Fundaments of Optical Remote Sensing

# Fundaments of Optical Remote Sensing

## Remote sensing or what ?

In our case we will consider only Space Technologies suitable for:

Telerilevamento (from the Vocabulary of Italian Language, Treccani, 2000)
Télédétection in French language

"...long-distance detection of the appearance and the situation of a territory, particularly its weather and environmental conditions ... its natural resources (forests, water, etc..) usually by air or, now, almost exclusively from artificial satellites... "equipped with sensors sensitive to different types of electromagnetic waves.

## Solid angle Ω

The solid angle $$\Omega$$ is the ratio of the area $$\sigma$$ intercepted on the surface of a sphere of radius R by a cone with the vertex in the center of the sphere and the square of the sphere radius ($$R^2$$). The measuring unit is the steradian [sr]

$$\bbox[5px, border: 2px solid #2D5595]{\Omega = \frac{\sigma}{R^2}}$$

$$\Omega=\frac{\sigma}{R^2}=\frac{\sigma '}{R^{ '2}}$$

In the case of a sphere:

$$\Omega = \frac{\sigma}{R^2}=\frac{4\pi R^2}{R^2}=4\pi sr$$

The monochromatic radiance represents the energy collected in a unit of time and solid angle when the unit of surface of a detector is crossed by e.m. radiation of wavelength in between $$\lambda$$ and $$\lambda + d\lambda$$ and directed orthogonally to the surface A.

$$\bbox[5px, border: 2px solid #2D5595]{I_\lambda = \frac{dE_\lambda}{dt \cdot dA \cdot d\lambda \cdot d\Omega}}$$ $$[I_\lambda]=[Watt \cdot m^{-2} \mu m^{-1} sr^{-1}]$$

$$dE_\lambda=$$energy, [J]; $$dt=$$time[s]; $$dA=$$collecting area [$$m^2$$];
$$d_\lambda=$$wavelength interval [$$\mu m$$]; $$d\Omega=$$collecting solid angle [ster]

In general $$\Theta \ne 0$$

$$\bbox[15px, border: 2px solid #003366]{I_\lambda = \frac{dE_\lambda}{dt \cdot dA \cdot \cos{\Theta} \cdot d\lambda \cdot d\Omega}}$$

## From sensors to radiation sources

 To the sensor From the source EXEMPLE:THE SUN RADIANT POWER $$f = dE / dt$$ [W] LUMINOSITY $$f = dE / dt$$ [W] Ex.: Sun Luminosity $$f = 3.90 \times 10^{26} W$$ IRRADIANCE $$F = dE / dt / dA$$ $$[W m^{-2}]$$ (RADIANT) EXISTANCE $$F = dE / dt /dA$$ $$[W m^{-2}]$$ Ex: Sun Exitance for a Sun radius $$R = 7 \times 10^8m$$ $$F(Sun) = \frac{3.90 \times 10^{26}}{4\pi (7 \times 10^8)^2} = 6.34 \times 10^7 W m^{-2}$$ RADIANCE $$I = dE / dt / d\Omega$$ $$[W m^{-2}st^{-1}]$$ RADIANCE (BRILLANCE) $$I = dE / dt / dA / d\Omega$$ $$[W m^{-2}st^{-1}]$$ Ex.: Radiance leaving from Sun surface (assumed Lambertian i.e. isotropic) $$I = F/\pi = 2.02 \times 10^7 W m^{-2} st^{-1}$$

## Inverse square law

$$f_\lambda=\frac{dE_\lambda}{dt \cdot d\lambda \cdot}=\int F_\lambda dA = \int dA \int I_\lambda(\Theta, \phi) cos\Theta d\Omega$$

$$f_\lambda =$$ cost $$\leftarrow$$ cons. of the Energy

$$f_\lambda = F_\lambda \int dA = F_\lambda \cdot 4\pi d^2 = F_\lambda^{'} \cdot 4\pi d^{'2} = f_\lambda^{'}$$

