Optical Depth

8-1 ECE 583 8-2 Radiant flux at the surface Lecture 8 Measurements at the ground observe an attenuated Atmospheric scattering and optical thickness, Aerosol size and radiance in the the radiant flux Mie scattering, Diffuse-light effects •If the wavelength interval is small then

• The exponential contains the optical depth , • Optical depth is related to the attenuation properties of the material • Optical depth is additive, but only if the spectral dependence is uniform across the pass band. • The above is satisfied for spectral bandpasses less than about 10 nm • If spectral regions with absorption are avoided then optical depth is reasonably constant within a 10-nm bandpass • Have neglected diffuse light in determining the radiant flux • The is the quasi-monochromatic radiant flux measured at the top of the atmosphere • Earth-sun distance here is one AU

8-4 8-3 Working voltage relation Effect of the Atmosphere on Radiation

Putting everything together to this point gives us our working Atmosphere can refract, scatter, and absorb energy voltage relation for a solar photometer observation (transmit means nothing happened) •This is the voltage output of a sensor viewing the sun from the bottom of •Refraction refers to the bending of radiation from its original path the atmosphere •Narrow bandpass •Weather radar and horizontal view remote sensing must consider it •Mean earth-sun distance •Only a major factor in long path lengths through the atmosphere •Mirages are a refractive phenomenon •The exponential term is the transmittance •We will not be concerned with refraction in this course •Scattering and absorption together also referred to as attenuation or extinction •Scattering refers to the redirection of radiation after interaction with a particle in the atmosphere •Molecular scattering •Aerosol scattering •Absorption refers to the incorporation of the radiation within the particle to alter the energy state of the particle •Gaseous absorption •Aerosol absorption 8-5 Atmospheric attenuation 8-6 Extinction coefficient and optical depth

Recall the exponential law of attenuation Extinction coefficient is an inherent property of a given material related to how efficiently it scatters or absorbs •Optical depth includes the distance that a photon must travel to determine the probability that a photon is scattered or absorbed •Often referred to as Beer’s Law, Lambert-Bouget Law •Extinction coefficient has units of per distance [m-1] • For most of the applications in solar radiometry a 10-nm wide bandpass •Optical depth is unitless can be considered monochromatic • Strictly speaking Beer’s Law is only for monochromatic case •The exponential term gives us the transmittance • Then

8-7 Extinction coefficient for Ozone 8-8 Extinction coefficient for Ozone

Thus, transmittance is based on the optical depth which is based on the extinction coefficient which includes absorption •Absorption by ozone in the Chappuis spectral region •Figure here based on Vigroux data measured for 1 cm of ozone at 273 K and 1013 mb 8-9 Extinction coefficient versus altitude 8-10 More on Optical Depth

Model atmospheres have been developed that give extinction Optical depth describes the attenuation along a path in the coefficient versus height for different materials atmosphere

•Based on both scattering and absorption Rayleigh Aerosol Aerosol •Broken down by component material •Valid only for a specific wavelength •Based on measurements •Graphs here are for 600 nm

1.000E+0 Rayleigh Ozone Extinction 1.000E- Aeroso 1 l Volcanic 1.000E-2 Enhanced Aerosol

1.000E-3

1.000E-4

1.000E-5

1.000E-6 0 10 20 30 40 50 Altitude

8-11 Optical depth versus height 8-12 Optical depth

Can convert the extinction coefficients to optical depths from a The previous spectral graphs have shown how transmittance given height to the surface or top varies with wavelength •Optical depth is typically an integral quantity •Now examine the variation in optical depth as a function of wavelength without strong absorption •Extinction includes all three (aerosol, ozone, and Rayleigh) •Plot here shows the total, molecular, aerosol, and ozone optical depths •These types of data sets are useful in instrument design and predictions of expected values 8-13 Optical depth - log/log scale 8-14 Log/log scale

Putting the data on log/log scale produces interesting results Focus on the VNIR allows more details to be seen •Molecular and aerosol optical depths become linear 0 •Ozone absorption is easier to see -1

1 -2 Aerosol Total -3

0. -4 1 -5

-6 0.01 Ozone Molecular -7 -8 0.001 -9

-10 0.0001 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0.1 1 10 log Wavelength (micrometers) Wavelength (micrometers)

8-16 Scattering 8-15 Where are we so far?

