The overall goal of this work has been to develop a set of computational tools and media abstractions for the terrain bidirectional reflectance problem. The modeling of soil and vegetation surfaces has been emphasized with a gradual increase in the complexity of the media geometries treated. Pragmatic problems involved in the combined modeling of soil, vegetation, and atmospheric effects have been of interest and one of the objectives has been to describe the canopy reflectance problem in a classical radiative transfer sense permitting easier inclusion of our work by other workers in the radiative transfer field

Smith, J. A.

English

NASA Technical Reports Server

Dr. 3ms A. Smith 
College of .?orestry & Natural Resources 
Colorado State University 
Fort Collins, Colorado 80523 
The overall goal of our work has been t o  develop a t of computa- 
tional tools and media abstractions for the terrain bidirectional reflec- 
tance problem. The ndel ing  of so i l  and vegetation surfaces has been 
emphasized w i t h  a gradual increase in the comp1exit.y of the mdia 
geanetries treated. Pragmatic problems involved in the m-cbined modeling 
of so i l ,  vegetation, and atmospheric effects have been1 of interest and one 
of our objectives has been t o  describe the canopy reflectance problem in a 
classical radiative transfer sense prmit t ing easier inclusion of our work 
by other workers in  the radiative transfer field. 
Th? application of radiative transfer theory t o  so i l  and vegetation 
media requires an abstraction of the scattering and absorbing properties of 
the medium in terms of biogeophysical attributes, a specification of the 
geanetrical boundary of the medium, and a mthgnatical solution technique 
t o  the radiative transfer equations. We have abstracted our media t o  con- 
sist of a collection of discrete scatterers, e.?. leaf or so i l  facets, with 
individual diffuse and specular optical properties. Bulk p ropr t i e s  of the 
media are usually derived from suitable averaging ovex the s t a t i s t i ca l  dis- 
tribution of the individual elements within the mdium. For sirrp?le plane 
parallel  nledia gecmetries, we are using classical solutions t o  the radia- 
t ive transfer equations, e.g. the discrete ordinates/ f i n i t e  difference 
method of Qlandrasekhar, the Adding method of Van de Hulsk, or the two 
stream method of Kaufm. For complex geanetries with irregular rough sur- 
faces, we are employing Monte Carlo techniques. 
We w i l l  briefly sumr~rize three areas which have been addressed since 
the last SRAEC annual meeting. These are orthogonal expansion of our phase 
functions in terms of to ta l  scattering angle and example use with standard 
radiative transfer aode, e.g. Kaufm's two stream approximtion; our Monte 
Carlo reflectance rnodeling for oomplex s o i l  surfaces which now include 
F r s n e l  facets; and our current work i n  modeling complex vegetation sur- 
f aces. 
Orthouonal, m s i o n  a Utilization of C.axgy .Phase Functions 
W e  have previously described our cakculational technique for averaging 
over leaf angle distributions and leaf optical properties t o  e s t i m t e  the 
canopy scattering diagram as a function of the global incident and exitant 
https://ntrs.nasa.gov/search.jsp?R=19860004326 2020-03-20T15:53:27+00:00Z
angles. The phase functions we presented were given in  terms of incident 
angles, p,rd and exitant angles, p1,8'. As Ross has indicated, it *pars 
that  canopy phase functions, unlike m e  of the more symmetrical atmos- 
pheric cases, m y  not necessarily be explicitly formulated i n  t e r n  of only 
a total  scattering angle, 8. We have q e n t  sane time pondering t h i s  cir- 
cumstance. Recently, we  have adapted an approach outlined by Liou for ice 
crystals whereby "averaget1 phase functions can be cornput&. The procedure 
essentf a l ly  averages over ranges of p, 18 and p1 ,rdl which yield the same 
to ta l  scattering angle, @, i.e. 
cos 8 - p p' + (l-p11/~ ( l - y ' ~ ~ / ~  cos [I& 4' )  (9) 
We tern the resulting scattering diagrams, nstylized" phase functions. 
We do not -believe the results are s t r i c t l y  correck in the fine details, 
