REMOTE ATMOSPHERIC SOUNDING

Owen E. Thompson
Department of Meteorology
University of Maryland

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BOOK OUTLINE (9/1/97)
Copyright 1997, O.E. Thompson
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CHAPTER 1.
CONCEPTS OF REMOTE SENSING AND INFERENCE -
What is measured and What is Infered

1.1 Introduction
1.2 Fundamental Concepts
1.3 Remote Sensing Signals
1.4 Satellite Orbital and Observational Strategies

1.4.1 Inclined Polar Orbits

1.4.2 Geosynchronous Orbits

1.4.3 Low Equatorial Orbits

1.4.4 Orbital Plane Spin Stabilization Viewing Mode

1.4.5 Cartwheel Viewing Mode

1.4.6 Spin Scan Viewing Mode

1.4.7 Three-Axis Stabilization
1.5 Sorting Out Atmospheric Variability
1.6 Remote Atmospheric Sounding
1.7 Brief (Brief) Review of US Environmental Satellite Programs

1.7.1 TIROS

1.7.2 TOS

1.7.3 ITOS

1.7.4 NIMBUS

1.7.5 ATS

1.7.6 SMS

1.7.7 GOES
1.8 Brief (Brief) Review of Infrared and Microwave Sounding Instruments

1.8.1 Introduction

1.8.2 Refraction and Diffraction Spectrometers

1.8.3 Interferometer Spectrometer

1.8.4 Filter Wedge Spectrometer

1.8.5 Transmission Filter Spectrometer

1.8.6 Selective Chopper Spectrometer (Radiometer)

1.8.7 Microwave Spectrometers
1.9 Earth Observing System (EOS)

CHAPTER 2.
THE FORWARD PROBLEM OF RADIATIVE TRANSFER -
How the Earth System Produces Signal Information

2.1 Introduction
2.2 The Radiative Transfer Equation for Nadir Mode Instruments

2.2.1 Absorption Effect

2.2.2 Thermal Emission Effect

2.2.3 Special Case - Absorption Only

2.2.4 Application to Earth's Atmosphere

2.2.5 Additional Contributions
2.3 Slant Path Observations
2.4 Atmospheric Absorption and Emission in the InfraRed and Microwave Spectrum
2.5 Approximations to the Planck Function

2.5.1 Rayleigh-Jeans Approximation

2.5.2 Wien Approximation
2.6 Equivalent Brightness Temperature

2.6.1 Rayleigh-Jeans Approximation

2.6.2 Wien Approximation
2.7 Random Measurement Noise and its Temperature Equivalence
2.8 Limb Sounding
2.9 Sounding Trace Gases by Solar Occultation
2.10 Ozone Sounding by Backscattered Ultraviolet
2.11 Atmospheric Sounding Using the Global Positioning System
2.12 Surface-Based Remote Atmospheric Sounding

CHAPTER 3.
Spectral "Fingerprints" of the Earth System

3.1 Introduction

3.1.1 Natural Broadening

3.1.2 Collision Broadening

3.1.3 Doppler Broadening

3.1.4 Combined Collision-Doppler Broadening
3.2 Transmittance Function for Constant Absorption Coefficient
3.3 Transmittance Function for Narrow Spectral Interval in the Far Wing of a Lorentz Line
3.4 Transmittance Function for a Broad Elsasser Band
3.5 Transmittance Function for a Kaplan Band of Randomly Spaced Absorption Lines
3.6 Comparison of Analytic Models of Transmittance Weighting Function
3.7 Instrument Bandpasses and Filter Functions
3.8 Multiple Absorbers
3.9 Numerical Models of Atmospheric Transmittance Weighting Function
3.10 Transmittance Functions for the TIROS Operational Vertical Sounder (TOVS)
3.11 Transmittance Functions for the Next Generation Sounders

3.11.1 Atmospheric Infrared Radiation Sounder (AIRS)

3.11.2 High resolution Interferometer Sounder (HIS)

3.11.3 Advanced Microwave Sounding Unit (AMSU)

3.11.4 Other Advanced Sounding Systems

CHAPTER 4.
RADIATIVE TRANSFER SIGNALS
- Information in the Upwelling Radiation

4.1 Introduction
4.2 Approach
4.3 Signals
4.4 ...

CHAPTER 5.
RADIATIVE TRANSFER KERNELS
- Linearization of the Forward Problem

5.1 Introduction
5.2 Separation of Variables Within the Planck Function
5.3 Frequency Normalization
5.4 Conversion to Equivalent Brightness Temperature
5.5 Linearization for Moisture Profiling
5.6 Linearization for Simultaneous Temperature and Moisture Profiling
5.7 Kernels Derived from Atmospheric Transmittances

