IEEE TRANSACTION MANUSCRIPT 1 First Principles Simulation ... · Simulation of a ICESat-2-like lidar system consisting of mod-els for the laser transmitter and multiple pixel PMT

IEEE TRANSACTION MANUSCRIPT 1

First Principles Simulation of SpaceborneMicropulse Photon-Counting Lidar Performance on

Complex SurfacesJiashu Zhang and John Kerekes, Senior Member, IEEE.

Abstract—To advance the science of lidar sensing of complexsurfaces as well as in support of the upcoming ICESat-2 mission,this paper establishes a framework that simulates the perfor-mance of a spaceborne micropulse photon-counting detectorsystem on a complex surface. A first principles 3-D Monte Carloapproach is used to investigate returning photon distributions.The photomultiplier tube (PMT) detector simulation takes intoaccount detector dead-time and multiple pixels based on the latestICESat-2 design, as well as photon detection efficiency (PDE)for probabilistic modeling. To explore system behavior, Fouriersynthesis is introduced to create a synthetic surface based onparameters derived from a real data set. A radiometric modelusing bidirectional reflection distribution functions (BRDF) isalso applied in the synthetic scene. Such an approach allowsthe study of surface elevation retrieval accuracy for landscapeswhich have different shapes as well as reflectivities. As a casestudy, returning photon detection on an example snow surfaceis explored. Based on the simulation results for lidar sensingon synthetic complex surfaces with an elevation range of 10 macross the scene, the spaceborne photon-counting lidar systemconsidered here is seen to have a derived elevation bias of upto 2 cm and a error standard deviation of 10 cm. Furtherstudy on multiple pixel PMT performance for complex surfacesdemonstrates that a less rough surface will result in higheraccuracy, and a surface with a smaller diffuse albedo will resultin smaller bias.

Index Terms—ICESat-2, Lidar, complex surface, PMT.

I. INTRODUCTION

IN RECENT YEARS satellite and aircraft have been pro-viding observations on the remarkable changes in polar ice

sheets [1][2][3]. These changes, including ice loss from icesheets and rapid declines in Arctic sea ice, could contributea large part to sea level rise and affect global climate change[4]. Hence, accurate knowledge of surface elevation is requiredto monitor the amount of ice sheet balance and sea levelchange [5][6]. To serve that purpose, the Ice, Cloud and landElevation Satellite (ICESat) was launched by NASA in 2003.Since then, by providing data on a global scale, ICESat hasmade great contributions on understanding ice sheets [7][8][9].The successor of ICESat, the Ice, Cloud and land ElevationSatellite-2 (ICESat-2) is currently scheduled for launch in2016 [10]. ICESat-2 is designed to provide elevation data

J. Zhang is with the Chester F. Carlson Center for Imaging Science,Rochester Institute of Technology, Rochester, NY, 14623 USA e-mail:[email protected].

J. Kerekes is with the Chester F. Carlson Center for Imaging Science,Rochester Institute of Technology, Rochester, NY, 14623 USA e-mail:[email protected].

Manuscript received April, 2013

to determine the temporal and spatial change of ice sheetelevation as well as sea ice freeboard. It is also intended tomeasure land topography and vegetation characteristics [4].These objectives will be achieved through the use of theAdvanced Topographic Laser Altimeter System (ATLAS) onboard ICESat-2, which employs 532 nm micropulse photon-counting detection.

In waveform laser altimeters, such as the Geoscience LaserAltimeter System (GLAS) on board ICESat, multiple pho-tons reflected from anywhere within the illuminated spot arerecorded by a waveform digitizer and deconvoluted usingcomplex algorithm in order to obtain a single range mea-surement [11]. As a result of this design, spacecraft primepower typically restricts spaceborne operations to low repe-tition rates, which limits the along-track spatial sampling toone sample every few hundred meters. Therefore, it appearsto be not practical to obtain higher along-track resolutionusing simple scaling of the laser. However, it has beentheoretically demonstrated that spaceborne lidar performancecan be enhanced when operating in a photon-counting mode[12]. Photon-counting lidars increase the surface return rateby emitting laser pulses in a high frequency (∼ kHz) trainand employing single photon detection. This improvementthen enables photon-counting topographic lidars to providedense along-track sampling as well as centimeter level rangingresolution. Previous work has demonstrated the practicality ofairborne photon-counting laser altimetry [13]. One exampleis a compact 3D photon-counting imaging lidar operated atrates up to 22 kHz [14]. It uses a 532 nm laser pulse toproduce a 100 pixel volumetric 3D image with 10×10 pixelmultiple stop detector, as well as a 1064 nm infrared laser.Due to the use of a Diffractive Optical Element (DOE) tosplit the laser beam into a 10x10 array of beamlets, theactual surface measurement rate for this lidar could reach 2.2million multistop 3D pixels per second. Another example is theMultiple Altimeter Beam Experiment Lidar (MABEL) laseraltimeter on board NASA’s high-flying ER-2 airborne scienceaircraft [15]. MABEL has used a photon-counting detector tocollect data over the Greenland ice cap and surrounding seaice fields.

