first surface scatteredphotons, followed by a slowly decaying tail due to thearrival of multiply scattered photons. We then calcu-lated the root mean square (RMS) pulse width ofeach waveform and also calculated the integratednumber of photons falling within each bin. Thephotons-per-bin values were then normalized withrespect to the area of each bin lying within the vir-tual canopy’s elliptical footprint.A contour plot of the integrated and normalized

number of photons arriving within each bin (i.e.,the simulated scattered beam footprint), is shownin Fig. 11. For clarity, this figure has been scaledto a peak value of unity and cropped to highlightthe region of greatest interest. Note that all photonsare initially propagating in the positive range (x) di-rection and are aimed at the primary target locatedat the coordinate system origin at the top of the fig-ure. The center of the large spot on the ground is thenpositioned beneath the location where photons firstenter the canopy (i.e., at ðx; yÞ ¼ ð−96:66; 0Þ). Recallfrom Fig. 5 that the majority of leaves, facing nomin-ally upward, will be distributed near the top of thecanopy. Therefore, most photons are scattered down-ward just after entering the canopy, which, in turn,causes the dominant beam footprint to be found be-neath the entrance location. This suggests that, inpractice, it might be prudent to illuminate the can-opy directly above an anticipated target, rather thanat an oblique angle.

We then integrated the beam footprint data acrossboth the range and the cross-range dimensions, indi-vidually, and calculated a best-fit Gaussian curve todescribe the results. Figure 12 contains the data(dots) and the best-fit Gaussian curves (solid curve)for both the (a) range and (b) cross-range dimensions.The standard deviations of the best-fit curves (i.e.,the RMS ground spot dimensions) were found tobe 17:28m for the range dimension and 14:52mfor the cross-range dimensions, with RMS fittingerrors of 9.1e-5 and 2.0e-4, respectively.

Next, we created a contour plot of the RMS tempor-al pulse width (i.e., temporal dispersion) as a func-tion of spatial location, as shown in Fig. 13, wherethe gray-scale legend has units of nanoseconds. Var-iations in path length lead to a broad spectrum oftransit times through the canopy. Therefore, wave-forms measured near the canopy entrance locationhave smaller pulse widths than do those measuredfurther away. Notice, for example, in Fig. 13, thatthe dispersion has a minimum directly beneaththe entrance location and increases as the distancefrom this spot increases.

Additionally, we examined the temporal data forphotons incident upon a 25 cm diameter virtual de-tector pupil located in the receiver plane. The one-dimensional (1-D) arrival time PDF for all photonshitting the pupil is shown in Fig. 14. The initial sharp

Fig. 10. Characteristic temporal waveform based upon thosesimulated photons that strike the virtual canopy floor.

Fig. 11. Simulated ground plane beam footprint for an illumina-tion angle of 80°.

Fig. 12. Simulated beam footprint cross sections and best-fit Gaussian distributions in the (a) range and (b) cross-rangedimensions.

Fig. 13. Spatial distribution of simulated RMS pulse widths onthe virtual canopy floor.


rise in the first peak is due to the influx of photonsbackscattered from high within the canopy, while theslowly decaying tail results from multiply scatteredphotons. Photons that first hit the secondary targetunder the canopy entrance location also arrivewithin the initial peak, but are inseparable from thecanopy backscatter noise. Additionally, there is anarrow, sharp peak found near an arrival time of1:7 μs, which corresponds to those photons thatpropagate to and from the primary target withoutscattering by leaves. These “ballistic” photons arescattered from the primary target back toward thereceiver pupil with very little deviation (i.e., a verynearly retroreflection). The temporal delay betweenthe initial peak and the unscattered peak exists be-cause photons that propagate to the primary targetand back travel a much greater round-trip distancethan do those that are down scattered after enteringthe canopy.

5. Experimental Verification

To validate our model, we collected waveform datafrom a nearby stand of sugar maple trees duringthe summer of 2008. We performed waveform mea-surements by illuminating the canopy at an approxi-mately 10° angle above the horizon (i.e., 80° zenithangle) and then sampling the beam footprint onthe ground with a grid of avalanche photodiodes(APDs). We sent two beams from the tower, one at785nm and the other at 1550nm.We established a reliable trigger signal by placing

a 1m diameter, high-reflectance Spectralon target ina clearing so that it was clearly visible from thetower. This we illuminated with the 1550nm beam.An amplified PIN diode was then stationed 0:5m infront of the Spectralon target and aimed in such a

way that the entire target fell within the field of viewof the detector. We then relayed the trigger signalthrough a 500m coaxial cable to an oscilloscope sta-tioned under the canopy. Keeping the trigger signalat a constant location allowed synchronization of thetemporal delays between waveforms measured atdifferent grid point locations.

