## Project Description

### 1. Single-variable retrievals using Channel 1 only

1) Single-variable retrievals using ONLY channel 1 of the simulated AMSU-A
data (LINEAR):
a) surface pressure
b) surface skin temperature
c) surface emissivity

Please use each of three methods:
i) unity gain
ii) minimum variance
iii) statistical regression

Here and throughout assume the following

• AMSU channel TBs have noise variance of $$\sigma_\varepsilon^2=$$(0.12 K)$$^2$$=0.0144 K$$^2$$.
• Channel noise is uncorrelated with other channels or with the variable(s) to be retrieved (noise covariance is a diagonal matrix)
• UPDATE:  See email of 4/17 for clarification of how to handle noise in this case as well as part 2.

Prepare scatter plots of your results (retrieval as y-axis, truth as x-axis).  Compute the following validation statistics for each and display them as annotations on your scatter plots:
I) RMS error
II) Correlation coefficient

### 2. Single-variable retrievalS using channels 1-3.

a) Repeat the exercise described above but using the FIRST THREE channels of the simulated AMSU-A data (LINEAR) as your inputs.

b) (Update: This step is now optional, because the ratio of effort to insight seems too low for this particular setup.) Repeat (a), but this time performing a two-step iterative retrieval. The first step will be identical to your previous linear retrieval methods but this time applied to the EXACT brightness temperatures (not the LINEAR Tbs).  The second step requires you to use the first-step results as your baseline and to evaluate a new 3×3 Jacobian for that baseline.  Don’t  change the covariance though!

Writeup:  Discuss the quality of your single-variable retrievals using all the different methods, comparing your retrieval error variance with the prior climatological variance.  Also look for any systematic biases (over- or underestimates) that may occur under certain circumstances.  If the retrieval quality is poor (as it probably will be in some cases), discuss the factors that you think contributed to the poor retrieval.  If two different methods give very similar results, try to explain why.  If you prefer the results of any one of the three methods, explain why.

### 3. Complete profile retrievals.

Perform a linear (single-step) minimum variance retrieval of the complete temperature profile using
a) all AMSU-A channels (LINEAR)
b) all AMSU-A channels (EXACT)

Choose a couple of representative individual retrieved profiles plotted together  with the “true” profile to illustrate retrieval quality (e.g.,   ability to resolve vertical structure such as inversions, etc.).

Compute and plot profiles of RMS error for the complete testing data set. On the same figures, plot the profile of climatological standard deviation.

Writeup:  Discuss the quality of your retrievals. Include a discussion of how your linear retrievals applied to the linear data compare with your linear retrievals applied to the nonlinear (exact) data.  Is non-linearity an important source of error in this particular application?

### Technical notes

The complete data sets and sample python programs for certain tasks are provided here. These include

• A set of 3,725 soundings taken from around the globe throughout calendar year 2012.  These have been extrapolated using climatology to 1 hPa and interpolated to a fixed set of 68  Z-levels, where  $$Z \equiv -\log(p/p_0)$$.  Random surface emissivities and surface skin temperatures have been assigned, and microwave brightness temperatures have been calculated for 8 AMSU-A frequencies.   These brightness temperatures are provided in two forms:  a) The EXACT values computed from a complete forward radiative transfer model; b) APPROXIMATE values computed via a linear approximation evaluated for the global mean state.  In both cases Gaussian noise with standard deviation 0.12 K has been added to all channels.
• The global mean profile (68 levels) and mean surface parameters (3 variables: $$p_0$$, $$T_\mathrm{skin}$$, and emissivity $$\varepsilon$$.
• The global covariance of the above 71 parameters.
• The forward-calculated microwave brightness temperatures corresponding to the global mean scene, for 8 AMSU-A channels.
• The Jacobian of the 8 AMSU-A channels evaluated for the global mean state.
• The forward microwave radiative transfer model for use in the non-linear (iterative) retrievals.
• Various functions for open and reading various files.

Note that for the one- and three-channel retrievals, you can extract the Jacobian and covariances you need  from the full $$71\times 8$$ Jacobian and $$71\times 71$$ covariance matrix provided.

#### Numerical operations

You will need to use NumPy routines to efficiently perform matrix manipulations, including multiplication, inverse, and transpose.  The matrix class and associate methods are described here.