IEEE Geoscience and Remote Sensing Magazine - June 2016 - 57
illustrative examples. In particular, important problems
for land, ocean, and atmosphere monitoring are considered, from accurately estimating oceanic chlorophyll
content and pigments to retrieving vegetation properties
from multi- and hyperspectral sensors as well as estimating atmospheric parameters (e.g., temperature, moisture,
and ozone) from infrared sounders.
Unprecedented data Stream for Land, ocean,
and atmoSphere monitoring
The spatiotemporally explicit, quantitative retrieval methods for Earth's surface and atmosphere characteristics are
required in a variety of Earth system applications. Optical Earth-observing satellites that are endowed with high
temporal resolution enable the retrieval and, hence, the
monitoring of climate and biogeophysical variables [1], [2].
With the forthcoming superspectral Copernicus Sentinel-2
(S2) [3] and Sentinel-3 missions [4], as well as the planned
EnMAP [5], HyspIRI [6], PRISMA [7], and the European
Space Agency's candidate FLEX [8], an unprecedented data
stream for land, ocean, and atmosphere monitoring will
soon become available to a diverse user community. This
vast data stream requires enhanced processing techniques
that are accurate, robust, and fast. Additionally, the statistical models should capture plausible physical relationships
and explain the problem at hand.
A wide variety of biogeophysical retrieval methods have
been developed over the last few decades, but only a few
of them have made it into operational processing chains,
and many are still in their infancy [9]. Essentially, there are
two main approaches to the inverse problem of estimating
biophysical parameters from spectra: 1) parametric physically based models and 2) nonparametric statistical models. On one hand, parametric, physically based models are
commonly used to model biological processes and climate
variables in Earth monitoring. These models rely on established physical relationships and implement complex
combinations of scientific hypotheses. Unfortunately, they
do not exploit empirical data to constrain simulation outcomes; thus, despite their solid physical foundation, they
are becoming more obscure because more complex processes, parameterizations, and priors need to be included.
These issues give rise to too-rigid solutions and large-model
discrepancies (see [10] and the references therein). Alternatively, nonparametric statistical models are typically only
concerned with developing data-driven models, paying
little attention to the physical rules governing the system.
The field has proven to be successful in many disciplines
of science and engineering [11], and, in general, nonlinear
and nonparametric model instantiations typically lead to
a more flexible and improved performance over physically
based approximations [12].
In the last decade, machine learning has attained outstanding results in estimating climate variables and the related biogeophysical parameters on local and global scales
[13]. For example, the current operational vegetation prodJUNE 2016
ieee Geoscience and remote sensinG
sensing maGazine
magazine
ucts, such as leaf area index (LAI), are typically produced
with neural networks [14], [15]; gross primary production,
the largest global CO2 flux driving several ecosystem functions, is estimated using ensembles of random forests and
neural networks [16], [17]; biomass has been estimated
with stepwise multiple regression [18]; principal component analysis (PCA) and piecewise linear regression have
been used for sun-induced fluorescence (SIF) estimation
[19]; support vector regression (SVR) showed high efficiency in modeling LAI; fractional vegetation cover (fCOVER),
evapotranspiration [20], [21], and relevance vector machines (RVMs) were successful in estimating ocean chlorophyll [22]; and, recently, GPs [23] provided excellent results
in estimating vegetation properties [24]-[27].
The family of Bayesian nonparametrics, and of GPs in
particular [23], has been given great consideration in remote sensing data analysis in recent years because they are
endorsed with important properties that are relevant to
common problems in our field. GPs can provide excellent
accuracy estimations as well as error bars (i.e., uncertainties) for the predictions. Also, and very importantly, they
can easily accommodate different data sources (e.g., multimodal data, multiple sensors, multitemporal acquisitions)
and can be designed to deal with different noise sources.
The use of GPs in problems involving large data has traditionally been problematic, but recently advanced sparse,
variational, and distributed computing techniques allow
training models in almost linear cost. This article studies
the modern approaches to tackle these issues.
Beyond these interesting features of GPs, statistical inference methods should be able to fit data well (i.e., focus
only on data exploitation) but should also show something
about the physical rules governing the problem (i.e., data
exploration). Therefore, these too-flexible models should
be constrained to provide physically plausible predictions,
which is why, in recent years, combining machine learning
and physical models seems promising, either via data assimilation, hybrid approaches, or the emulation of physically
based RTMs. In this respect, GPs can be used to learn about
the relevance of the problem features, as they can adapt to
anisotropic data distributions, the derivatives of the predictive mean and variance can be computed in closed form,
and they are ideal for use in empirical (i.e., noninterventional) causal inference. Additionally, GPs have been the
first choice in emulating RTMs to endorse these statistical
models with physically meaningful constraints [28].
gaUSSian proceSS regreSSion
Regression, function approximation, and function emulation
are old, largely studied problems in statistics and machine
learning. The problem boils down to optimizing a loss (e.g.,
cost or energy) function over a class of functions. In particular, a large class of regression problems are defined as the
joint minimization of a loss function accounting for errors of
the function f ! H to be learned and a regularization term,
X ( f 2H ), that controls its capacity (i.e., excess of flexibility).
59
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