Because of their high efficiency, photon-counting lidarsscale much more easily to orbital altitudes than conventionalmultiphoton lidars. There has been significant scientific inter-est in estimating the performance of such lidar systems onelevation retrieval. Previous work has addressed the impact ofclouds on surface altimetry from spaceborne photon-counting

IEEE TRANSACTION MANUSCRIPT 2

lidars [16][17]. It was demonstrated that cloud attenuationlowers the average number of arriving photons, and the cloudforward scattering makes the surface appear further away fromsatellite. However, the performance of spaceborne photon-counting lidars and the accuracy of elevation retrieval oncomplex surface still remains uncertain. To advance scientificunderstanding of theory behind topographic lidar sensing aswell as in support of the ICESat-2 mission, a model whichsimulates the behavior of spaceborne photon-counting lidarsystem on complex surfaces has been pursued in our presentwork.

In this paper, a framework is established to theoreticallyexplore performance of a spaceborne micropulse lidar systemwith photon-counting detectors. Within this framework thebehavior of a multiple channel photomultiplier tube (PMT)receiver is studied, as well as its performance on surfaceelevation, roughness, and diffuse albedo. Different surfaces,including flat surfaces, slopes and complex scenes, are testedusing a first principles simulation to evaluate system perfor-mance. The impact of sub-beam terrain characteristics on theretrieved elevation is also discussed.

The rest of this paper is organized into the following parts.First the framework is established for modeling spacebornephoton-counting lidars consisting of the laser transmitter andreceiver. Then based on parameters derived from a real dataset, complex surfaces are produced using frequency synthe-sis, and attributed with bidirectional reflection distributionfunctions (BRDFs) implementation. Finally, returning photondetection and quantitative elevation retrievals for ICESat-2-like lidar sensing are obtained. Performance of a spacebornephoton-counting lidar system is evaluated for flat surfaces,slopes and complex surfaces, including its accuracy of eleva-tion retrieval. At last, the results are summarized and discussedin the conclusion.

II. PHOTON-COUNTING LIDAR MODELING

Like most remote sensing lidars, the detector on boardICESat-2 will measure the time-of-flight of reflected energyto derive surface elevation. ICESat-2 also adopts a high-pulse-repetition laser with dense along-track sampling andphotomultiplier tube (PMT) technology with single photonsensitivity. With this significant improvement, ICESat-2 willhave the ability to retrieve the elevation of underlying terrainwith high along-track resolution. In this section, details onspaceborne photon-counting lidar modeling will be discussed.Simulation of a ICESat-2-like lidar system consisting of mod-els for the laser transmitter and multiple pixel PMT receiveris addressed.

A. Transmitter

ICESat-2’s laser transmitter source uses a Master OscillatorPower Amplifier (MOPA) that emits 532 nm micropulse lightwith 10 kHz pulse rate. Transmission through a lens array andreflection off the primary mirror toward the earth’s surfaceyields six footprints. These beams are grouped into 3 sets oftwins with 2.5 km spacing between twin beams and 3.3 kmspacing between sets. Each set of twin beams follow nominally

the same groundtrack with a seperation distance of 90 m.Meanwhile, each twin set contains two different energies withthe strong beam having 4 times the power of the weak one.

ICESat-2’s nominal orbital altitude is 496 km. As can beseen in Fig. 1, for each laser firing, each photon packetemitted by the laser transmitter is spatially modeled as circularGaussian with a 1/e2 diameter of 20.2 microradians, or 10 mon the ground, which yields a σ of 2.5 m. For a given ICESat-2 footprint, the location of the footprint center is known, butthe point of origin of any recorded photon within a footprintwill not be known and is assumed to be random. As such,all received photons are effectively collapsed in space to thefootprint center [18]. Meanwhile, the temporal shape of laserphotons is modeled with Gaussian distribution, with a 1 nsFull Width Half Maximum (FWHM) pulse width. Laser along-track sampling is 0.7 m based on the latest design of ICESat-2.For the reason of simplicity, the simulation only considers onelaser beam and the underlying terrain it reaches, rather thanthe six beams that will be the case for ICESat-2. However, thedifferences between strong and weak spot will be considered.

1.pdf

Transmitter Receiver

Multiple pixel PMT

Spatially

Temporally

Fig. 1. Illustration of photon-counting lidar modeling. The transmitter emitslaser pulse at a high frequency of 10 kHz, which enables a dense along-tracksampling of 0.7 m. The nominal spatial distribution of laser energy withineach spot is a circular Gaussian with a 1/e2 diameter of 10 m, and temporaldistribution of laser is also modeled as Gaussian with 1 ns FWHM pulsewidth. Meanwhile the receiver on board satellite uses a multiple pixel PMT.

B. Receiver

In the ATLAS receiver a telescope and aft optics form animage of the earth surface in a focal plane, where it consists of6 fibers for each individual field of view of laser beam. Eachfiber captures the signal and background light and carries it tothe Optical Filter Assembly (OFA). The OFA rejects most ofthe light collected by the receiver, passing on to the DetectorArray Assembly (DAA) only that light within a 30 picometerbandwidth around the laser wavelength. The detector optics (6sets) take light from each of the 6 fibers coming from the OFAand re-forms it to match the geometry of the detectors [19].Note that the receiver Instantaneous Field of View (IFOV) is

IEEE TRANSACTION MANUSCRIPT 3

83.3 microradians, which yields a circle with a diameter of41.3 m on the ground.

