Page 126 - Maxwell House

106 Chapter 2

resonance (MRI) and X-ray imaging, ultrasonic imaging and night vision, seismic active and
passive imaging, detection and localization of lightning , underwater exploration, crop and
forestry inventorying for yield prediction, weather forecasting, satellite mapping, active and
passive tags, etc.

In general, using the remote sensing, we are trying to solve the so-called inverse problem: how
to reconstruct some physical object or phenomenon at a distance based on the set of more or
less complete measured data. Since it is rather seldom known how much data are enough, such
inverse problems are regarded as one of the toughest in mathematics and electrodynamics.
Nevertheless, let us show without much detail how a quite modest remote sensing problem can
be formulated and solved using the K-K relations.

Suppose we need to measure the salinity of water remotely in the middle of the deep sea. For
simplicity, assume that the weather conditions are so ideal that the sea surface is perfectly flat.
From the literature survey, we know that the complex permittivity of sea salt water strongly
depends on its salinity. If so, our identification mission is to measure this relative permittivity
remotely. One of the ways to do this is to send a broadband signal from radar installed on
satellite or aircraft in the direction of tested sea area and measure the energy reflected from the
sea surface. Another source of the signal in such test might be solar radiation. In both cases, we
can estimate relatively straightforward the power of reflected from the sea surface signal and
the magnitude of reflection coefficient |()| at multiple frequencies. Later we will show
33
that the complex reflection coefficient and permittivity () is related as

1−� ()
() =
1+� ()
� (2.116)
2 () 2
1−() 1−|()|
∴ () = � � = � �

1+() 1+|()| ()
Clearly, our real number data sample of |()| is not sufficient because the salt water absorbs
energy and its permittivity () certainly is a complex number. Therefore, we need to

determine the phase factor () of reflection coefficient. But any phase measurements are
quite challenging, typically not very accurate and have the phase ambiguity problem or 2π-
problem. Look back at (2.111) and you will see that the phase measurement might be replaced
by analytical or numerical calculations using the second equation in (2.111) rewritten in the
form of (2.113) as

2 ∞ ln(|()|/|()|)
() = − ∫ (2.117)
0 − +
As soon the phase factor is found, the complex permittivity is fully determined. Surely on our
way to a solution, we cut the problem to bare bones and omitted many essential details. But one
problem persists in all of the K-K relations applications. According to (2.113) the knowledge
of |()| is required over the entire spectrum from 0 to infinity. Since such data are practically
not available, the integral in (2.113) is evaluated over a truncated frequency band that leads to
more or less accurate approximation for the phase factor. In case of materials with properties
described by (2.111) we can get some help.

33 Numerically, the reflection coefficient is the portion of transmitted and striking a sea surface signal
that is reflected back.