We then illuminated the tree grove with the785nm beam, which, due to diffraction, had a widthof about 10m at the top of the canopy. This beam wasdirected toward a reflexite (primary) target locatednear the center of the grove. Note that, althoughour model uses maple leaf BSDFs measured at1064nm, the scattering functions of these leavesare expected to be very similar under 785nm illumi-nation [20]. We then cleared an 88m path from theprimary target toward the tower, with 12 additional16m paths cleared orthogonal to the beam propaga-tion axis and spaced in uniform 8m intervals. Alongthese paths, we then established five uniformlyspaced measurement locations to create a 12 × 5measurement grid. We observed that the beamappeared to enter the canopy roughly above the11th transverse path, approximately 80m from theprimary target.

We then measured the waveforms at each gridlocation with a Hamamatsu silicon APD, whichwas mounted on a tripod, aimed directly upward,and moved from point to point throughout the test.This detector had a field of view of approximately45°. We then relayed the detected signals to the os-cilloscope along with the trigger signal. At each gridpoint we collected 256 pulses, which we averagedinto a single mean waveform and stored for later pro-cessing. A sample mean waveform captured near theentrance location of the canopy is shown in Fig. 15.There are two large, narrow spikes at the front edgeof the pulse, which we attribute to the direct reflec-tion from some unidentified scatterer, followed by aslowly decaying tail, as was seen in the pulse (Fig. 10)generated in our simulation. In general, these cap-tured waveforms have more structure than the simu-lated waveforms, presumably due to the clumpingand clustering of leaves and branches. However,the general shape of the measured waveforms is con-sistent with that of the simulation.

The normalized beam footprint, created from theintegrated energy contained within each waveform,is shown in Fig. 16. Note that all photons are initiallypropagating in the positive direction along the range(x) axis and are aimed at the primary target locatedat the coordinate system origin. As predicted by thesimulation, there is a spot on the ground centerednear the entrance location (i.e., at ðx; yÞ ¼ ð−76:0Þ)due to the large amount of down scattering just afterphotons enter the canopy. It appears that there was aslight misalignment between the path we clearedand the beam trajectory, however, as the spot onthe ground is seen to be sloping off a bit to the rightside in the plot. We attribute this to our cleared pathbeing skewed slightly with respect to the actual

Fig. 14. Simulated 1-D temporal PDF measured in pupil plane ofthe virtual detector.

Fig. 15. Characteristic temporal waveform measured by anupward-looking APD placed on the canopy floor.


beam trajectory. Additionally, there appears to be asmall discrepancy in our illumination angle, as weexpected the entrance location to be approximately96:6m in front of the target. At the primary targetrange we are considering, however, this correspondsto an illumination angle error of only 1:10°.We then integrated the measured beam footprint

data across both the range and cross-range dimen-sions, individually, and calculated a best-fit Gaussiancurve to describe the results. Figure 17 contains thedata (dots) and the best-fit Gaussian curves (solidcurvesw) for both the (a) range and the (b) cross-range dimensions. The standard deviations of thebest-fit curves were found to be 12:31m for the rangedimension and 8:79m for the cross-range dimension,with RMS errors of 1.16e-4 and 5.01e-3, respectively.To a first order, these data closely match the pre-dicted beam footprint, shown in Fig. 12, in bothshape and size, with discrepancies likely due tothe fact that our simulated data represents an aver-age over billions of canopy realizations, whereas ourexperimental data arises from only one actual can-opy realization.Next, the temporal dispersion, or RMS pulse

widths of the detected signals, is shown in Fig. 18,

where the gray-scale legend has units of nanose-conds. As predicted by our simulation, the measureddispersion has a minimum directly beneath the en-trance location and increases as the radial distancefrom this spot increases. The misalignment betweenour cleared path and the beam trajectory can also beseen in this figure. Once again, discrepancies be-tween our simulated and experimental dispersionare likely due to the fact that our simulated data re-presents an average over billions of canopy realiza-tions, whereas our experimental data arises fromonly one actual canopy realization.