The detectors are segmented-anode photomultiplier tubes(PMTs). Each PMT has a 4×4 segmented anode in whicheach segment can be regarded as an independent detector.Light carried through DAA will spread evenly across eachPMT. For each PMT for a strong spot, the signal from eachanode segment is processed independently, resulting in up to16 digital outputs. Meanwhile, for each PMT for a weak spot,the signals from groups of 4 anode segments are summedbefore the discrimination function, resulting in up to 4 digitaloutputs.

Each photon incident on a detector results, with a certainprobability, in a digital pulse coming out of the detectorelectronics. The photon detection efficiency (PDE) is assumedto be 50%, which represents the triggering probability forarriving photons. Meanwhile, the PMT timing jitter (variationin delay between the absorption of a photon and the generationof an output electrical pulse) can affect the result of lidarelevation ranging. However, this effect will not be consideredin our simulation. In addition, not all the light will be spreadevenly across the PMT. Therefore, a binomial probabilityrepresenting fill factor is combined to simulate the reductionof photons in the process. In the rest of this paper, a PDE=50%and fill factor=80% are utilized unless pointed out explicitly.

To simulate the behavior of the ATLAS receiver, a prob-abilistic model is used to calculate the number of photonshitting the detector and triggering a current pulse and formingdigital pulses. As a stochastic process, the number of pho-tons arriving at the detector can be modeled with a Poissondistribution:

pk(λ) =λk

k!e−λ (1)

where pk(λ) is the probability to have k arriving photons whenthe average number is λ. Based on the ATLAS radiometrymodel for ice sheets and glaciers [20], the mean receivedphotoelectrons per shot is set as 2.04 for a weak spot and8.17 for a strong spot.

Due to detector dead time, once triggered the photon-counting detector will not register any additional arrivingphotons until after a period of time, typically 3 ns. Thus, thederived surface elevation will be biased toward the photonwhich arrives early, and this will make the surface appearhigher than reality [16]. This effect is called first photonbias. To mitigate the bias created by dead time, multiplepixels are utilized in the PMT. The detector is then ableto record more signals since these pixels can independentlyregister returning photons and the returning photon distributioncan be used to retrieve the surface elevation. Compared to asingle pixel detector, multiple channel designs quantitativelyimprove detection accuracy on elevation retrieval, as will bedemonstrated in detail in the Results and Discussion section.

For a satellite mission, noise such as solar scattered photonsand detector dark current will also be recorded by the receiver.For simplicity, a clear sky is assumed in our simulation. Sincethe objective of this study is to investigate detectability ofspaceborne photon-counting lidar systems on complex surface

and the accuracy on elevation retrieval, only photons reflectedby the surface are simulated at the receiver in this paper. Inother words, the simulation below assumes an algorithm willbe applied in data analysis that separates surface returns fromnoise returns. Therefore, the impact of noise and atmosphericscattering in ICESat-2-like system modeling will be not beconsidered here, but will be studied in future work.

C. Elevation retrieval statistics

To evaluate the performance of spaceborne lidar systems, aprecise definition for the surface elevation is thus required inour framework. In our simulation, the recorded time betweenlaser firing and photons arriving at the detector is then trans-lated into apparent surface elevation zi based on the speedof light. Therefore, the data train consisting of the retrievedelevation for each returned photon is achieved. For a givenlidar footprint, the location of the footprint center (xi, yi) isknown, but the point of origin of any recorded photon withina footprint will not be known. Therefore, the output is givenas a 2-D projection of the cloud of single-photon reflectionsversus along-track distance of a ground track [18].

Here we define “reference elevation” zr as the Gaussianweighted average of a circle area within the laser transmitterbeam width for each laser shot. Obviously, for horizontal flatsurfaces, the reference elevation is always a constant. Forsloped surfaces, the reference elevation denotes the altitudefor the laser beam center on ground. When it comes tocomplex surfaces, the reference elevation will vary along track.Then the accuracy of the lidar derived elevation is determinedby comparing the retrieved elevation zi with the referenceelevation zr [21]. Statistically, the mean and standard deviationfor differences between the retrieved and reference elevationwill be computed. Hereafter, we refer to the mean differenceas “elevation bias.”

III. MODELING OF SYNTHETIC SCENES

In our framework, the simulation for the aforementionedspaceborne micropulse lidar system requires a synthetic sur-face. In this section, different types of surfaces will be createdand tested: horizontal surface, sloped surface, and complexsurface. The horizontal and sloped surfaces are trivial toconstruct, but the complex surfaces used in this researchrequire some explanation. In this section, the algorithm usedfor scene synthesis will be introduced and some examplecomplex surfaces will be produced, as well as including adescription of the implementation of the surface reflectiveproperties.

A. Algorithm for scene synthesis

Our objective in this work is to study the photon returnsfor a realistic complex surface. The art of scene synthesishas progressed from line drawing to shaded polygon tilingto fractal surfaces [22]. Among these methods, fractal ter-rains [23] represent a major step in creating natural-lookinglandscapes. Landscapes are regarded as self-similar becausethey demonstrate the same characteristics at different scales.

IEEE TRANSACTION MANUSCRIPT 4

This self-similarity can be characterized by power law filteringwhich introduces correlation over a wide range of scales [22].Voss [24] presented a method of generating fractal scenes byfiltering a background consisting of white noise with a 1/f filterin the spatial frequency domain. In our framework, a similarapproach, which is called frequency synthesis, will be utilizedto create a synthetic scene.

