An Introduction to Optimal Estimation Theory
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An Introduction to Optimal Estimation Theory Chris O´Dell AT652 Fall 2013. The Bible of Optimal Estimation : Rodgers, 2000. The Retrieval PROBLEM. DATA = FORWARD MODEL ( State ) + NOISE. OR. State Vector Variables: Temperatures Cloud Variables Precip Quantities / Types

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An Introduction to Optimal Estimation Theory

Chris O´Dell

AT652 Fall 2013

The Retrieval PROBLEM



State Vector Variables:


Cloud Variables

Precip Quantities / Types

Measured Rainfall

Et Cetera


Instrument Noise

Forward Model Error

Sub grid-scale processes




Radar Backscatter

Measured Rainfall

Et Cetera

Forward Model:

Cloud Microphysics


Surface Albedo

Radiative Transfer

Instrument Response

Example: Sample Temperature Retrieval (project 2)

Temperature Profile

Temperature Weighting Functions

  • We must invert this equation to get x, but how to do this:

  • When there are different numbers of unknown variables as measured variables?

  • In the presence of noise?

  • When F(x) is nonlinear (often highly) ?

Retrieval Solution Techniques

  • Direct Solution (requires invertible forward model and same number of measured quantities as unknowns)

  • Multiple Linear Regression

  • Semi-Empirical Fitting

  • Neural Networks

  • Optimal Estimation Theory

Unconstrained Retrieval

Inversion is unstable and solution is essentially amplified noise.

Retrieval with Moderate Constraint

Retrieved profile is a combination of the true and a priori profiles.

Extreme Constraint

Retrieved profile is just the a priori profile, measurements

have no influence on the retrieval.

Best” x

x from y alone:

x = F-1(y)

Prior x = xa

Mathematical Setup

State Vector Variables:

Temp Profile

(n,1) colvector



(m,1) col vector

Forward Model:

(m,n) matrix

  • Ifweassumethaterrors in eachyaredistributedlike a Gaussian, thenthelikelihoodof a givenxbeingcorrectis proportional to

  • This assumesnopriorconstraint on x.

  • Wecanwritethe2 in a generalizedwayusingmatricesas:

  • This is just a number (a scalar), andreducestothefirstequationwhenSyis diagonal.

Generalized Error Covariance Matrix

  • Variables: y1, y2 , y3 ...

  • 1-sigma errors: 1 , 2 , 3 ...

  • The correlation between y1 and y2 is c12 (between –1 and 1), etc.

  • Then, the Measurement Error Covariance Matrix is given by:

Diagonal Error Covariance Matrix

  • Assumenocorrelations (sometimesistruefor an instrument)

  • Hasvery simple inverse!

Prior Knowledge

  • Prior knowledge about x can be known from many different sources, like other measurements or a weather or climate model prediction or climatology.

  • In order to specify prior knowledge of x, called xa , we must also specify how well we know xa; we must specify the errors on xa .

  • The errors on xa are generally characterized by a Probability Distribution Function (PDF) with as many dimensions as x.

  • For simplicity, people often assume prior errors to be Gaussian; then we simply specify Sa, the error covariance matrix associated with xa .

Example: Temperature Profile Climatology for December over Hilo, Hawaii

P = (1000, 850, 700, 500, 400, 300) mbar

<T> = (22.2, 12.6, 7.6, -7.7, -19.5, -34.1) Celsius

Correlation Matrix: Hilo, Hawaii

1.00 0.47 0.29 0.21 0.21 0.16

0.47 1.00 0.09 0.14 0.15 0.11

0.29 0.09 1.00 0.53 0.39 0.24

0.21 0.14 0.53 1.00 0.68 0.40

0.21 0.15 0.39 0.68 1.00 0.64

0.16 0.11 0.24 0.40 0.64 1.00

Covariance Matrix:

2.71 1.42 1.12 0.79 0.82 0.71

1.42 3.42 0.37 0.58 0.68 0.52

1.12 0.37 5.31 2.75 2.18 1.45

0.79 0.58 2.75 5.07 3.67 2.41

0.82 0.68 2.18 3.67 5.81 4.10

0.71 0.52 1.45 2.41 4.10 7.09

A priori error covariance matrix s a for project 2
A priori error covariance matrix S Hilo, Hawaiia for project 2

  • Asks you to assume a 1-sigma error of 50 K at each level, but these errors are correlated!

