Introduction to Remote Sensing

Guido Visconti

doi:10.1007/978-3-319-29449-0_6

Fundamentals of Physics and Chemistry of the Atmosphere pp 161-180 | Cite as

Introduction to Remote Sensing

Authors
Authors and affiliations

Guido Visconti

Chapter

754 Downloads

Abstract

In this chapter, we will introduce what is essentially new for this book and that these are the principles of remote sensing techniques. Remote sensing in general includes all the techniques that can obtain information about the nature of the surface and the atmospheres of the Earth and planets. The most ancient techniques are visual and telescopic observations on other planets well before sophisticated instrumentation would be orbited around them. In this chapter, we will concentrate on observations of the atmosphere considering that they can be made either from space or from the ground. It is very instructive to go through this introduction because in practice it applies most of the things we introduced in the earlier chapters. We will omit in this first, preliminary approach many of the technicalities and details that the interested reader could find in books on remote sensing.

Keywords

Optical Thickness Total Ozone Transmission Function Radio Occultation Total Ozone Column

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, to check access.

Supplementary material

Open image in new window

E.6 Examples

E.6.1 Refractive Index of Air

We have worked out this problem at microscopic level in Chap. 2, but here we would like to explain how to arrive to Eq. ( 2.18). We start by considering that inside a dielectric the effective electric field is given by

$$ {\mathbf{E}}^{\mathbf{\prime}}=\mathbf{E}+\mathbf{P}/3{\varepsilon}_0 $$

(E.6.1)

where P is the polarization vector and ε₀ is the dielectric constant. We then have for the polarization vector

$$ \mathbf{P}=\alpha N{\mathbf{E}}^{\mathbf{\prime}}=\alpha N\left(\mathbf{E}+\mathbf{P}/3{\varepsilon}_0\right) $$

where N is the number of molecules per unit volume and α is the polarizability. And then it follows that

$$ \mathbf{P}=\alpha N{\varepsilon}_0\mathbf{E}/\left(1-\alpha N/3\right) $$

(E.6.2)

To define the refractive index, we have to find the dielectric constant ε using the electric displacement vector:

$$ \mathbf{D}={\varepsilon}_0{\varepsilon}_r\mathbf{E}={\varepsilon}_0\mathbf{E}+\mathbf{P}\Rightarrow {\varepsilon}_r=1+\mathbf{P}\cdot \mathbf{E}/{\varepsilon}_0{E}^2 $$

(E.6.3)

The index of refraction n is then

Open image in new window

(E.6.4)

Substituting for P obtained from (E.6.2), we obtain

$$ N\alpha =3\frac{n^2-1}{n^2+2}\approx 2\left(n-1\right) $$

(E.6.5)

considering that n is close to unity. This is called the Clausius–Mossotti equation. We get immediately

$$ n=1+N\alpha /2 $$

This means that the refractive index of a gas depends on the volume concentration of the gas. For a gas mixture, it follows that

$$ n-1=\frac{1}{2}N{\displaystyle \sum_i{y}_i}\left({n}_i-1\right) $$

(E.6.6)

where y_i is the mole fraction of species i and m_i is its refractive index. The quantity n − 1 is known as refractivity. For air, (E.6.6) can be written as

$$ n-1={k}_1\frac{p}{T}{z}_a^{-1}+{k}_2\frac{e}{T}{z}_w^{-1}+{k}_3\frac{e}{T^2}{z}_w^{-1} $$

where e is the partial pressure of water vapor, p is the partial pressure of dry air, T is the absolute temperature, k_i are constants, and z_a,w are deviations from the ideal gas law.

E.6.2 The Abel Transform

The Abel transform can be introduced as the projection of a circularly symmetric object as shown in Fig. E.6.1. Suppose the different shades of gray represent density which is a function only of the distance from the center of the circles, r. An observer in O will see

Fig. E.6.1
The observation geometry of an object with spherical symmetry

$$ F(y)={\displaystyle {\int_{-\infty}^{\infty }}f(r)dx=}{\displaystyle {\int_{-\infty}^{\infty }}f\left({\left({x}^2+{y}^2\right)}^{1/2}\right)dx} $$

(E.6.7)

where

$$ dx=rdr/{\left({r}^2-{y}^2\right)}^{1/2} $$

Being f(r) symmetric, (E.6.7) can be written as

$$ {\displaystyle {\int_{-\infty}^{\infty }}f(r)dx=}2{\displaystyle {\int_0^{\infty }}f(r)dx} $$

Again substituting the expression for dx as a function of r, we obtain the definition of the Abel transform:

$$ F(y)={\displaystyle {\int_y^{\infty}}\frac{f(r)rdr}{\left({r}^2-{y}^2\right)}} $$

(E.6.8)

The inverse Abel transform can be obtained from

$$ f(r)=-\frac{1}{\pi }{\displaystyle {\int_r^{\infty}}\frac{dF}{dy}}\frac{dy}{{\left({y}^2-{r}^2\right)}^{1/2}} $$

(E.6.9)

A verification of this relation can be obtained by integrating by parts (E.6.8). We have

$$ F(y)=-2{\displaystyle {\int_y^{\infty }}{f}^{\prime }(r)\sqrt{\left({r}^2-{y}^2\right)}}dr $$

where f′(r) is the derivative of f(r). Differentiating we get

$$ \frac{dF}{dy}=2y{\displaystyle {\int_y^{\infty}}\frac{f^{\prime }(r)}{\sqrt{r^2-{y}^2}}}dr $$

that can be substituted in (E.6.9) to have

$$ -\frac{1}{\pi }{\displaystyle {\int_r^{\infty}}\frac{dF}{dy}}\frac{dy}{{\left({y}^2-{r}^2\right)}^{1/2}}=\frac{1}{\pi }{\displaystyle {\int_y^{\infty }}{\displaystyle {\int_r^{\infty}}\frac{-2y}{{\left[\left({y}^2-{r}^2\right)\left({s}^2-{y}^2\right)\right]}^{1/2}}{f}^{\prime }(s) dsdy}} $$

The last integral can be solved with Fubini’s theorem:

$$ {\displaystyle {\int_r^{\infty }}{\displaystyle {\int_r^s}\frac{-2y}{\pi {\left[\left({y}^2-{r}^2\right)\left({s}^2-{y}^2\right)\right]}^{1/2}} dy{f}^{\prime }(s)ds}}={\displaystyle {\int_r^{\infty}}\left(-1\right){f}^{\prime }(s)ds=f(r)} $$

References¹

Books

Liou KN (1980) An introduction to atmospheric radiation. Academic, New YorkGoogle Scholar
Stephens GL (1994) Remote sensing of the lower atmosphere: an introduction. Oxford University Press, OxfordGoogle Scholar
Visconti G et al (eds) (2007) Observing system for atmospheric composition. Springer, New YorkGoogle Scholar

Copyright information

Authors and Affiliations

Guido Visconti
- 1

1.Dipartimento di Scienze Fisiche e ChimicheUniversità dell’AquilaCoppito, L’AquilaItaly

Introduction to Remote Sensing

Abstract

Keywords

Supplementary material

References¹

Books

Copyright information

Authors and Affiliations

Personalised recommendations

Cite chapter

Introduction to Remote Sensing

Abstract

Keywords

Supplementary material

References1

Books

Copyright information

Authors and Affiliations

Personalised recommendations

References¹