1) Frequency synthesis: First the two dimensional whitenoise w(x, y) is created on a N × N grid. Then it is trans-formed into the spatial frequency domain using the DiscreteFourier Transform (DFT):

W (u, v) =1

N2

N−1∑x=0

N−1∑y=0

w(x, y)e−i2π(ux/M+vy/N) (2)

The next stage is to scale each value with a 1/fp filter, wherep controls how rough or smooth is the surface. This filter isapplied to both the real and imaginary values of the harmonicsin the frequency domain:

H(u, v) =W (u, v)

(u2 + v2)p/2(3)

Finally each value in the frequency domain is transformedto the spatial domain. The surface elevation map h(x, y) isachieved using the inverse DFT:

h(x, y) =

N−1∑u=0

N−1∑v=0

H(u, v)ei2π(ux/M+vy/N) (4)

Note that as the Fourier method assumes periodic bounds, theelevation map can tile perfectly. Therefore, a large area canbe produced using this approach.

2.png

Fig. 2. Plot of resampled ATM data for a 860 m segment collected in northernGreenland (76◦46′48′′N, 66◦12′0′′W). Arbitrarily one corner is moved tocoordinate origin by applying an offset for each point.

Obviously, a good approximation of p is important for re-alistic scene simulation using this method. Here the simulatedterrain is compared to an empirical model based on an airbornetopographic mapper (ATM) data set [25] to estimate realisticvalue for p. Our work will use a resampled ATM data set

[26], containing ice sheet elevation data collected in northernGreenland by ATM operated in a profiling mode on a NASAP-3 aircraft. An example point cloud for a 860 m segment(512 sampling points) is shown in Fig. 2. This data set hasapproximately 1.6 meter along-track sampling, and the flighttrack is almost a straight line.

3.png

Fig. 3. Simulated power spectrum with comparison to that of the ATM dataset. p is the parameter used in 1/fp filter, here the simulated curve is themultiplication of 1/fp filter and white noise background.

Comparing a fractal model simulated power spectrum withthat of the ATM data set, it is shown in Fig. 3 that pbetween 1.5 and 2.0 will fit the empirical power spectrum best.Therefore, a synthetic surface model similar to an empiricalscene can be created using an appropriate p value.

2) Example results for synthetic scenes: As can be seenbelow in Fig. 4, two scenes (1024×1024 m) are created usingfrequency synthesis. The total elevation range is approximately10 m, which is comparable to the range in the ATM dataset. One of the benefits of using a 1/fp filter is that p canbe adjusted to create scenes with similar shape but differentroughnesses. Visually, a larger p results in a smoother surface.

4-a.png(a) p=1.6

4-b.png(b) p=2.0

Fig. 4. Synthetic surfaces created using different p values in 1/fp filter.

To quantify the effect of p on the surface roughness, a curve

IEEE TRANSACTION MANUSCRIPT 5

of root mean square (RMS) height (σ) [27] for sample along-track profiles versus p is plotted in Fig. 5. For a discrete one-dimensional surface profile consisting of N points with heightzi, the RMS height (σ) is calculated as:

σ =

√√√√ 1

N[

N∑i=1

z2i −Nz2] (5)

where,

z =1

N

N∑i=1

zi (6)

It is shown that a larger p value represents a less roughsurface, which is consistent with the surface visualizationshown in Fig. 4.

5.png

Fig. 5. Curve of root mean square (RMS) height (σ) versus p for syntheticsurface studied.

B. BRDF implementationModeling of the radiometry using synthetic surfaces re-

quires assignment of reflective properties to the surface ma-terials. This can be achieved by using the function whichdetermines how reflected radiance is distributed in terms ofincident irradiance [28], which is the BRDF. Here a quickand simple way introduced by Ward [29] is utilized.

As an empirical model, the Ward BRDF has several advan-tages over prior BRDF models and has become widely used inthe computer graphics community [30]. It is easy to controlbecause only a few parameters are used in this model. TheWard BRDF consists of two main components:

fr(i,o) = fd(i,o) + fs(i,o) (7)

The diffuse term fd(i,o) can be simply described as:

fd(i,o) =ρdπ

(8)

where ρd controls the reflectivity. The specular componentfs(i,o) has a Gaussian anisotropic gloss lobe defined as shownbelow [30]:

fs(i,o) =ρs

4παxαy√

(i · n)(o · n)

exp(− (h · x/αx)2 + (h · y/αy)2

(h · n)2) (9)

Here, i, o and n denote incident, out and normal directionvectors, respectively. Half direction h is defined to lie midwaybetween i and o. ~x is a unit vector in the surface slope while~y is a unit vector in the surface plane perpendicular to ~x. Thematerial properties are given by the specular reflectance ρs andthe roughness values αx and αy that characterize the standarddeviation of the surface slopes in the perpendicular directions~x and ~y.

In our simulation, we simply assume that the lobe isisotropic, which makes αx = αy . As the ATM data were col-lected in northern Greenland, the surface material is modeledas snow. Previous work has shown that the snow reflectance isclose to Lambertian, while the largest reflectance is observedin the forward scattering direction, particularly at large viewingangles [31]. Therefore, the Ward BRDF will be used to sim-ulate that characteristic. Based on review of previous papers[32][31], it is selected that ρs = 0.05 and αx = αy = 0.3,respectively. Meanwhile, a range of diffuse albedo ρd (from0.6 to 0.9), will be tested later so that the surface materialcan also be modeled as different types of snow. ρd variesdue to snow grain size and absorption caused by impurities[33]. However, the range of diffuse albedo is mostly narroweraround 0.9 for visible 532 wavelength. Ward BRDF plots fortwo different incident light angles are shown below.

