IEEE Geoscience and Remote Sensing Magazine - June 2016 - 72

were practically impossible with these models and extend
them to practical problem sizes.
A particular requirement in many RTMs is the prediction of spectral reflectance over the solar reflective
domain (i.e., broadly from 400 to 2,500 nm) so instrument bandpass functions can be applied to the data. To
emulate full spectra, the idea of the PCA of hyperspectral data can be extended, where there are large degrees of
spectral redundancy. Let the output y be given a stacking
of Nt spectra. Each of these spectra can be approximately
reconstructed from
yi .

/ v j w j,

(11)

j=1

where only the first L principal components are considered, and vj is the jth score associated with the wj principal component. In PCA, the principal components are
orthogonal over the input set, so one strategy is to emulate the scores v 1, f, v L with independent emulators, and
then use these emulators to reconstruct a full spectrum
[uncertainties and gradients can also follow through
quite easily due to the linearity of (11)].

Reflectance (-)

0.40
0.32
0.24
0.16
0.08
0.00
(a)
Mean
Median

Reflectance (-)

0.006

5-95% IQR
25-75% IQR

0.003
0.000
-0.003
-0.006
500

1,000
1,500
2,000
Wavelength (nm)
(b)

2,500

figUre 9. An example of RTM emulation with GPs. The PROSAIL
soil-leaf-canopy RTM is emulated spectrally. (a) The complete
model (full lines) and the emulated reflectance (dashed lines) for
ten random input parameter sets. (b) The mean, median, 5-95%,
and 25-75% interquartile ranges for the residuals of the full
model minus the emulator. This example assumes a sun zenith
angle of 30c, a view zenith angle of 0c, and a relative azimuth
of 0c, and the validation is done with a set of 1,000 uniformly
independent samples.

AN ILLUSTRATIVE EXAMPLE
As an example, consider a coupled, soil-leaf-canopy RTM
over the solar reflective domain, PROSAIL [75]. A simple
linear spectral mixture RTM for the soil (therefore assuming the soil properties are isotropic), the leaf optical properties spectra (PROSPECT) model, and the scattering by arbitrarily inclined leaves (SAIL) canopy RTM will be used. The
aim is to map from a state made up of soil, leaf, canopy, and
parameters such as LAI and chlorophyll content to top-ofcanopy reflectance. This is an important example because
the coupled model can be used within a DA system to infer
the properties of the land surface (i.e., vegetation structure
and biochemistry) from the atmospherically corrected directional surface reflectance. A validation of the emulation
approach is shown in Figure 9, where the emulator has been
trained with 250 input parameter-reflectance pairs, which
were chosen using a Latin hypercube sampling design. Using
the approach outlined in the previous section for multivariate output, L in (11) was chosen to be 11 so as to encompass
99% of the variance in the training set. It can be immediately
seen that the emulator is virtually indistinguishable from the
original model, with negligible bias in the validation, and
a very small RMSE. Although PROSAIL is a fast RTM, this
emulator is some 5,000-fold faster than the original in a contemporary PC. In evaluating the GP, the PROSAIL gradient
is also calculated.
concLUSionS and fUrther WorK
This article provides a comprehensive survey of GPs in
the context of remote sensing data analysis, particularly
for statistical biophysical parameter estimation. The GPs'
main properties and their advantages over other estimation methods were summarized to find that GPs can essentially provide competitive predictive power, give error bars
for estimations, allow design and optimization of sensible
kernel functions, and analyze the encoded knowledge in a
model via ARD kernels. The GP models also offer a solid
Bayesian framework to formulate new algorithms that are
well suited to signal characteristics. For example, it can be
seen that, by incorporating proper priors, signal-dependent noise can be encompassed and parametric forms of
warping the observations as an alternative to either adhoc filtering or linearization, respectively, can be inferred.
A downside for GPs is the scalability issue, which is that,
essentially, the optimization of GP models require computing determinants and invert matrices of size n × n,
which runs cubically in computational time and quadratically in memory storage. In recent years, however, great
advances have been made in machine learning, and it is
now possible to train GPs with millions of points in almost
linear time.
All of the developments were illustrated on local and
global scales through a full set of illustrative examples in
geosciences and remote sensing. In particular, addressed
were important problems of ocean, land, and atmospheric
sciences, from accurately estimating oceanic chlorophyll
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JUNE 2016

Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - June 2016