for a punctual Lambertian source

$$F_\lambda (d) = \frac{f_\lambda}{4\pi d^2}$$

$$F_\lambda(d) \propto \frac{f_\lambda}{d^2}$$

$$\frac{F_\lambda (d_2)}{F_\lambda (d_1)} = \frac{d_1^2}{d_2^2}$$

Planck's function

$$\bbox[5px, border: 2px solid #003366]{B_\lambda(T) = \frac{2~~h~~c^2}{\lambda^5(e^{\frac{h~~c}{\lambda~~k~~T}}-1)}}$$

$$B_\lambda(T)=$$ monochromatic radiance emitted at the
wavelength $$\lambda$$ by a blackbody at the absolute
temperature $$T [W/m^3]$$
$$c=2.998 \cdot 10^8 m/s$$ speed of light in the vacuum
$$h=6.626 \cdot 10^{-34} Js$$ Planck’s constant
$$k=1.380 \cdot 10^{-23} J/K$$ Boltzman constant

## Reyleigh-Jeans approximation

$$B_\lambda=\frac{2~~h~~c^2}{\lambda^5{(e^{\frac{h~~c}{\lambda~~k~~T}}-1)}}$$ if $$\lambda>0.001m$$      $$T$$~$$300K$$

Planck

$$x=\frac{h~~c}{\lambda~~k~~T}\rightarrow 0$$

$$e^x = 1 + x + ....$$ Taylor

$$e^{\frac{h~~c}{\lambda~~k~~T}}\approx 1 + \frac{h~~c}{\lambda~~k~~T}$$

$$B_\lambda(T)\approx \frac{2~~h~~c^2}{\lambda^5(1+\frac{h~~c}{\lambda~~k~~T}-1)}$$

$$\bbox[5px, border: 1px solid #003366, #CCCCFF]{B_\lambda(T) \approx \frac{2 k c T}{\lambda^4} \propto T}$$
Reyleigh-Jeans

## Emissivity

$$B_\lambda(T) = \frac{2~~h~~c^2}{\lambda^5(e^{\frac{h~~c}{\lambda~~k~~T}}-1)}$$
Thermally emitted radiation $$\bbox[5px, border: 2px solid #008080]{I_\lambda(T)=\epsilon_\lambda B_\lambda(T)}$$
Emissivity $$\bbox[5px, border: 2px solid #2D5595]{\epsilon_\lambda = \frac{I_\lambda(T)}{B_\lambda(T)} \leq 1}$$

## Brightness Temperature

Planck's function
$$\bbox[5px, border: 2px solid #003366]{B_\lambda(T) = \frac{2~~h~~c^2}{\lambda^5(e^{\frac{h~~c}{\lambda~~k~~T}}-1)}}$$

Blackbody Temperature
$$\bbox[5px, border: 2px solid #003366]{T = \frac{hc}{\lambda k} \frac{1}{\ln[1+\frac{2~hc^2}{\lambda^5 B_\lambda(T)}]}}$$

Brightness Temperature
$$\bbox[5px, border: 2px solid #003366]{T_B = \frac{hc}{\lambda k} \frac{1}{\ln[1~+~\frac{2~hc^2}{\lambda^5 I_\lambda(T)}]}}$$

$$\bbox[5px, border: 2px solid #2D5595]{T_B \leq T}$$

## Exercise: Moon surface temperature

$$T_s = \frac{hc}{\lambda~k} \frac{1}{\ln[1~+~\frac{2~hc^2}{\lambda^5~I_\lambda(T)}]}$$

$$\bbox[5px, border: 2px solid #003366]{I_{\lambda=10\mu} = 3 \cdot 10^4 erg \cdot s^{-1} \cdot cm^{-2} \mu m^{-1} sr^{-1}}$$

wavelength $$\lambda = 10\mu$$ by he Moon surface
$$c\sim 3 \cdot 10^{10} cm/s$$ speed of light in the vacuum
$$h\sim 6,6 \cdot 10^{-27} erg \cdot s$$ Planck's constant
$$k\sim 1,4 \cdot 10^{-16} erg/K$$ Boltzman constant