6-a.png(a) θi=0◦

6-b.png(b) θi=20◦

Fig. 6. Ward BRDF plots in polar coordinates for (a) θi=0◦ and (b) θi=20◦.Test scene is set as isotropic surface with ρd = 0.9 and ρs = 0.05, androughness of αx = αy = 0.3. Black cross is incident direction.

Since our framework assumes a clear sky, multiple scatter-ing in the atmosphere is not considered here. Therefore, amongall the directions that photons are reflected into, only those inthe back scattering direction will arrive at the receiver. Hence,for a specific photon, the BRDF function is used to determine

IEEE TRANSACTION MANUSCRIPT 6

whether it will be reflected in its own back scattering direc-tion. The combined effects on returning photons of complexsurfaces modeled with BRDF and the multiple pixel PMT onreturning photons is then complicated as compared to returnsfrom flat surfaces.

IV. RESULTS AND DISCUSSION

Having completed the framework for system modeling,simulation performance for spaceborne photon-counting lidarscan now be studied. Results and discussion are presentedbelow for lidar system performance on horizontal, sloped andcomplex surfaces.

A. Horizontal flat surfaces

As discussed above, for those photons which arrive at thedetector during dead time (modeled as 3 ns in our simulation),a photon-counting detector is not able to record that event.Therefore, if there are multiple photons arriving at the detector,the derived surface elevation will be biased towards the earlierphotons. To mitigate that impact, the PMT receiver in oursimulation uses a multiple pixel design where each pixel canbe triggered independently.

7.png

Fig. 7. Number of normalized triggers versus flat surface elevation retrievalsfor different numbers of PMT pixels.

To demonstrate the improvement on elevation retrieval usinga multiple channel PMT, N = 200, 000 trials of laser firingare tested for a horizontal flat surface normal to the beamusing the Monte Carlo method. Parameters for a weak spot areapplied here with surface diffuse albedo ρd=0.9 and ρs=0.05.Fig. 7 shows the histogram of surface elevation derived fromarriving photons, with different curves representing differentnumbers of pixels in the detector. Since all photon eventscan be registered by an ideal detector (i.e. one with infinitelyfast rise time and zero dead time), the retrieved elevation forthe ideal case shows no bias. For a realistic photon-countingdetector, the surface elevation results are biased toward thelidar system by the aforementioned first photon effect. As aresult, the derived surface elevation is higher than the actualone. However, this bias can be largely eliminated with amultiple pixel PMT design. As the different lines in Fig. 7demonstrate, the averaged elevation bias can be closer to 0when more pixels are utilized in the detector. However, it is

not always an advantage using multiple pixel PMT comparedto single detector, especially when signal is very weak, (e.g.number of mean photoelectron per shot is much smaller than1.) In this situation, multiple measurement will yield the samerange distribution as a single detector.

B. Sloped surfacesIt is interesting to explore PMT performance on the retrieved

elevation for a sloped surface, as a case intermediate betweenflat and complex surfaces. Previous work has shown the slopealtimetry using photon-counting lidar for quasi-Lambertiansurfaces [12]. For the nadir sensing of sloped surfaces, theincident light angle increases as the slope angle goes up. Thus,the probability distribution for photons being reflected in theback scattering direction can vary for different slopes. TheWard BRDF suggests that fewer photons will arrive at thedetector as slopes become steeper.

A histogram plot of photon triggers for a strong spot versusretrieved elevation for different sloped surfaces is shown inFig. 8. The Monte Carlo simulation is done with N = 50, 000trials of laser shots for the same sloped surface. Since slopesraise elevation uncertainty within a sampled area, the standarddeviation for returning photons increases as the slope anglegoes up for these photon-counting detectors. Hence, comparedto horizontal flat surfaces, uncertainty in elevation retrieval forslopes appears to result from a combined effect of spatial andtemporal laser pulse width, as well as surface BRDF variation.

8.png

Fig. 8. Number of normalized triggers on sloped surface elevation retrievalsfor different slope angles. Test sloped scene is set as isotropic surface withρd = 0.9 and ρs = 0.05, and roughness of αx = αy = 0.3 in Ward BRDF.

Quantitative sensitivity of retrieved elevation versus surfaceslope angle is shown below in Fig. 9. Average bias andstandard deviation are computed using N = 50, 000 trialson a sloped surface, with the slope angle ranging from 0◦ to10◦. As shown in Fig. 9, the mean elevation bias for arrivingphotons increases as the slope angle increases. Meanwhile, thestandard deviation keeps going up with slope angle increasingfrom 0◦ to 10◦. Note these results are specific to the systemparameters assumed here. A more realistic, complex surfacewill be discussed next.

C. Complex surfacesBefore presenting results for complex surfaces, we need

to review the details for calculating the retrieved elevation

IEEE TRANSACTION MANUSCRIPT 7

9.png

Fig. 9. Elevation retrieval of (top) mean elevation and (bottom) standarddeviation versus slope angle. Note the reference elevation in this case is 0cm.

and associated errors. For a spaceborne photon-counting lidarsystem, there is no way to tell where each detected photoncomes from within a given laser footprint area on the ground.Hence, it is not practical to derive an exact surface elevationfor each laser shot [16]. A statistical approach for many lasershots is necessary for elevation retrieval. So far we havefocused on understanding the general behavior of a multiplepixel PMT system. That is why a large number N was usedin our analysis for horizontal flat (N = 200, 000) and slopedsurfaces (N = 50, 000).