Finally, from the tower, we collected a series of 511range-gated images of the entire illuminated stand oftrees using a gate width of 2ns and a gate delay se-quence ranging from 1.20 to 1:99 μs. For each frame,we set a lower threshold to reject detector noise andan upper threshold to eliminate any high-intensityfirst surface reflections. Then, by integrating the pix-el values within each thresholded frame, we createda 1-D PDF for the photon transit time through thecanopy. This is shown in Fig. 19. The shape of thisplot is very similar to that of the simulated PDF,shown in Fig. 14. Both are characterized by an initialpeak due to the high density of leaves near the top ofthe canopy, each with a width of approximately100ns. Following the initial peak is a period of verylow return, as the few photons that pass through thetop of the canopy experience mostly free-space propa-gation until they reach the target. Finally, there is

Fig. 17. Measured beam footprint cross-sectional data andbest-fit Gaussian curves in the (a) range and (b) cross-rangedimensions.

Fig. 18. Spatial distribution of actual RMS pulse widths mea-sured on the canopy floor.

Fig. 19. Actual 1-D temporal PDF measured with a range-gatedintensified CCD camera located in the tower.

Fig. 16. Measured ground plane beam footprint for an illumina-tion angle of 80°.


a sharp peak around 1:75 μs, corresponding to thereturn of ballistic photons that strike the targetand are retroreflected back toward the receiver.

6. Summary and Conclusions

We have presented a Monte Carlo model used tosimulate the propagation and scattering of lightthrough a dense tree canopy. The model is character-ized by the PDFs governing several parameters, suchas the leaf-number density, the angular orientationof leaves, and the bidirectional scattering distribu-tion functions of individual leaves. We then mea-sured the physical dimensions and the leaf-areadensity of a nearby grove of trees, as well as the angleand distance between our tower and the target loca-tion within this tree grove. We then applied thesevalues to the model.We ran our simulation for 80° illumination and ex-

amined the expected beam footprint and pulse widthprofile on the ground, as well as the temporal returnswe would expect to measure at the receiver. Then, inorder to validate the model, we illuminated our treegrove with a 785nm beam. We measured waveformson the canopy floor with a 5 × 12 grid of upward-facing APDs and collected range-gated images usingan ICCD camera located in the tower.As predicted by the simulation, we experimentally

verified that a large number of photons are downscattered just after entering the canopy, creating alarge spot on the ground beneath the entrance loca-tion. We then found that a Gaussian distribution fitthe range and cross-range cross sections of the simu-lated and measured beam footprints with great accu-racy. We also found that the standard deviations ofthe best-fit Gaussian distributions were of the sameorder in both simulation and experiment. Addition-ally, we observed that the pulse widths of the wave-forms measured on the ground were similar in bothshape and magnitude to those predicted by our simu-lation. Finally, we found that the measured 1-D tem-poral PDFof photons returning to the receiver closelymatched the PDF predicted by the simulation.We believe that any mismatch between our simu-

lation and experimental results can be attributed totwo factors. First, in the experiment, there was aslight misalignment between the trajectory of the in-itial beam axis and the path cleared in the tree grove,which resulted in a small angular error in the point-ing of our beam. Second, and most importantly, thesimulation results represent an average over billionsof realizations of the canopy, while the experimentconsiders only one real forest realization. Consider-ing these factors, we believe our experimental resultscorrelate very well with our simulation, leading us toconclude that our model is valid.

Appendix A

Nearly all technical computing software has a built-in function to generate uniformly distributed randomvariables between 0 and 1. Often, however, we de-sire to select specific values of a random variable

described by a nonuniform PDF. The procedure fordoing so is actually quite simple.

In general, consider a random variable Y that canbe described as a function of another random vari-able X [i.e., Y ¼ f ðXÞ]. If X is a random variable uni-form on ½0; 1Þ, and if we know the PDF pYðyÞ for Y,then, for any given randomly chosen value of X,the corresponding random value selection for Y isgoverned by the relationship

y ¼ f ðxÞ ¼ F−1Y ðxÞ; ðA1Þ

where x and y represent specific values of the randomvariables X and Y , respectively, and where FyðyÞ,known as the cumulative distribution function(CDF) of Y, is the indefinite integral of f YðyÞ [21].