### Brightness Temperature

$$T_B = \frac{hc}{\lambda k} \frac{1}{\ln[1~+~\frac{2~hc^2}{\lambda^5 I_\lambda(T)}]} = \frac{6,6 ~\cdot~ 10^{-27} erg ~\cdot~ s ~\cdot~ 3 ~\cdot~ 10^{10} cm~/~s}{10 ~\cdot~ 10^{-4} cm ~\cdot~ 1,4 ~\cdot~ 10^{-16} erg~/~K} \frac{1}{\ln[1~+~\frac{2 ~\cdot~ 6,6 ~\cdot~ 10^{-27} erg ~\cdot~ s ~\cdot~ (3 ~\cdot~ 10^{10})^2 cm^2 / s^2}{10 \mu m ~\cdot~ (10 ~\cdot~ 10^{-4} cm)^4 ~\cdot~ 3 ~\cdot~ 10^4 erg ~\cdot~ s^{-1} ~\cdot~ cm^{-2} \mu m^{-1} sr^{-1}}]}=$$
$$=\frac{6,6~\cdot~3~\cdot~10^2}{1,4}\frac{1}{\ln[1~+~\frac{2~\cdot~6,6~\cdot~9}{3}sr]}K= \frac{6,6~\cdot~3~\cdot~10^2}{1,4}\frac{1}{3,7}K \approx 382~K$$
$$\bbox[5px, border: 2px solid #2D5595]{T \geq T_B}$$

## Exercise: Earth's surface temperature

$$\bbox[5px, border: 2px solid #003366]{I_{\lambda=10\mu} = 10^4 erg \cdot s^{-1} \cdot cm^{-2} \mu m^{-1} sr^{-1}}$$

wavelength $$\lambda = 10\mu$$ by he Moon surface
$$c\sim 3 \cdot 10^{10} cm/s$$ speed of light in the vacuum
$$h\sim 6,6 \cdot 10^{-27} erg \cdot s$$ Planck's constant
$$k\sim 1,4 \cdot 10^{-16} erg/K$$ Boltzman constant

### Brightness Temperature

$$T_B = \frac{hc}{\lambda k} \frac{1}{\ln[1~+~\frac{2~hc^2}{\lambda^5 I_\lambda(T)}]} = \frac{6,6 ~\cdot~ 10^{-27} erg ~\cdot~ s ~\cdot~ 3 ~\cdot~ 10^{10} cm~/~s}{10 ~\cdot~ 10^{-4} cm ~\cdot~ 1,4 ~\cdot~ 10^{-16} erg~/~K} \frac{1}{\ln[1~+~\frac{2 ~\cdot~ 6,6 ~\cdot~ 10^{-27} erg ~\cdot~ s ~\cdot~ (3 ~\cdot~ 10^{10})^2 cm^2 / s^2}{10 \mu m ~\cdot~ (10 ~\cdot~ 10^{-4} cm)^4 ~\cdot~ 1 ~\cdot~ 10^4 erg ~\cdot~ s^{-1} ~\cdot~ cm^{-2} \mu m^{-1} sr^{-1}}]}=$$
$$=\frac{6,6~\cdot~3~\cdot~10^2}{1,4}\frac{1}{\ln[1~+~\frac{2~\cdot~6,6~\cdot~9}{1}sr]}K= \frac{6,6~\cdot~3~\cdot~10^2}{1,4}\frac{1}{4,8}K \approx 296~K$$
$$\bbox[5px, border: 2px solid #2D5595]{T \geq T_B}$$

## Brightness Temperature

### (Reyleigh-Jeans approximation)

Planck's function
$$\bbox[5px, border: 2px solid #003366]{B_\lambda(T) \approx \frac{2~kcT}{\lambda^4} ~\propto~T}$$

Blackbody Temperature
$$\bbox[5px, border: 2px solid #003366]{T \approx \frac{\lambda^4}{2~kc} B_\lambda(T)}$$

$$T_B \approx \frac{\lambda^4}{2kc}I_\lambda(T)\approx\frac{\lambda^4}{2kc}\epsilon_\lambda B_\lambda(T)\approx\frac{\lambda^4}{2kc}\epsilon_\lambda\frac{2kc}{\lambda^4}T=\epsilon_\lambda T$$

$$\bbox[5px, border: 2px solid #2D5595]{T_B\sim\epsilon_\lambda T}$$ $$\bbox[5px, border: 2px solid #2D5595]{T_B\leq T}$$

