The development of the radiative transfer equation and its solution techniques in the chapter on
radiative transfer theory has been concerned with the forward or direct problem of radiative
transfer theory. The rules of the game were simple: given the inherent optical properties of the
water and the physical properties of the boundaries, find the radiance distribution throughout
and leaving the water. This problem has a unique solution, which means that a given set of IOPs
and boundary conditions yields a unique radiance distribution. The only limits on the accuracy
of computed radiances are the accuracy with which we specify the IOPs and the boundary
conditions, and the amount of computer time we wish to devote to the numerical solutions. In
this sense the direct problem of computing radiances can be regarded as solved.
Figure 1 shows the conceptual process of solving the forward problem. In principle, we can start
with the fundamental physical properties of the particles and dissolved substances in the ocean
and derive the water IOPs from the physical properties (e.g., using Mie theory to compute the
VSF from the particle properties and size distribution). In optical oceanography, we often being
with direct measurements of the IOPs. We then apply suitable boundary conditions and solve the
very complicated RTE to obtain the radiance distribution. Any other quantities of interest, such
as irradiances or AOPs can then be computed from the radiances.
Figure:
Fig. 1. The conceptual process involved in solving the forward radiative transfer problem.

This page introduces the inverse problem of radiative transfer theory: given radiometric
measurements of underwater or waterleaving light fields, determine the inherent optical
properties of the water. This is very much an unsolved problem. Both conceptual and practical
limits are encountered in inverse problems. Unfortunately, remote sensing is an inverse problem.
The first problem we encounter is uniqueness of the solution. Consider the following situation.
A body of water with a particular set of IOPs and boundary conditions has an underwater
radiance distribution
. If the boundary conditions now change, perhaps because the
sun moves, there will be a different radiance distribution
within the water, even
though the IOPs remain unchanged. Can we correctly recover the same set of IOPs from the two different light fields? Can we distinguish between
because of a change in boundary conditions, as opposed to
because of a change in IOPs? Because the same set of IOPs
can yield different radiance distributions, as we just saw, we are led to ask if two different sets of
IOPs and boundary conditions can lead to the same radiance distribution. In other words, is there
even in principle a unique solution to the inverse problem stated above?
Another problem often encountered with inverse solutions is the stability of the solution, or its
sensitivity to errors in the measured radiometric variables. In direct problems we usually find
that a small error (say 5%) in the IOPs or boundary conditions leads to a correspondingly small
error in the computed radiance. With inverse problems we often find that small errors in the
measured radiometric quantities lead to large errors, or even unphysical results, in the retrieved
IOPs. Sensitivity of the inversion scheme to small errors in the input data often renders inversion
algorithms useless in practice, even though they appear in principle to be quite elegant and
satisfactory.
It can be shown that if the full radiance distribution is measured with perfect accuracy, there is in
principle a unique inverse solution to the RTE to obtain the full set of IOPs. But from a practical
standpoint, if we have to measure the entire radiance distribution throughout the water body with
high accuracy to obtain the IOPs, we could measure the IOPs themselves just as easily. An
inverse method is useful only when it saves us time, money, or effort. What is desired is a
recovery of at least some of the IOPs from a limited set of imperfect radiometric measurements.
We have seen one example of this in Gershun's law, which allows us to recover the absorption
coefficient from measured values of the plane and scalar irradiances, if there are no internal
sources present.
In remote sensing, we have a very limited set of imperfect light field measurements, namely just
the water leaving radiance or remotesensing reflectance, from which we want to retrieve as
much information as possible about the water body. Our input measurements fall far short of
measuring the full radiance distribution, and the measurements we do have may contain
substantial errors due to poor atmospheric correction or inaccurate radiometer calibration. Thus
we expect a priori that we will not be able to recover a full set of water IOPs, and that what is
recovered may contain large errors. Oceanic remote sensing is thus a very difficult inverse
problem. The various inversion algorithms discussed on the following pages show the wide
range of techniques developed over the years to address the inherent difficulties of inverting
remote sensing measurements to obtain information about the ocean. Each of these techniques
has its strengths and weaknesses, and each is imperfect, but each still has demonstrated great
value to oceanographers.