This effort was supported in part by the U.S. AirForce and General Dynamics, of Dayton, Ohio,through contract F33601-02 F-A581, and by theLadar and Optical Communications Institute (LOCI)at the University of Dayton.

References1. R. M. Marino and W. R. Davis Jr., “Jigsaw: a foliage-

penetrating 3D imaging laser radar system,” Lincoln Lab.J. 15, 23–36 (2005).

2. B. W. Schilling, D. N. Barr, G. C. Templeton, L. J. Mizerka,and C. W. Trussell, “Multiple-return laser radar for three-dimensional imaging through obscurations,” Appl. Opt. 41,2791–2799 (2002).

3. J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges,and W. Austin, “Advanced 3D polarimetric flash ladarimaging through foliage,” Proc. SPIE 5086, 84–95 (2003).

4. J. Liu, R. A. Melloh, C. E. Woodcock, R. E. Davis, and E. S.Ochs, “The effect of viewing geometry and topography ongap fractions through forest canopies,” Hydrol. Process. 18,3595–3607 (2004).

5. S. Jacquemoud and S. Ustin, “Leaf optical properties: a stateof the art,” in 8th International Symposium Physical Measure-ments and Signatures in Remote Sensing (CNES, 2001),pp. 223–232.

6. T. W. Brakke, “Specular and diffuse components of radiationscattered by leaves,” Agric. Forest Meteorol. 71, 283–295(1994).

7. E. A. Bucher, “Computer simulation of light pulse propagationfor communication through thick clouds,”Appl. Opt. 12, 2391–2400 (1973).

8. C. Kittel and H. Kroemer, “Collision cross sections and meanfree paths,” in Thermal Physics (Freeman, 1980), pp. 395–397.

9. E. F. Gilman andD. G.Watson, “Acer saccharum sugarmaple,”Fact Sheet ST-51, Environmental Horticultural Department,Florida Cooperative Extension Service, Institute of Food andAgricultural Sciences (1993).

10. R. B. Myneni, J. Ross, and G. Asrar, “A review of the theory ofphoton transport in leaf canopies,” Agric. Forest Meteorol. 45,1–153 (1989).

11. M.A.KaramandA.K.Fung, “Acanopyscatteringmodeland itsapplication to a deciduous forest,” in International Geoscienceand Remote Sensing Symposium (IEEE, 1990), pp. 137–140.

12. D. S. Falster and M. Westoby, “Leaf size and angle vary widelyacross species: what consequences for light interception?,”New Phytol. 158, 509–525 (2003).


13. M. Abramowitz and I. A. Stegun, “Laplace transforms,” inHandbook of Mathematical Functions with Formulas, Graphsand Mathematical Tables (Dover, 1972), pp. 1020–1021.

14. J. Ross, V. Ross, and A. Koppel, “Estimation of leaf area and itsvertical distribution during growth period,” Agric. ForestMeteorol. 101, 237–246 (2000).

15. B. Lalic and D. T. Mihailovic, “An empirical relation describingleaf-area density inside the forest for environmental model-ing,” J. Appl. Meteorol. 43, 641–645 (2004).

16. M. Greiner, B. Duncan, and M. Dierking, “Bidirectional scat-tering distribution functions of maple and cottonwood leaves,”Appl. Opt. 46, 6485–6494 (2007).

17. T. W. Brakke, “Goniometric measurements of light scat-tered in the principle plane from leaves,” in International

Geoscience and Remote Sensing Symposium (IEEE, 1992),pp. 508–510.

18. J. L. Privette, R. B. Myneni, W. J. Emery, and B. Pinty, “Inver-sion of a soil bidirectional reflectance model for use withvegetation reflectance models,” J. Geophys. Res. 100,25497–25508 (1995).

19. S. F. Ray, “The fisheye lens and immersed optics,” in AppliedPhotographic Optics (Focal, 2002), pp. 327–332.

20. L. Claustres, Y. Boucher, and M. Paulin, “Wavelet-based mod-eling of spectral bidirectional reflectance distribution functiondata,” Opt. Eng. 43, 2327–2339 (2004).

21. S. L. Miller and D. Childers, “Operations on a single randomvariable,” in Probability and Random Processes (Elsevier,2004), pp. 100–107.


Documents

Monte Carlo simulation of multiple photon scattering in sugar maple tree canopies