Title: SUMMARY
1A Gaussian Process Approach to Quantifying the
Uncertainty of Biospheric Parameters from Remote
Sensing Observations Roberto Furfaro1, Robin D.
Morris2, Athanasios Kottas3 , Matt Taddy3, Barry
D. Ganapol1 1Aerospace and Mechanical Engineering
Dept, University of Arizona, Tucson, AZ
2Universities Space Research Association, RIACS,
Mountain View, CA 3Department of Applied Math
and Statistics, UC Santa Cruz, CA
SUMMARY
RADIATIVE TRANSFER MODELING
PRELIMINARY RESULTS CONCLUSIONS
We describe a methodology that can have a major
impact on estimating the uncertainties involved
in using biogeochemical models that take remote
sensing data as inputs. It allows a full
probabilistic uncertainty analysis of a complex
computational model, such as those used in
modeling light reflectance from vegetation or
carbon fluxes. We show an example of the
methodology applied to a radiative transfer model
(RTM), producing estimates of parameters
important to carbon budget determination. Modeling
the biosphere requires inputs of biospheric
parameters over extended regions. The only
practical measurement technology is satellite
remote sensing. Generating estimates of the
biospheric parameters requires inverting the
physical process between the parameters and the
observations. These models, for example, plant
growth and the radiative transfer of energy
within the canopy to produce a model of the
upwelling radiation. Inverting the model gives
an estimate of plant growth, and hence carbon
sequestration. Here we focus on Leaf Area Index
(LAI). LAI can be related to MODIS data using a
physically-based approach, radiative transfer
modeling. Recently a nested Leaf-Canopy RTM
(LCM2) has been developed to simulate the
interaction between light and vegetation. The
model computes wavelength-dependent hemispherical
reflectance as function of canopy morphological
(e.g. LAI) and biochemical (e.g. chlorophyll
concentration) parameters. Retrieval algorithms
based on the canopy equation have proven to be
efficient in determining LAI using remote
measurements much work is still required to
effectively quantify the uncertainty of the
retrieved parameters. Hence, we propose a
Bayesian statistical analysis of the LCM2
computer model output. Gaussian processes (GPs)
provide the foundation for the statistical model
framework. The GP defines a prior for the
functional input-output relationship generated by
the LCM2, and the prior-to-posterior analysis
yields a flexible statistical model approximation
to the LCM2 output. To illustrate the Bayesian
statistical approach in characterizing and
mapping biospheric parameters, we will present
results of the GP approximation for LAI and
wavelength as input. This work will set the stage
for further developments, including uncertainty
analysis of the LCM2 output and probabilistic
sensitivity analysis for the full set of input
parameters. This will demonstrate the
applicability of the methodology in the analysis
of models important to carbon budget estimation.
The GP methodology was first applied with LAI as
the single input for each of 12 different
wavelength values. The number of input values for
LAI was n 10 the set of values was
0.1,0.5,0.8,1.0,1.5,2.0,3.0,5.0,8.0,10.0.
Figure 1 includes posterior point and interval
predictive estimates. Because the GP model
approximation essentially interpolates the data,
the length of the interval estimates decreases as
the LAI values get closer to the LAI input values
above. The large uncertainty depicted in some of
the panels in Figures 1 can be reduced through a
small increase in the number of input values, or
by choosing a different non-linear mean
function. The next step was to add wavelength to
the inputs. The GP model approximation now
provides estimates for the output surface over
the two-dimensional input space. The pairs of
input values were chosen based on D-optimal
designs, which maximize an isotropic Gaussian
correlation. Results based on 200 and 500 input
points are given in Figures 2 and 3. In each
case, the top panel provides the bivariate
posterior predictive surfaces, whereas the bottom
panel shows posterior predictive estimates for
the output as a function of LAI only, by
conditioning on a specified wavelength value.
Modeling the radiative regime inside the canopy
Overview Goal create a true radiative transfer
model to simulate the radiative regime within and
at the top of the canopy. First principles used
in modeling the interaction light-vegetation.
Leaf level and canopy level approach to generate
a nested leaf-canopy model.
Level 1 Radiative regime within the
leaf Simulation of the radiative regime inside
the leaf to determine the optical properties. The
leaf is modeled as a transversely infinite
medium with uniform absorption and isotropic
scattering coefficient (natural averaging). The
balance of photons will generate the mathematical
equation to determine reflectance and
transmittance.
Level 2 Radiative regime within the canopy The
canopy is modeled as a turbid medium with
leaves as scattering/absorbing elements. The
medium is anisotropic. Leaves are considered
bi-Lambertian. The balance of photons determines
the mathematical model describing the radiant
intensity within and at the top of the canopy
FIGURE 1
Note Black lines Posterior Mean Prediction Red
Lines 90 Prediction Interval Green Lines True
Output Curves
Level 3 Coupling Leaf and Canopy Models
LEAFMOD construct the scattering and
absorption profile for the single leaf and
compute the optical properties. CANMOD Take
reflectance and transmittance of the single leaf
(LEAFMOD outputs) and compute the reflectance at
the Top-of-the-Canopy RESULT Nested
Leaf-Canopy Model (LCM)
GAUSSIAN PROCESS MODELING
MODELING THE FORWARD RELATIONSHIP BETWEEN
HEMISPHERICAL REFLECTANCE AND LAI AND/OR
WAVELENGTH USING GAUSSIAN PROCESS
We start by modeling the output of the
simulations as a realization of a Gaussian
Process (GP). Specifically, with f(z) denoting
the computer code output for a given input vector
z (z1,..,zq), we are assuming that the prior
model for the random function f() is a GP. The
GP prior is described by its mean function µ(z)
E(f(z)), and its covariance function, which is
taken to be isotropic with constant variance s2,
and correlation function that depends only on the
distance between input values. In particular, we
used the Gaussian Correlation function C(z, z)
and linear functions for µ(z) for situations
where LAI is the only input parameter and with
LAI and wavelength as input parameters. Here, we
used LCM2 code to generate the observations
(hemispherical reflectance factors) yi f(zi), i
1,..,n with z1,..,zn is the collection of
inputs (LAI and/or wavelength) at which the LCM2
is executed. Based on the finite dimensional
distributions induced by the GP, we have that y
(y1,..,yn) follows an n-variate normal
distribution with mean vector (µ(z1),..,
µ(zn)) and covariance matrix s2S, where S is the
correlation matrix with (i j)-th element C(zi,
zj)
Two- Angle Formulation
CANMOD Transport Equation
Boundary Conditions
FIGURE 2
CONCLUSIONS FUTURE EFFORTS The GP process
approach to modeling the forward functional
relationship between LCM output
(top-of-the-canopy hemispherical reflectance) and
LAI and wavelength has been demonstrated.
Preliminary results show the potential of the
methodology as applied to the problem of
estimating uncertainty prediction of critical
canopy biophysical parameters. Future efforts
will include extending the GP process forward
modeling to multiple inputs (e.g. chlorophyll,
water content), performing a global probabilistic
sensitivity analysis, validating the model using
the GP process approach and developing a GP
methodology for fast solution of the inverse
problem (i.e. given the hemispherical
reflectance, estimate canopy biochemical and
morphological parameters). ACKNOWLEDGMENT This
work have been supported by NASA Applied
Information System Research (AISR) program (NRA
NNH05ZDA001N)
Scattering Area Function (Two-Angle)
COVARIANCE FUNCTION PRIORS
Case 1 only LAI as LCM input
Case 2 LAI and Wavelength as LCM inputs