  • The correlation falls off exponentially with the distance between two levels, with a decorrelation length l.

  • A couple ways to parameterize this. In the project, use:

The Hilo, Hawaii2 with prior knowledge:

Yes, thatis a scarylookingequation.

But you just codeitup & everything will work!

Minimizing Hilo, Hawaiithe2

  • Minimizing the 2 is hard. In general, you can use a look-up table (if you have tabulated values of F(x) ), but if the lookup table approach is not feasible (i.e., it’s too big), then:

  • If the problem is linear (like Project 2), then there is an analytic solution for the x the minimized 2, which we will call

  • If the problem is nonlinear, a minimization technique must be used such as Gauss-Newton iteration, Steepest-Descent, etc.

Linear Forward Model with Prior Constraint Hilo, Hawaii

The x the minimizes the cost function is analytic and is given by:

Use for P2!

The associated posterior error covariance matrix, which describes the theoretical errors on the retrieved x, is given by:

Use for P2!

A simple 1D example Hilo, Hawaii

You friend measurements the temperature in the room with a simple mercury thermometer.

She estimates the temperature is 20 ± 2 °C.

You have a thermistor which you believe to be pretty accurate. It measures a Voltage V proportional to the temperature T as:

y=V = kT

where k = 3 V/°C, and the measurement error is roughly ± 0.1 V, 1-sigma.

Sy=0.12=0.01 V2


Voltage=1.8± 0.1 Hilo, Hawaii

Express Measurement Knowledge as a PDF



Prior Knowledge is PDF

Measurement is a PDF

 Posterior is a (narrower) PDF





A Simple Example: Hilo, Hawaii

Cloud Optical Thickness and Effective Radius

water cloud forward calculationsfrom Nakajima and King 1990

  • Cloud Optical Thickness and Effective Radius are often derived with a look-up table approach, although errors are not given.

  • The inputs are typically 0.86 micron and 2.06 micron reflectances from the cloud top.

  • x = Hilo, Hawaii{reff,  }

  • y = { R0.86 , R2.13 , ... }

  • Forward model must map x to y. Mie Theory, simple cloud droplet size distribution, radiative transfer model.

Simple Solution: The Look-up Table Approach

  • Assume we know .

  • Regardless of noise, find such that

  • But is a vector! Therefore, we minimize the 2 :

  • Can make a look-up table to speed up the search.

is minimized.

Example: Approach

xtrue : {reff = 15 m,  = 30 }

Errors determinedbyhowmuchchange in eachparameter (reff ,  ) causesthe2tochangebyoneunit(formally: find theperimeterwhichcontains 68% ofthevolumeoftheposterior PDF)

BUT Approach

  • What if the Look-Up Table is too big?

  • What if the errors in y are correlated?

  • How do we account for errors in the forward model?

  • Shouldn’t the output errors be correlated as well?

  • How do we incorporate prior knowledge about x ?

Gauss Approach-Newton Iteration forNonlinear Problems


In general these are updated on every iteration.

Iteration in practice: Approach

  • Not guaranteed to converge.

  • Can be slow, depends on non-linearity of F(x).

  • There are many tricks to make the iteration faster and more accurate.

  • Often, only a few function iterations are necessary.

So, again, what is optimal estimation? Approach

Optimal Estimation is a way to infer information about a system, based on observations. It is necessary to be able to simulate the observations, given complete knowledge of the system state.

  • Optimal Estimation can:

  • Combine different observations of different types.

  • Utilize prior knowledge of the system state (climatology, model forecast, etc).

  • Errors are automatically provided, as are error correlations.

  • Estimate the information content of additional measurements.

Applications of Optimal Estimation Approach

  • Retrieval Theory (standard, or using climatology, or using model forecast. Combine radar & satellite. Combine multiple satellites. Etc.)

  • Data Assimilation – optimally combine model forecast with measurements. Not just for weather! Example: carbon source sink models, hydrology models.

  • Channel Selection: can determine information content of additional channels when retrieving certain variables. Example: SIRICE mission is using this technique to select their IR channels to retrieve cirrus IWP.