## Wien's Displacement Law

$$\bbox[5px, border: 2px solid #2D5595]{\lambda_{MAX}(\mu m)=\frac{2898}{T(K)}}$$

## Exercise: choosing the best spectral range

Earth surface $$T=300K$$

$$\lambda_{MAX}=10\mu m=$$TIR

fires, lava flows $$T=800~~-1000~K$$

$$\lambda_{MAX}=3-4\mu m =$$MIR

Sun photosphere $$T=6000~K$$

$$\lambda_{MAX} = 0,5\mu m =$$ VIS

$$\lambda_{MAX}(\mu m)\approx\frac{3000}{T(K)}$$

Conservation of the Energy
$$\bbox[5px, border: 1px solid black]{E_i = E_{rif} + E_{ads} + E_{tras}}$$

Conservation of the energy
$$\bbox[5px, border: 1px solid black]{E_{rif} + E_{ads} + E_{trans} = E_i}$$

$$\bbox[5px, border: 1px solid black]{I_\lambda^R + I_\lambda^A + I_\lambda^T = I_\lambda^I}$$

$$\bbox[5px, border: 1px solid black]{\frac{I_\lambda^R}{I_\lambda^I}+\frac{I_\lambda^A}{I_\lambda^I}+\frac{I_\lambda^T}{I_\lambda^I}=\frac{I_\lambda^I}{I_\lambda^I}=1}$$

$$\rho_\lambda=\frac{I_\lambda^R}{I_\lambda^I}$$

$$\alpha_\lambda=\frac{I_\lambda^A}{I_\lambda^I}$$

$$\Im_\lambda=\frac{I_\lambda^T}{I_\lambda^I}$$

Reflectance

Absorbance

Transmittance

monochromatic

$$\bbox[5px, border: 1px solid black]{\rho_\lambda+\alpha_\lambda+\Im_\lambda = 1}$$

## Spectral Signatures in the laboratory

$$\bbox[5px, border: 1px solid black]{\rho_\lambda=\frac{I_\lambda^D}{I_\lambda^S}}$$

$$\bbox[5px, border: 1px solid black]{\epsilon_\lambda=\frac{I_\lambda^D}{B_\lambda(T)}}$$

$$\bbox[5px, border: 1px solid black]{\Im_\lambda=\frac{I_\lambda^D}{I_\lambda^S}}$$

## Kirchoff law

at the thermodynamic equilibrium:

$$\bbox[15px, border: 2px solid blue, white]{\epsilon_\lambda = \alpha_\lambda}$$
$$\bbox[15px, border: 1px solid black]{\rho_\lambda+\alpha_\lambda + \Im_\lambda = 1}$$ $$\bbox[15px, border: 1px solid black]{\rho_\lambda+\epsilon_\lambda + \Im_\lambda = 1}$$
for not transparent materials: $$\Im_\lambda \approx 0$$ $$\bbox[5px, border: 1px solid black]{\rho_\lambda + \alpha_\lambda = 1}$$

at the termodynamic equilibrium (Kirchoff)

$$\bbox[15px, border: 1px solid black]{\epsilon_\lambda = \alpha_\lambda = 1 - \rho_\lambda}$$

### Our daily remote sensing experience in the visible (VIS) spectral range

Newton prism and light colors

## Measuring reflectances in the laboratory

$$\bbox[5px, border: 1px solid black]{\rho_\lambda=\frac{I_\lambda^R}{I_\lambda^I}}$$ Reflectance

## Measuring reflectances in the laboratory

$$\bbox[5px, border: 1px solid black]{\rho_\lambda = \frac{I_\lambda^R}{I_\lambda^I}}$$ Reflectance

# Beyond visible.

## TUTORIAL RGB Landsat-ETM

• RGB true color on MODIS image

• RGB false color to enhance vegetation (RED → NIR)

Which "colors"
beyond visible?