Inversions are always based on an assumed model that relates what is known to what is desired.
The inversion is then effected by using the known quantities as inputs to the model, whose output
is an estimate of the desired quantities. In some cases the model is simple. For example, if
historical data relating the chlorophyll concentration to the ratio of remote sensing reflectance at
two wavelengths are used to find a bestfit function of the form
, then the model is that function. Inserting a newly measured reflectance at
and
then gives an estimate of the chlorophyll concentration. The accuracy of
that estimate will depend on the scatter in the original data, and on whether the water body being
studied is similar to the one used to determine the function. In other cases the model is
complicated. A neural network with may layers is a complex model in which it is often not
obvious how a particular input is related to a particular output. The accuracy of a neural network
inversion depends on how well the neural network represents nature and on the data used to train
the network.
Because of the limited measurements available for remote sensing, inversion algorithms usually
require constraints to limit the possible solutions obtained from a give remote sensing
reflectance. Constraints can be "built in," for example as simplifications to the RTE. They can
also be external, typically as additional required measurements (such as a measurement of the
water leaving radiance or bottom depth at one point in an image). They can also be implicit
constraints, such as a limitation of retrieved values to the range of values found in the data set
used to predetermine certain parameters in the inverse model.
Figure 2 summarizes the conceptual issues involved with inverting remotely sensed data to
obtain estimates of oceanic properties.
Figure:
Fig. 2. The conceptual process involved in solving remotesensing inverse radiative transfer

Classification of inverse problems
There are many kinds of inverse problems. For example, there are medium characterization
problems, for which the goal is to obtain information about the IOPs of the medium, which in our
case is the water body with all of its constituents. This is the type of problem that we consider in
this chapter. There are also hiddenobject characterization problems, for which the goal is to
detect or obtain information about an object imbedded within the medium, for example a
submerged submarine. We shall not discuss this type of problem. Inverse problems may use
optical measurements made in situ, as with the use of Gershun's equation to obtain the
absorption coefficient. Remote sensing uses measurements made outside the medium, typically
from a satellite or aircraft.
Another type of inverse problem seeks to determine the properties of individual particles from
light scattered by single particles. Such problems usually start with considerable knowledge
about the particles (for example, the particles are spherical and have a known radius) and then
seek to determine another specific bit of information (such as the particle index of refraction).
The associated inversion algorithms usually assume that the detected light has been singly
scattered. Even these highly constrained problems can be very difficult. We shall not discuss
these "individualparticle" inverse problems. In the ocean, there is no escaping multiple
scattering, which greatly complicates our problem, and we do not generally have the requisite a
priori knowledge of the individual particle properties needed to constrain the inversion.
Solution techniques to inverse problems fall into two categories: explicit and implicit. Explicit
solutions are formulas that give the desired IOPs as functions of measured radiometric quantities.
A simple example is Gershun's law when solved for the absorption in terms of the irradiances.
Implicit solutions are obtained by solving a sequence of direct or forward problems. In crude
form, we can imaging having a measured remotesensing reflectance (or set of underwater
radiance or irradiance measurements). We then solve direct problems to predict the reflectance
for each of many different sets of IOPs. Each predicted reflectance is compared with the
measured value. The IOPs associated with the predicted reflectance that most closely matches
the measured reflectance are then taken to be the solution of the inverse problem. Such a plan of
attack can be efficient if we have a rational way of changing the IOPs from one direct solution to
the next, so that the sequence of direct solutions converges to the measured reflectance or
radiance.
The follow pages show examples of both explicit and implicit inversions, with algorithms based
on analytical simplifications of the RTE, on statistical fitting of functions to historical data sets,
and on combinations of the two. Sometimes the underlying models and constraints will be
obvious, and sometimes not. In any case, studying a remotesensing algorithm in the general
context of inverse models helps to ascertain what model underlies the algorithm, what its
limitations and constraints are, and what is its expected domain of applicability.