A window containing 1000 shots will be used for statisti-cally estimating the surface elevation for complex surfaces. Toevaluate system behavior, 1000 laser shots with 0.7 m along-track sampling is simulated on a synthetic surface produced us-ing p = 1.8 with the reflective properties assumed previously.Here, other parameters for the PMT detectors are: 3 ns deadtime and 50% probability of detection. For the transmitter, thelaser temporal pulse width is 1 ns, with a footprint diameterof 10 m on ground.

In Fig. 10, each red point represents an arriving photonand its altitude with respect to the actual surface profile asshown in the blue line. As discussed before, the accuracy ofthe ICESat-2 derived elevation is determined by comparing theretrieved elevation with the reference elevation, which is themean value of a circle area within the laser beam for each lasershot. The elevation bias, also known as accuracy, is statisticallycalculated as the mean of differences between the retrieved andreference elevations. In addition, the standard deviation forthe elevation difference denotes the precision of the elevationretrieval.

To investigate the impact of complex surface roughness on

10.png

Fig. 10. Returning photon point cloud for elevation retrieval with a complexsurface. Red points represent returning photons and the blue line representsthe along-track profile (reference surface).

elevation retrieval, the synthetic surfaces discussed previouslyusing different p values are tested (surfaces for p = 1.6 andp = 2.0 are shown in Fig. 4). The rest of the parametersfor system modeling remain the same. To reduce statisticaluncertainty, the simulation is done for a flight track containing1000 laser shots for each individual scene. Note that laserfootprint is assumed to have a 1/e2 diameter of 10 m. Sincethe synthetic surfaces have 1 m resolution, the impacts of sub-beam terrain characteristics on the returning lidar signal arestudied here.

11.png

Fig. 11. Standard deviation for retrieved elevation bias for a weak spot versusp showing its value for individual pixel and average over 4 outputs.

The result of the simulation for ATLAS detection using theweak beam on a complex surface with different roughnessis shown in Fig. 11. Each set of bars for a specific pvalue represents the bias standard deviation for each of the4 individual outputs, while the red line shows the averageresult. As stated before, the signal of a weak spot from groupsof 4 anode segments are summed before the discriminationfunction, resulting in 4 digital outputs, labeled as pixel A, B,C and D in Fig. 11. The standard deviation of retrieved errorbecomes smaller as p increases. This confirms that a less roughsurface will result in higher precision in elevation retrieval.

The comparison between elevation retrieval for weak andstrong spots is shown in Fig. 12. For the weak spot, each resultis achieved by averaging 4 individual outputs, while for thestrong spot it is done by averaging 16 outputs. Note that for a

IEEE TRANSACTION MANUSCRIPT 8

12-a.png(a) Weak spot

12-b.png(b) Strong spot

Fig. 12. Elevation bias and standard deviation of derived elevation differenceversus p for (a) weak spot and (b) strong spot.

strong spot, the mean photoelectrons per shot is 8.17 [20]. Asthe curves indicate, retrieved elevation bias is approximatelysimilar for the two laser intensities. This is because the ratioof pixel numbers of strong spot to weak spot is similar to theratio of laser intensities of the two.

Another test is done by changing the diffuse albedo forthe same synthetic surface. Here p is set to be 2.0, whilediffuse albedo ρd changes from 0.6 to 0.9. Since Lambertianreflectance increases, the number of returning photoelectronswill also increase proportionally. The mean and standarddeviation for elevation retrieval in this case are shown inFig. 13 for a strong spot. It can be seen that the two curvesvary slightly for this range of albedo. However, a smallerdiffuse albedo will result in a smaller bias average. It ispartially because that for the Ward BRDF model, a decreasein diffuse reflection will increase the quasi-specular reflectionsimultaneously, which will potentially increase the number ofback-scattered photons for small slopes. This result confirmsthat ATLAS will yield less bias on snow surfaces with asmaller diffuse albedo, such as those with larger snow grainsizes or melting snow.

However, remaining uncertainties in solar background noiseand dark current may have additional impacts on the accuracyof retrieved elevation data. Future research will address thesenoise sources and their impact on ice sheet elevation change,so that ICESat-2 like system performance can be confidently

13.png

Fig. 13. Standard deviation for retrieved elevation bias for a strong spotversus diffuse albedo.

derived.

V. CONCLUSION

In this paper we have presented a framework for the simu-lation of ICESat-2-like spaceborne photon-counting detectorperformance on a complex surface. A multiple pixel PMTmodel was constructed to test detectability on flat and slopedsurfaces, quantifying the improvement on elevation retrievalaccuracy using a multiple pixel design. We then createdsynthetic complex terrains using fractal filters in the spatialfrequency domain. A versatile BRDF was then implemented toaccurately model returning photon flux on a complex surface.Even without considering atmospheric and background noise,simulation on a ICESat-2-like lidar system shows that theretrieved elevation bias is about 2 cm. It was also demonstratedthat the high-rep laser and multiple pixel PMT on boardICESat-2 will achieve lower elevation bias on smoother ter-rain. In addition, snow surfaces with smaller diffuse albedoeswill result in higher accuracy of elevation retrieval. Thisstudy provides advanced understanding on the performance ofspaceborne photon-counting lidar systems sensing on complexsurfaces.