## Spectral signature of water

$$\bbox[5px, border: 1px solid black]{\rho_\lambda + \alpha_\lambda + \Im_\lambda = 1}$$

## Spectral signatures of snow and Clouds (in reflectance)

• TUTORIAL
RGB MODIS FOR
DISCRIMINATING SNOW FROM
CLOUDS

## Spectral signatures of vegetation

$$\bbox[5px, border: 1px solid black]{\rho_\lambda + \alpha_\lambda + \Im_\lambda = 1}$$
Total
Chlorophyll
Concentration
($$\mu g \, cm^{-2}$$)

## Spectral signatures of vegetation

$$\bbox[5px, border: 1px solid black]{\rho_\lambda + \alpha_\lambda + \Im_\lambda = 1}$$

## Spectral signatures of vegetation (reflectance) and water (absorption)

Water depth
[$$cm^{2}$$]

## Spectral signatures of vegetation in different condition of water stress water content %

### Measuring reflectances in the lab & in the field

#### (Field Spec ASD Spectrometer)

$$\bbox[15px, yellow]{NDVI = \frac{R_{NIR}-R_{VIS}}{R_{NIR} + R_{VIS}}}$$

## NDVI

### (Normalized Difference Vegetation Index)

• TUTORIAL

on NDVI computation and applications

• ### Example of NDVI reduction after fires

##### Low
Burned Areas Wooded (ha) Mixed Covers (ha)
A - Motta S.Giovanni e Melito di Porto Salvo 141 3320
B - Bova e Bova Marina 3600 3300
C - Grotteria e Roccella Ionica 1150 2413
D - Bivongi 500 1000

## Applications

### Vegetation (Normalised Difference Vegetation Index)

$$\bbox[15px, yellow]{NDVI = \frac{R_{NIR}-R_{VIS}}{R_{NIR} + R_{VIS}}}$$

### Maximum Value Composite (MVC)

1 2 3 4 5
MAX (1,2,3,4,5)

## SEASONAL CHANGE AT REGIONAL SCALE

JUNE     NOVEMBER

CLOUDS 0.0 0.15 0.25 0.35 0.45 0.55 0.60 0.70

$$\bbox[15px, yellow]{NDVI = \frac{R_{NIR}-R_{VIS}}{R_{NIR} + R_{VIS}}}$$

## Applications

### MONITORING OF DEFORESTATION

$$\bbox[15px, yellow]{NDVI = \frac{R_{NIR}-R_{VIS}}{R_{NIR} + R_{VIS}}}$$

## Applications

### MONITORING OF DESERTIFICATION

From the analysis
of long (ten-year) time
series of NDVI

### Seasonal change at continental scale

A. April 12-May 2, 1982;
B. B. July 5-25, '82;
C. Sept. 27-Oct. 17, '82;
D. Dec. 20, '82-Jan. 9, '83

$$\bbox[15px, yellow]{NDVI = \frac{R_{NIR}-R_{VIS}}{R_{NIR} + R_{VIS}}}$$

## Spectral signatures of vegetation

$$\bbox[5px, border: 1px solid black]{\rho_\lambda + \alpha_\lambda + \Im_\lambda = 1}$$

## Atoms and molecule adsorb and emits at specific wavelength

atomic spectra molecular spectra

$$\frac{Ze^2}{r^2} = \frac{mv^2}{r} ~~~\Rightarrow~~~ r=\frac{Ze^2}{mv^2}$$

$$l_n = mvr_n = n\hbar ~~~\Rightarrow~~~ r_n = \frac{n^{2}\hbar^{2}}{me^2Z}$$

$$E=-\frac{Ze^2}{r}+\frac{1}{2}mv^2=-\frac{1}{2}\frac{Ze^2}{r}$$

$$E_n = -\frac{m_e e^4}{2\hbar^2} \cdot \left(\frac{1}{n^2}\right)Z^2$$

$$E_n = -\frac{m_e e^4}{2\hbar^2} \cdot \left(\frac{1}{n^2}\right)Z^2~~~$$energy of Bohr orbit

$$v_{jk} = R \cdot \left(\frac{1}{n_j^2}-\frac{1}{n_k^2}\right)Z^2$$

with      $$R=$$      Rydberg constant $$~~~=\frac{2 \cdot \pi^2 \cdot m_e \cdot e^4}{h^3}$$