At this point, we have developed a set of equations relating an instrument measurement to the atmospheric optical depth and the incident solar irradiance •Have equations for radiant flux and voltage

•A calibrated instrument allows conversion of voltage to radiant flux •Knowledge of the solar irradiance model then allows determination of atmospheric transmittance and optical depth •Will use the spectral characteristics of optical depth to retrieve atmospheric composition 8-17 Scattering 8-18 Junge Aerosol Size Distribution

Scattering is the process by which a particle that redirects the A common size distribution used to describe aerosols is the incident energy into a new direction Junge distribution •Particle must be of a different index of refraction than the surrounding •Size distribution gives the number of particles at a given particle size medium •Simplest distribution is monodisperse distribution where all particles are •Size of the particle plays a role in the amount and directionality of the the same size scattering •Junge distribution is slightly more complicated •Size also plays a role in what models can be used to determine the •Based on measurements of particles in the stratosphere (10-50 km) scattering properties •Also referred to as a Power Law Distribution •Typical particle sizes are -3 -2 •Aitken nucleus - 10 -10 :m -2 •Haze particle - 10 -1 :m -4 where ν is the Junge parameter with typical values between 2 and 4 •Air molecule - 10 :m •Major advantage is it simulates distributions with a single parameter •Fog and cloud droplets - 1 - 10 :m 2 4 •Raindrop - 10 -10 :m •More realistic models are log-normal and sums of log-normals •Junge model has problems at small radii •Must keep in mind that models are simply that

8-19 Extinction coefficient - scattering 8-20 Backscatter coefficient 8-21 Scatter cross-section 8-22 Rayleigh Scatter

8-23 Rayleigh scatter 8-24 Molecular optical depth 8-25 Mie scattering 8-26 Mie scattering The angular scattering coefficients Aerosol particles are too large to fall under the Rayleigh assumption for scattering in the VNIR/SWIR depend in a complex •Aerosols are approximately the same size as the wavelength of light in this manner on particle parameters spectral range •Mie scattering is a general approach to describing scattering by particles •Particle radius - in reality an effective radius •Relies on Bessel function approach •“Exact” solution under Maxwell’s equations since we really don’t have •Rayleigh scattering is a subset of Mie scattering (best way to test a Mie computer code is to put in small particles) spheres •Spherical shape •Atmospheric aersols are not spheres •Index of refraction for the particles •Now feasible to determine scattering for non-spheres •More complicated particles than columns, oblate spheroids, plates are •Complex number in general not readily soluble •Imaginary component relates to the absorption by •Will rely on Mie scattering concepts for this class because it allows for a calculation of the needed coefficients to describe the scattering the particle •Real part has larger effect on angular aspect •mparticle=mreal-imimag •Wavelength

8-28 8-27 Mie scattering coefficients Scattering Rayleigh scattering 8-30 Mie scattering example e −ik(R−ct) E = E k 2α Polarizability r 0r R p=αE Computers provide an opportunity to model the effects of Mie scattering •Example here is from two wavelengths for Spherical wave the L model haze distribution below form •Show the polarization components e −ik(R−ct) = 2α Θ El E0l k cos •Note the change in R the scattering coefficients with k 4 α 2 wavelength Ir = I0r 2 •Haze particles R do not cause as much 4 2 k α 2 polarization as the I = I cos Θ l 0l R 2 molecules π λ 4 →λ-4 Unpolarized radiation (2 / ) 1 k 4 α 2 I = (I + I ) = I (1 + cos2 Θ) 2 r l 0 R 2 I − I C P(Θ) r l = sca LP(Θ) = I I0 2 I + I 4πR r l

8-31 Mie scattering, another example 8-32 Radiative modeling

Phase function and degree of polarization for three size Aerosol model used to model parameter cases and two aerosol types scattering in the atmosphere •m=1.33 similar to water droplets, m=1.50 would be similar to dust or •Results are for a specific pollutant wavelength •Phase function is related to the angular scattering coefficient •Optical depth used to define amount of particles •Solar zenith angle also defined •Plot shows the radiance (labeled here as intensity) as a function of view angle •Observer is at the ground •Peak shown is from scattered light only •The direct solar beam is not included in this plot 8-33 Example of forward scattering 8-34 What does all this mean?

Images below show the solar aureole due to forward scattering by aerosols Third image illustrates the brightness one obtains by including the solar beam in the image as opposed to only the aureole

8-35 Diffuse-light effects 8-36 Diffuse/Direct versus wavelength Very difficult to empirically determine the diffuse light Given sensor and sun component geometry •Detectors do not know whether a photon has been scattered •Four aerosol types •Easier at some level to model the effect to determine what cases we including model values encounter will suffer the largest error and two “measured” cases •Common practice in remote sensing (and other fields) •July and August models •Sensitivity studies will determine when it is necessary to be concerned based on Mt. Pinatubo volcanic dust about the diffuse light effect •Larger Junge value •Method is to use a radiative transfer model (developed at the UofA) o relates to smaller determine scattering based on given aerosol models particles •Determine the ratio of the diffuse to direct energy •Optical depths are given •Later use this to examine the impact on sensor calibration for a reference wavelength •For now, simply compute expected values under certain conditions 8-37 Diffuse/Direct effect with angle 8-38 Diffuse-direct versus airmass Same cases as Airmass is related to the path previous viewgraphs through the air •Single aerosol case •Will see more about airmass based on July, 1992 shortly (next lecture) volcanic model •For now consider that it is related to •Each curve represents a the sun angle separate wavelength •High airmass means sun is closer to horizon

•Same aerosols as previous viewgraph •Wavelength is 451 nm •Longer path allows for more opportunity of scattering