ACKNOWLEDGMENT

This work is supported by NASA under award numberNNX11AK77G. The authors would like to thank Dr. BeataCsatho, Dr. Toni Schenk, Dr. Scott Brown, Dr. Adam Goode-nough, Dr. Anthony Martino and Sudhagar Nagarajan for theirhelpful discussions.

REFERENCES

[1] J. Bufton, J. Robinson, M. Femiano, and F. Flatow, “Satellite laseraltimeter for measurement of ice sheet topography,” IEEE Trans. Geosci.Remote Sens., vol. 20, no. 4, pp. 544–549, October 1982.

[2] A. Brenner, J. DiMarzio, and H. Zwally, “Precision and accuracy ofsatellite radar and laser altimeter data over the continental ice sheets,”IEEE Trans. Geosci. Remote Sens., vol. 45, no. 2, pp. 321–331, February2007.

[3] T. Cossio, C. Slatton, W. Carter, K. Shrestha, and D. Harding, “Predict-ing topographic and bathymetric measurement performance for low-snrairborne lidar,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 7, pp.2298–2315, 2009.

IEEE TRANSACTION MANUSCRIPT 9

[4] W. Abdalati, H. Zwally, R. Bindschadler, B. Csatho, S. Farrell,H. Fricker, D. Harding, R. Kwok, M. Lefsky, T. Markus, A. Marshak,T. Neumann, S. Palm, B. Schutz, B. Smith, J. Spinhirne, and C. Webb,“The icesat-2 laser altimetry mission,” Proc. of IEEE, vol. 98, no. 5, pp.735–751, May 2010.

[5] D. Quincey and A. Luckman, “Progress in satellite remote sensing ofice sheets,” Progr. Phys. Geography, vol. 33, no. 4, pp. 547–567, August2009.

[6] R. Alley, P. Clark, P. Huybrechts, and I. Joughin, “Ice-sheet and sea-levelchanges,” Science, vol. 310, no. 5747, pp. 456–460, October 2005.

[7] D. Slobbe, R. Lindenbergh, and P. Ditmar, “Estimation of volumechange rates of greenlands ice sheet from icesat data using overlappingfootprints,” Remote Sens. Environ., vol. 112, no. 12, pp. 4204–4213,December 2008.

[8] D. Yi, H. Zwally, and X. Sun, “Icesat measurement of greenland icesheet surface slope and roughness,” Ann. Glac., vol. 42, no. 1, pp. 83–89, August 2005.

[9] C. Shuman, H. Zwally, B. Schutz, A. Brenner, J. DiMarzio, V. Suchdeo,and H. Fricker, “Icesat antarctic elevation data: Preliminary precisionand accuracy assessment,” Geophys. Res. Lett., vol. 33, no. 7, pp. 10–13, 2006.

[10] http://icesat.gsfc.nasa.gov/icesat2/.[11] J. Degnan, J. McGarry, T. Zagwodzki, P. Dabney, J. Geiger, R. Chabot,

C. Steggerda, J. Marzouk, and A. Chu, “Design and performanceof an airborne multikilohertz photon-counting, microlaser altimeter,”International Archives of Photogrammetry and Remote Sensing, vol.XXXIV3-W4, pp. 9–16, 2001.

[12] J. Degnan, “Photon-counting multikilohertz microlaser altimeters for air-borne and spaceborne topographic measurements,” J. Geodyn., vol. 34,no. 3-4, pp. 503–549, 2002.

[13] P. Dabney, D. Harding, J. Abshire, T. Huss, G. Jodor, R. Machan,J. Marzouk, K. Rush, A. Seas, C. Shuman, X. Sun, S. Valett, A. Vasilyev,A. Yu, and Y. Zheng, “The slope imaging multi-polarization photoncounting lidar: development and performance results,” IGARSS, July2010.

[14] J. Degnan, R. Machan, E. Leventhal, D. Lawrence, G. Jodor, andC. Field, “Inflight performance of a second-generation photon-counting3d imaging lidar,” in Proc. SPIE, vol. 6950, 2008, p. 695007.

[15] M. McGill, T. Markus, V. Scott, and T. Neumann, “The multiplealtimeter beam experimental lidar (mabel): An airborne simulator forthe icesat-2 mission,” J. Atmos. Oceanic Technol., vol. 30, pp. 345–352,2013.

[16] Y. Yang, A. Marshak, S. Palm, T. Varnai, and W. Wiscombe, “Cloudimpact on surface altimetry from a spaceborne 532 nm micro-pulse pho-ton counting lidar: system modeling for cloudy and clear atmospheres,”IEEE Trans. Geosci. Remote Sens., vol. 49, no. 12, pp. 4910–4919,December 2011.

[17] Y. Yang, A. Marshak, S. Palm, Z. Wang, and C. Schaaf, “Assessmentof cloud screening with apparent surface reflectance in support of theicesat-2 mission,” IEEE Trans. Geosci. Remote Sens., vol. 51, no. 2, pp.1037–1045, February 2013.

[18] U. C. Herzfeld, B. W. McDonald, B. F. Wallins, T. A. Neumann,T. Markus, A. Brenner, and C. Field, “Algorithm for detection of groundand canopy cover in micropulse photon-counting lidar altimeter data inpreparation for the icesat-2 mission,” IEEE Transactions on Geoscienceand Remote Sensing, 2013.