5.7.1 Perturbation Surface Emission Term

5.7.2 Perturbation Atmospheric Term

5.7.3 Solar Reflection Term

5.7.4 Computing Grid for Perturbation Terms

5.7.5 Conversion to Equivalent Brightness Temperature
5.8 Kernels Derived from Upwelling Spectral Radiances
5.9 Empirical Kernels Derived from Co-Located Radiances and Radiosondes

CHAPTER 6.
ERRORS OF LINEARIZATION -
How Accurate if the Forward Problem to be Solved?

6.1 ... Errors of Linearization

CHAPTER 7.
THE INVERSE PROBLEM OF RADIATIVE TRANSFER -
Formulating a Tractable Inference Problem

7.1. Introduction
7.2 Difficulties with the Surface Contribution Term in the Radiative Transfer Equation

7.2.1 Clouds

7.2.2 Dirty Windows

7.2.3 Surface Emissivity

7.2.4 Surface Reflection

7.2.5 Influence of Surface Temperature Errors on Atmospheric Soundings
7.3 Difficulties with the Atmospheric Contribution Term in the Radiative Transfer Equation

7.3.1 Transmittance Modeling

7.3.2 Tropopause and Stratospheric Radiances

7.3.3 Upper Boundary Condition

7.3.4 First Guess Profile Errors

7.3.5 Distribution of Profile Inference Errors
7.4 Basic Features of the Inverse Problem

7.4.1 Integral Equations

7.4.2 Relationship between values of t(z) and vaues of r()

7.4.3 Discrete Radiance Measurements

7.4.4 Influence of Retrieval Algorithm on Vertical Resolution
7.5 The Ill-Posed Nature of the Inverse Problem

7.5.1 Non-Uniqueness of the Solution

7.5.2 Very Different Solutions

7.5.3 Ill-Posedness as Ill-Conditioning of the Linear System

7.5.4 Effect of Additional Instrument Channels on Ill-Posedness

7.5.5 Ill-Posedness in Nature?
7.6 Concept of Regularizing the Ill-Posed Problem

7.6.1 Placing External Constraints on the Problem

Mathematical Constraints

Statistical Constraints

Hydrodynamic Constraints
7.7 Classes of Mathematical Approach to the Inverse Problem

7.7.1 Matrix Inverse Approach

7.7.2 Statistical Approach

7.7.3 Direct Relaxation Approach

CHAPTER 8.
INFERENCE OF SEA SURFACE TEMPERATURE -
An Inverse Problem for Earth's Surface

8.1 Introduction
8.2 The Difficulties

8.2.1 The Atmosphere Effect

8.2.2 Cloud Effects
8.3 Sea Surface Temperature Inference Using a Single Atmospheric Window Channel

8.3.1 Atmospheric Corrections for Clear Sky Conditions

8.3.2 Atmospheric Corrections for Overcast Conditions

8.3.3 Atmospheric Corrections from Measured Window Brightness Temperatures

8.3.4 Correcting for Broken Cloudiness - The Histogram Method
8.4 Sea Surface Temperature Inference Using Two Atmospheric Window Channels
8.5 Sea Surface Temperature Inference Using Three Atmospheric Window Channels
8.6 Sea Surface Temperature Inference Using Many Atmospheric Window Channels

CHAPTER 9.
INFERENCE OF ATMOSPHERIC PROFILES
- Optimized Inverse Methods

9.1 The Direct Matrix Inverse Solution
9.2 The PseudoInverse Solution
9.3 The Pseudo Inverse Solution as the a Formal Optimization Calculation
9.4 Noise Scaling
9.5 The Concept of Constrained, Regularized, or Optimized Inverse Solutions
9.6 Constraint of Mathematical Smoothness

9.6.1 Mathematical Smoothness - Example 1

9.6.2 Mathematical Smoothness - Example 2

9.6.3 Mathematical Smoothness - Example 3

9.6.4 Application to Satellite Sounding

9.6.5 The Mathematically Smoothed Solution

9.6.6 Using Incomplete Expansion Functions as Smoothers
9.7 Constraint with a Good First Guess from a Hydrodynamic Model
9.8 The Statistical Regularization Solution

9.8.1 The Concept of Statistical Regularization

9.8.2 Statistical Definitions and Notation

9.8.3 Derivation of the Statistical Regularization Solution

9.8.4 Other Derivations of Statistical Regularization

Foster

Strand and Westwater ("Minimum Variance Solution")

