Plot the reflectivity and the attenuation for each radar as a function of rain rate using the data provided in the files pr.txt and cpr.txt. The relationships provided in this file are the radiative transfer model that you will be inverting in this project.

NOTE: The reflectivity and attenuation are given in decibel (dB) units. For reflectivity this gives dBZ = 10log₁₀(Z). Likewise attenuation in units of dB km^-1 is given by [20/ln(10)]*k_ext where k_ext is in units of km^-1. The radar beam must travel a two-way path from the instrument to the target volume and thus suffers a two-way attenuation. The value given in this test file already takes into account the two-way path taken by the radar beam. These relationships were derived using Mie scattering calculations and an assumption of an exponential precipitation drop size distribution (Marshall and Palmer, 1948) and therefore do not represent a unique mapping between reflectivity and rain rate.

Plot the radar reflectivity profile observed by each instrument provided in the files PR_DATA.TXT and CPR_DATA.TXT. Based on the plots you made in part (1) explain the differences between the two profiles. The radar bin volumes have a depth of 250 meters and the heights given in the data files are the bin centers.

NOTE: The radar profiles provided in these files are not real data. For reference, the true depth of the CPR radar bins is 240 meters. Furthermore the bottom 2-4 bins cannot typically be seen by spaceborne radar due to contamination by surface clutter.

You will now perform a retrieval of the surface precipitation using the classic reflectivity approach. This method begins by estimating the rain rate in the top bin and then proceeding to the next bin working your way down to the surface bin while correcting for attenuation of the radar beam.

Step 1: Use the reflectivity in bin #1 to estimate the rain rate in that bin.
Step 2: Find the attenuation [in dB] associated with that rain rate. For this and the previous step, using some kind of interpolation scheme is recommended.
Step 3: Correct for the effect of attenuation in bin #1 on the observed reflectivity in bin #2. You now have an attenuation corrected reflectivity for bin #2.
Step 4: Repeat Steps 1-3 for bin #2. Continue working your way down to the surface correcting for attenuation as you go.

(a) Plot the retrieved rain rate profiles.
(b) Plot the observed and the attenuation corrected reflectivities.
(c) What are the associated surface rain rates from each retrieval?

The noise in both the PR and CPR observations is 1 dBZ. To simulate this noise create a profile of observational noise filled with random numbers (mean = 0, standard deviation = 1) and add them to the observations.

(a) Repeat question (3) using the noisy observations.
(b) Which retrieval is affected more by the presence of noise? Why do you think that this is this the case? Note: recall the plots of attenuation that you made in part (1). It may be helpful to repeat step (a) several times using different noise realizations.
(c) Based on results to this point, what advantages does the PR have relative to the CPR in terms of retrieving rainfall?

The Path Integrated Attenuation (PIA) of a radar beam is defined as the 2-way attenuation of the beam from the TOA to surface and back to TOA, due only to hydrometeors. (See Haynes et al. (2009) for an PIA-based rainfall algorithm for CloudSat). The PIA of the CloudSat radar beam can be estimated to approximately ±1 dB. In the present case, the observed PIA was 8.9 dB.

(a) Assume that the rain column is uniform in height. Use the observed PIA and estimate the rain top height to derive the column mean rain rate and associated uncertainty from the CPR. Compare this answer with the reflectivity based answers from the CPR.
(b) What are the advantages and disadvantages of each method (reflectivity vs. PIA)?
(c) How might you combine the two methods to provide an optimal rain rate estimate from the CPR?

The minimum detectable signal of the PR is +17 dBZ whereas the CPR has a minimum detectable signal of -30 dBZ.

(a) Which radar do you believe to be more accurate for deriving surface rain rate?
(b) Estimate the minimum detectable rain rate of the PR. With this knowledge, how might the CPR complement the PR?