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Course outline & Intro

at721

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On the website:“Theoretical Topics In Radiative Transfer”

In Reality:“Construction of forward models of atmospheric remote sensing instruments, and their application to inverse techniques”

Bigger Picture Questions• Forward Modeling: How can we simulate different instruments,

in order to understand their sensitivity to different atmospheric (or surface) variables we care about?

• Inverse Modeling: How can we estimate atmospheric or surface variables from remote sensing observations? (i.e. radiances)

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Example: passive microwave ocean retrievals

• State: (ask students)– Surface Wind– water vapor path– Cloud liquid water path– Precip

• Observations:– Passive microwave at different frequencies /

polarizations

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• MODEL:– State Synthetic Observations

Best Match leads to our “best-guess answer”What are the known sources of error? (Ask Students)• Instrument noise• Errors in non-retrieved parameters

– (e.g., temperature profile, cloud fraction)• Forward model errors

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Before forward model errors, what goes into a typical forward model?

• Ie, how go from state to synthetic observations? (Ask Students)

Three Main Pieces:– Physical Optical properties– Radiative transfer (Radiances or TBs typically)– Instrument model (spectral response, spatial

response, polarization response)

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Sources of Forward model error?

• Physical assumptions: (Ask students)– “Ice is spherical” (mie theory applies)– No ice is present– No water-leaving radiances– Structure of cloud profile– Cloud is plane parallel– Etc.

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Errors in optical properties calculations

– Calculation of scattering properties of non-spherical aerosols

– Gas absorption spectroscopy– Water vapor continuum absorption– Surface reflectance parameterization (lambertian,

cox-munk, polarization?)– Imperfect / approximate SS calculations

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RT errorsAssumption-based

- Plane Parallel vs. 3D/spherical shell atmosphere

- Oriented vs. non-oriented particlesCalculation-based• Zenith/azimuthal resolution (“streams”)• Neglect of polarization (very common)• Truncation of phase function

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Assumptions vs. calculations

• In most cases, calculations can be made nearly perfect given sufficient computing power (though usually not practical)

• Physical / optical / RT assumptions usually dominate errors (“ice is spherical”) rather than 2 vs. 16 streams, e.g.

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Primary Focus of this class• How to construct forward models for a range of remote

sensing problems?

• Forward models can help with many things:– Sensitivity studies– Instrument / mission design– Training look-up based retrieval algorithms– Virtually all retrieval schemes– Testing retrieval schemes (usually with more accurate forward

model)– “Direct Radiance” Data Assimilation

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Secondary focus • How might different types of forward model errors affect

retrieval errors? This generally requires a full treatment of inverse theory.

• How can we use forward models to assess the sensitivity of different observations to different physical variables? (dy/dx)

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Identify primary forward model components

– Thermal infrared temperature retrievals– Vis/NIR Aerosol retrievals over land– Cloud Optical depth / Re retrieval (a la MODIS)– Sea ice retrievals

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Thermal IR Temperature RetrievalsForward model components

• Temperature-dependent abs. coefficient calculator for target gas

• Abs coeff for other gases• Surface Emissivity model (perhaps very simple, like

0.98)• RT model to simulate TBs at TOA at an arbitrary zenith

angle• Instrument Model! How does the instrument respond

spectrally? Is monochromatic good enough? Integrate across a small piece of spectrum?

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Solar Model:• Continuum: Fit to SolSpec• Disk-Integrated Linelist (G. Toon)• Sun-Earth Doppler shift included

Instrument Model:• Spectral Dispersion• Tabulated ILS • Earth-Instrument Doppler shift included• Explicit Treatment of Polarization Angle

Atmospheric/Surface Properties• Hi-resolution points at 0.01 cm-1 spacing• Rayleigh Scattering (fully polarized)• Spectroscopy: New Line Parameters + Line Mixing + Non-Voigt Line Shapes• Aerosol/Cloud properties: single scattering properties at band endpoints• Lambertian land surface• Fully polarized Cox-Munk ocean surface model

Atmospheric Radiative Transfer• Scalar multiple-scattering code• Exact first order of scattering • 2OS polarization correction• “Low-Streams Interpolator” acceleration method

Jacobians:• Analytic derivatives calculated for all SV elements.