$$E_{mol}=E_{trasl}+E_{ele}+E_{vib}+E_{rot}$$

$$E_{trasl}~\propto~KT$$

$$E_{vib}^v = \hbar\sqrt{\frac{K}{\mu}}\left(V+\frac{1}{2}\right)~~~\Delta V = \pm 1$$     with overtones

$$E_{rot}^J = \frac{\hbar^2}{2I}J(J+1)~~~\Delta J=\pm 1$$

$$v_{jk} = \frac{E_k-E_j}{h}$$

## Basic principle of multi-spectral remote sensing

• Every material emits, absorbs, transmits or reflects differently at different wavelengths, the em radiation, according to their chemical and physical characteristics

• The spectral response curve represents, in this sense, its spectral signature

• The availability of multi-spectral observations allows, in theory, to trace back to the chemical and physical characteristics of the material that has emitted, absorbed, transmitted or reflected measured em radiation.

# From the lab to the space.

## Multispectral digital image

EO Sensors
Image systems and multispectral images

EO Sensors
Image systems and multispectral images
• TUTORIAL
Spectral Signatures from
MODIS

• Natural sources of radiation for Earth Observation
The Sun(only daytime !)
Wien

$$\bbox[15px, border: 1px solid black]{\lambda_{MAX}=\frac{2898}{T(K)}\approx\frac{3000}{6000}=0,5\mu m}$$ $$T\approx 6000K$$

Natural sources of radiation for Earth Observation
The Earth itself (day and night !)
Wien

$$\bbox[15px, border: 1px solid black]{\lambda_{MAX}=\frac{2898}{T(K)}\approx\frac{3000}{300}=10\mu m}$$ $$T\sim 300K$$

 Peak Wavelength Max. Intensity Sun $$0.5$$ micrometers $$7.35 \times 10^7 W/m^2$$ Earth $$10$$ micrometers $$390 W/m^2$$

## Natural sources of radiation actually available for EO

 Peak Wavelength Max. Intensity Sun $$0.5$$ micrometers $$7.35 \times 10^7 W/m^2$$ Earth $$10$$ micrometers $$390 W/m^2$$

## Passive Earth Observing Techniques

Natural source of
from the target

Diagnostic parameter:
$$\rho_\lambda$$

Natural source of
coincident with the target

Diagnostic parameters:
$$\epsilon_\lambda, T$$

## Active Earth Observing Techniques

The signal is generated
by the observation
system itself

Diagnostic Parameter

$$I_\lambda^{back} \propto$$ roughness
$$\frac{c\Delta t}{2}=d=$$ distance

e.g. SAR (Synthetic

## Different spectral region different capabilities

This pair of images demonstrates some of the differences between passive and active sensors. The top image is an aerial photograph (which records reflected light) of Amundsen-Scott Station, a research facility built on the South Pole. The bottom image is the same area, at approximately the same scale and orientation, from RADARSAT. RADARSAT is an active remote sensing instrument. Microwaves are generated by the sensor, reflected from the Earth's surface and back to the sensor. The radar image reveals an abandoned cluster of buildings (to the lower left of the bright dome) that are now buried under Antarctic ice. (RADARSAT image courtesy Canadian Space Agency)

## Scattering in the atmosphere

In general we can have scattering for whatever
wavelength $$0 < \lambda < \infty$$
in the atmosphere we have scattering from particle
having dimension like:

$$10^{-8} < d < 1 \quad cm$$
$$[X~rays]~~~~~[microwaves]$$

different wavelengths by
common atmospheric constituents
and other particles (Source:
Tomlinson 1972)

## Extinction of solar radiation in the atmosphere

Atmospheric
Transmittance and
atmospheric windows

## AVHRR: all bands in atmospheric windows !

• The 5/6 AVHRR channels spectral positions compared
with corresponding atmospheric transmittance
• ## Penetration of e.m. radiation into the matter

More in depth

Penetration of em radiation in a medium
with complex refractive index

• TUTORIAL

sediments or shallow water?

• ## Sediments or shallow water ?

LANDSAT -MSS

Band 4 = Blue    Band 5 = Green    Band 7 = Red

Multitemporal approach (Di Polito et al., 2016)