[19] A. Martino, “Atlas instrument architecture document (draft),” November2011.

[20] ——, “Atlas performance spreadsheet,”http://icesat.gsfc.nasa.gov/icesat2/, April 2010.

[21] H. Fricker, A. Borsa, B. Minster, C. Carabajal, K. Quinn, and B. Bills,“Assessment of icesat performance at the salar de uyuni, bolivia,”Geophys. Res. Lett., vol. 32(2), L21S06, 2005.

[22] G. A. Mastin, P. A. Watterberg, and J. F. Mareda, “Fourier synthesisof ocean scenes,” Computer Graphics and Applications, IEEE, vol. 7,no. 3, pp. 16–23, 1987.

[23] A. Fournier, D. Fussell, and L. Carpenter, “Computer rendering ofstochastic models,” Communications of the ACM, vol. 25, no. 6, pp.371–384, 1982.

[24] R. Voss, “Fourier synthesis of gaussian fractals: 1/f noises, landscapes,and flakes,” in State of the Art in Image Synthesis, Tutorial No. 10. NewYork: SIGGRAPH, 1983.

[25] W. Krabill, W. Abdalati, E. Frederick, S. Manizade, C. Martin, J. Son-ntag, R. Swift, R. Thomas, and J. Yungel, “Aircraft laser altimetrymeasurement of elevation changes of the greenland ice sheet: Techniqueand accuracy assessment,” Journal of Geodynamics, vol. 34, no. 3, pp.357–376, 2002.

[26] B. Csatho, private communication, 2013.[27] F. Ulaby, R. Moore, and A. Fung, Microwave Remote Sensing: Active

and Passive, vol. II. Boston, USA: Artech House, 1982.[28] R. Montes Soldado and C. Urena Almagro, “An overview of BRDF

models,” 2012.[29] G. J. Ward, “Measuring and modeling anisotropic reflection,” in ACM

SIGGRAPH Computer Graphics, vol. 26, no. 2. ACM, 1992, pp. 265–272.

[30] B. Walter, “Notes on the ward brdf,” Program of Computer Graphics,Cornell University, Technical report PCG-05-06, 2005.

[31] A. A. Kokhanovsky and F.-M. Breon, “Validation of an analytical snowbrdf model using parasol multi-angular and multispectral observations,”Geoscience and Remote Sensing Letters, IEEE, vol. 9, no. 5, pp. 928–932, 2012.

[32] W. Li, K. Stamnes, H. Eide, and R. Spurr, “Bidirectional reflectancedistribution function of snow: corrections for the lambertian assumptionin remote sensing applications,” Optical Engineering, vol. 46, no. 6, pp.066 201–066 201, 2007.

[33] O. Meinander, S. Kazadzis, A. Arola, A. Riihela, P. Raisanen, R. Kivi,A. Kontu, R. Kouznetsov, M. Sofiev, J. Svensson et al., “Spectral albedoof seasonal snow during intensive melt period at sodankyla, beyond thearctic circle,” Atmospheric Chemistry and Physics, vol. 13, no. 7, pp.3793–3810, 2013.

Jiashu Zhang received the B.S. and M.S. degreesin Physics from Nanjing University, Nanjing, Chinain 2007 and 2010. He is currently working towardhis Ph.D. degree in Chester F. Carlson Center forImaging Science, Rochester Institute of Technology,Rochester, NY. Since 2011, he is a Graduate Re-search Assistant working with Digital Imaging andRemote Sensing Laboratory (DIRS). His researchinterests focus on modeling and analysis of remotesensing system performance in complex surface ge-ometries.

John P. Kerekes (S81-M89-SM00) received theB.S., M.S., and Ph.D. degrees in electrical engineer-ing from Purdue University, West Lafayette, IN, in1983, 1986, and 1989. From 1983 to 1984, he wasa Member of the Technical Staff with the Spaceand Communications Group, Hughes Aircraft Co.,El Segundo, CA, where he performed circuit designfor communications satellites. From 1986 to 1989,he was a Graduate Research Assistant, working withboth the School of Electrical Engineering and theLaboratory for Applications of Remote Sensing at

Purdue University. From 1989 to 2004, he was a Technical Staff Member withthe Lincoln Laboratory, Massachusetts Institute of Technology, Lexington,MA. In 2004, he became an Associate Professor in the Chester F. CarlsonCenter for Imaging Science, Rochester Institute of Technology, Rochester,NY. He is currently a Professor in the Center for Imaging Science. Hisresearch interests include the modeling and analysis of remote sensingsystem performance in pattern recognition and geophysical parameter retrievalapplications. Dr. Kerekes is a member of Tau Beta Phi, Eta Kappa Nu, theAmerican Geophysical Union, and the American Society for Photogrammetryand Remote Sensing. He is a Senior Member of the Optical Society ofAmerica and SPIE. From 1995 to 2004, he served as the Chair of the BostonSection Chapter of the IEEE Geoscience and Remote Sensing Society (GRSS),and from 2007 to 2010 he served as the founding Chair of the WesternNew York Chapter of GRSS. He was a Co-General Chair of IGARSS 2008held in Boston, MA. He is currently a member of the GRSS AdministrativeCommittee (AdCom) and is serving as the Vice President of TechnicalActivities of the GRSS.