The OCO-2 retrieval: Forward Model Components

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Eg: MODIS

• MODIS cloud retrievals– Temperature profiles (assumed?)– Solar model– Physical cloud model

• Vertically homogeneous or not?• Particle size distribution• Calculation of Index of Refraction• Cloud Phase

– Surface albedo (from previous clear-sky obs)

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Class Topics1. Introduction (1 class)2. Physical to Optical Properties (3 weeks)

• Geometry / scattering angle• Phase Function expansion• Single scattering / Mie theory• Non-spherical particles• Combination of optical properties• Surfaces (Cox-Munk, Lambertian)

3. Radiative Transfer (6 weeks)• Stokes Parameters, Polarization• The general RT equation & component terms• Nonscattering (emission-only) RT• Scattering Techniques

• Single-scattering approximation• Multiple scattering techniques

• Polarized RT• Fast techniques for non-monochromatic channels

4. Instrument Models (1 week)5. Inverse theory (3 weeks)

• Problem set-up• Sensitivity studies; the Jacobian• Baye’s theorem• Cost Functions & Covariance matrices• Solution for Gaussian statistics & linear problem• Solution diagnostics• Nonlinear solution techniques• Information theory / channel selection

6. Class Project Presentations (1 week)

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Optical Properties?1: Index of Refraction

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Optical Properties: Gas Absorption

Desired Quantity:Absorption Cross-

section per molecule (m^2)Function of

temperature, pressure, h2o concentration.

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Optical properties: Particle “single scattering properties”

• Extinction Efficiency Qe or Cross section σe

• Single Scattering Albedo ϖ0

• Phase Matrix (a 4x4 Quantity) P(Θ)– Intensity phase function P(Θ) is the (1,1) component– Asymmetry parameter g is sometimes used as a simple substitute

Derived Quantities:– Extinction Cross Section σe = Qe π r2

– (extinction) Optical Depth: τ = σe N(z) Δz– Scattering Cross Section : σs = σe ϖ0

– Backscattering Cross Section: Cb = σe ϖ0 P(Θ=180o)

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Single-scattering regimes

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Optical Properties: Surface reflectance

• Absorptivity (may be internally transmitted)

• Emissivity (=Absorptivity)

• Reflectivity = Albedo = 1 – Emissivity

• Bidirectional Reflectance Distribution Function (BRDF) How light incident on surface can be

reflected differently into ALL outgoing angles!

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Rough surfaces: Specular vs. Lambertianincident radiation can be reflected into any direction, but amount depends on

surface

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Radiative Transfer

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(Scalar) Radiative Transfer

extinction Scattering Source Emission Source

Very hard to Solve generally, and this doesn’t even include polarization!Special Cases:• Plane Parallel: can ignore 3D nature of the problem. This is a huge

assumption but often can be justified.• Scattering not important : Equation becomes (almost) trivial and is

simple to code• Single-scattering only : Equation becomes relatively simple (especially in

1D)• Strong multiple scattering: Directional dependence is weak, can invoke

certain approximations.

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RT Topics

• Emission-only• Single-scattering• Quadrature Schemes

– “Discrete Ordinates” (DISORT)– Adding-Doubling– Successive orders of Scattering– Similarity Transformations (“Delta Scaling”)

• Monte Carlo Techniques• Polarized RT

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Instrument Modeling Topics

• Spectral Response Function (how does it accept light spectrally). Ie “wavelength”

• Spatial Response Function (ie footprint size)• Polarization response

– Intensity only?– Single or multiple polarizations?

• Noise model– Often needed for inverse models

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AVIRIS spectral movie from 0.4 to 2.4 microns Fire scene

Smoke - small part.

CloudHot Area

Smoke -large part.

Fire

Shadow

GrassLake

Soil

The different spectral regions provide different signatures of the Earth below

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10 um ‘window’

6.3 um watervapor

0.6 um visible

FUV 0.1um

Penetration

10 um ‘window’

6.3 um watervapor

0.6 um visible

FUV 0.1um

Emission by clouds, surface

Emission by high clouds, upper trop water vapor

Scattering by clouds, aerosol, surface

Scattering by molecular atmosphere

We are going to create a model in this class that will allow you model the flow of radiation through the atmosphere from the UV to the microwave

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Non-uniqueness and Instability Estimation

Cost Function: = M [y-f(x)]

measurement Prediction of measurement

‘metric’ of length (e.g.least squares)

Unconstrained Constrained

x2

x1

Solution space non-unique

x2

x1

= M [y-f(x)] + (x)

M M

C

C(x)= initial or a priori constraint

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Information content background

Information is an augmentation of existing knowledge thus it is a relative concept

P(xa) Sa

P(xy) Sx

we propose that the measurement has added information if the ‘volume’ of the distribution is reduced - so Sx/Sa characterizes infoirmation

We start with some‘knowledge base’

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Shannon’s measure of informationEntropy is a measure of the # of distinct states of a system, and thus a measure of information about that system. If the system is defined by the pdf P(x), then

In our context, information is the change (reduction) in entropy of the ‘system’ after a measurement is made

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