Rodgers ("Maximum Probability Estimator Solution")

9.8.5 Properties of the Statistical Regularization Solution

Validity of the statistics

Interpretation as a Statistically "smoothed" solution

Noise Free Case

Maximum Information Case

Minimum Information Case
9.9 Truncated Physical Singular Value Decomposition Solution -

9.9.1 Constraint of the Problematic Physics

9.9.2 Regularization of the Physics versus Regularization by Statistics

9.9.3 Singular Value Decomposition of the Physics Operator

9.9.4 Interpretation of Regularization by Statistics

9.9.5 Regularization by Truncation of the Physical SVD

9.9.6 Relationship to the Pseudo Inverse Solution

9.9.7 Relationship to the Statistical Regularization Solution

9.9.8 Relationship to the Mathematically Smoothed Solution

CHAPTER 10.
INFERENCE OF ATMOSPHERIC PROFILES
- Statistical Methods

10.1 The Regression Solution (a.k.a. "Least Squares Best Fit Solution")

10.1.1 Blah Blah Blah

10.1.2 Relationship between Regression and Statistical Regularization
10.2 Empirical Orthogonal Function Solution

10.2.1 Blah Blah Blah

10.2.2 Examples of EOF Solution Results

10.2.3 Relationship Between Empirical Orthogonal Function and Regression Solutions

10.2.4 Relationship Between Empirical Orthogonal Function and Statistical Regularization Solutions
10.3 Relationship Between Radiance Eigenvectors and Profile Eigenvectors
10.4 Relationship Between Empirical Orthogonal Functions and Singular Vector Decomposition
10.5 Note on Partitioning Natural Variance Using Physical Singular Vectors

CHAPTER 11.
INFERENCE OF ATMOSPHERIC PROFILES
- Forward Relaxation Methods

11.1 Introduction
11.2 Using the Mean Value Theorem for Integrals

11.2.1 (Planck Function Retrieval)

11.2.2 Separation of Variables in the Planck Function - Approach of Chahine

11.2.3 Taylor Expansion of the Planck Function - Approach of Fleming and Smith
11.3 Using Perturbation Methods
11.4 An Equivalence Theorem for Forward Relaxation and Matrix Inverse Methods

CHAPTER 12.
INFERENCE OF ATMOSPHERIC PROFILES
- Pattern Recognition Methods

12.1 Introduction
12.2 The Basic Concept of Satellite-Based Pattern Recognition
12.3 The Analog Retrieval Method
12.4 The "3I" Method
12.5 Partitioning A Priori Data by Pattern Regognition

CHAPTER 13.
VERTICAL RESOLVING POWER OF PROFILE RETRIEVALS
- The Structural Fidelity of Inverse Solutions

13.1 Introduction
13.2 Theoretical Analysis of Vertical Resolving Power of a Satellite Sounding System
13.3 An Optimized Vertical Resolution Retrieval Solution
13.4 Empirical Analysis of Vertical Resolving Power

CHAPTER 14.
INFORMATION CONTENT OF PROFILE RETRIEVALS
- Transfer of Information Through Forward and Inverse Problems

14.1 The Twomey Approach
14.2 The Westwater and Strand Approach
14.3 The Effect of Incorrect A Priori Information
14.4 The Thompson and Dazlich Approach to information content

CHAPTER 15.
EXPECTED PROFILE RETRIEVAL ERRORS
- Transfer of Error Through the Forward and Inverse Problems

15.1 Introduction
15.2 A Model of Expected Retrieval Errors
15.3 Application to the Statistical Regularization Solution
15.4 Limiting Cases of Retrieval Error Covariance

15.4.1 Small S Limit

15.4.2 Large S Limit
15.5 Synthesizing the Range of Possible First Guess Errors
15.6 Kernel Matrices

15.6.1 Taylor Kernels

15.6.2 Jacobian Kernels

15.6.3 Empirical Kernels
15.7 Case Study of Predicted Retrieval Errors
15.8 Tentative Conclusions

ADD SECTIONS FROM reter.wpd SHOWING THE CASE STUDY CALCULATIONS USING AIRS SIMULATIONS.

CHAPTER 16.
IMPACT OF PROFILE RETRIEVALS ON WEATHER FORECASTING
- Assimilating Remote and In Situ Observations

CHAPTER 17.
IMPACT OF PROFILE RETRIEVALS ON CLIMATE STUDIES
- The Roles of Observations and Models of the Earth System