because of both fundamental reasons and numerical. problems. Howeverp as 
Van de Hulst has indicated, for  many calculations the fine details may not 
be a l l  that inportant. Theoretically, the phase functions are  milch more 
tractable and be expanded, in m e n d r e  polynanials, for example, a s  fol- 
laws: 
and, thence, into a variety of useful forms. The table below shows an 
example calculation for a spherical leaf angle distribution for visible and 
near infrared wavelengths. The phase funckion, and, hence, expansion me£-- 
f icients  vary with the vegetation optical proprt ies .  
WI'E: % should be 1.0; sane measure of the numerical precision of the 
calculatioils is indicated by the deviations from t h i s  value above. 
Such expansions can be easily incorporated into standard radiative 
transfer code. We have used the above values in Kaufman's published modi- 
f ied two strean method-the program aade was obtained from the Atmospheric 
Science Department a t  Colorado State University. Figure 1 is a plot of the 
(spline - smoothed) stylized phase functions a t  the three wavelengths 
above. 
A t  l a s t  year's SRAEX: sm preliminary resul ts  from a Monte Caiclo mxlel 
for  the reflectance of a rough s o i l  were presented. The t h r ~ s t  of our work 
i n  t h i s  area over the past year has been t o  extend the usefulness s f  t h i s  
model t o  surfaces having non-%ambertian e lmental  rdleetance properties, 
In particular,  we were concerned with the effect  of a distr ibution of 
Fresnel facets  superinposed on macroscopic surface i r regular i t ies  . The 
reflectance properties of an element of area of surface covered by these 
facets  was determined theoretically, and then this reflectance fuaction was 
substituted fo r  the original Limbertian one in the Monte Carlo mdel.  
In the substi tution of reflectance functions into  the Monte Carlo 
model it was desirable t o  allow the most general forms possible. For t h i s  
reason two alternatives were investigated: the  approximation of the reflec- 
tance functions in  t e ~ m  of spherical harmonics, and the  use of a lookup 
tab le  generated from the value of the reflectance functioris a t  discrete 
angles. M t e r  sane preliminary testing it appears tha t  the  former approach 
is sl ight ly  bet ter  when the reflectance function in  question is sinple, 
however for  m r e  complex functions, and i n  particular for  the  functions 
ar is ing from clistrjd3utions of Fresnel facets, the  lookup table approach is 
the only reasonable one. 
Thus far, w e  have only considered the case when the Fresnel facets 
have a uniforrn distribution of inclination angles. An exanqle of the  sor t  
of results obtained for  a surface with a sirnpLe cosine undulation is given 
in  Fig. 2a. I n  t h i s  case l i g h t  was incident on the surface at  a zenith 
angle of 45 degrees and an azimuth p rpend icaa r  t o  the direction of the 
rows. The resu l t s  a r e  given by plott ing mntours of equal r~,'lectance fac- 
tor on a polar p lo t  i n  which distance £ram the center indicates the zenith 
view angle, and the azimuth of the view corresponds direct ly  t o  t h a t  on the 
plot. Figure 2b shows the bidirectional reflectance distribution function 
for the same surface possessing purely Lambertian elemental scattering pro- 
perties. 
After reviewing the bidirectional reflectance distribution modeling 
work for vegetation media currently underway in the SRAEC fundamental 
research e f for t r  w e  concluded tha t  it would be useful t o  attack the  problem 
of reflectance from a rough surface vegetation layer from a multiple 
scattering, radiative transfer perspective. The approach w e  chose is an 
~utgrcrwth of Nom2's work and eornplements Strahler's geanetric optics 
approach ut i l iz ing stylized structures for  t r ee  crowns. Basically, we are  
adapting a Monte Carlo code by ut i l iz ing probability of gap expressions 
which take in to  acmunt the  varying optical path length through vegetation 
layers as a function of canopy densityr macro-structure and look angle. 
Gecmetrical routines a r e  used t o  compute required parameters f ran a canopy 
height profile. In practice we gnploy Fourier se r ies  expansion of canopy 
height profiles. Later we may examine s t a t i s t i c a l  structures inherent in 
the data via  power spectra analysis and formulate appropriate mdels. 
These ideas a r e  par t ia l ly  generated from same thermal d e l i n g  work w e  are  
performing over a forested site a t  Oak Ridge National Laboratory and we are  
capitalizing on the canopy s t ructure  &ta available for  t h i s  site. 
This work is i n  preliminary stages, but br ief ly  our approach follows: 
1. An adapted version of our Monte Carlo vegetation reflectance model 
w i l l  be used t o  handle the  multiple scattering calculations. 
2. Probability of gap expressions a r e  computed for  photon interactions by 
considering both micro (i.e. leaf angle distr ibutions) and macro 
(canom height profi le)  structure. Specifically, 
Po (8) = exp - IN*$ (8) *D (8) /h (3) 
where g (  8 1 is the mean canow projection i n  direction 8, LA1 is leaf 
area index, and D( 8 is the gemet r ic  path length from the  bottan of 
a canopy l a t e 1  t o  the top prof i le  i n  direction 8, and h is the canopy 
layer thickness. 
3. Appropriate geometrical computatioml. routines have been developed t o  
calculate D(8) for  an arbi t rary  canopy height profi le.  We have 
applied these routines t o  sample data transects from the Oak Ridge 
Site. In  practice, t h e  canopy height prof i le  data a re  f i r s t  
"detrended" using l e a s t  squares regression, and the the p i e r  s p c t r a  
computed fo r  the detrended data in  order t o  develop Fourier se r ies  
representations for  the  calculation routi.nes. Figure 3 shows a sample 
detrended height profi le,  and schematically i l l u s t r a t e s  the varying 
path lengths through t h i s  canopy layer as a function of look angle. 
Multiple voids i n  the canopy are  correctly handled. 
The m i n  significance i n  our work is an increased understanding, 
expressed via a variety of computational procedures, of t e r ra in  bidirec- 
t ional  reflectance properties. To our knawledge, there were no macro s o i l  
reflectance models for  amplex surfaces incorporating the complete rnultiple 
scattering solution and accounting fo r  shadowing effects.  Our calculation, 
treatment, and ut i l izat ion of canopy phase functions also appears t o  be 
smewhat novel, although we have bu i l t  upon the  e f fo r t s  of ea r l i e r  workers. 
Extensions could be made t o  our work t o  include, fo r  example, the  incor- 
poration of media subelement models t o  account for  physiological or polari- 
zation properties. 
In  the e igh t  months remaining of our th i rd  year of funding the f ina l  
s tep in our research focus w i l l  be t o  complete a tractable treatment of 
complex vegetation surfaces amendable t o  our Monte Carlo canorpy reflectance 
modeling techniques. I n  addition, we w i l l  more gracefully complete several 
loose ends i n  our completed work including the generation of orthogonal 
expansions of our canopy phase functions and the  ut i l izat ion of the mdels  
developed under this study t o  i l l u s t r a t e  soil/vegetation/atmospheric 
effects  in such applications as AWEU+derived vegetation index calcula- 
tions. 
I 1 1 
'h, 0.55 micrometers 
'1 0.80 
\ r. 
--\ 
5 s s 
Scattering Angle 
Figure 1 . Sty1 ized Canopy Phase Functions 3s a 
Function of Scattering Angle fur 
wavelengths of 0.55 and 0.80 micro- 
meters. (Normal ization i s  incomplete) 
g j i ~ u v @  2 d -  Do? J T  r s q ~ t  o f  hidifactiona'l  ref1 ectance d is t r ibut ion  
fur.re,t,jon froi;n ~ 0 2 3  so i l  ~ u r f a c e  ~ i t k  ~adrtia11y specularly 
ve9""8ee,t1nrj Fresnel f a c e t s ,  Sun i s  & t  45 degvees a&th 
d z  71'1tith zjerglt -4 i c ~ l  21' &rl me5, Kler$~en%al ~ e f  3 ectance 3 s 
285, , 
Figure 2b. ?olar plot of bidirectional reflectance distribution 
function from a row soil surface witn strictly 
Lambertian elemental ref1 ectance properties. 
Figure 3. Detrended height prof i le  fo r  fo res t  t ransect  and probability 
of gap calculation f o r  path length as a  function of look angle. 


Reflectance Modeling

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