# Presentation: 789723

## Profiler DSD Estimates and Uncertainties Retrieved using Ensemble Statistics and Optimal Estimation Theory

**Christopher R. Williams**
- Cooperative Institute for Research in Environmental Sciences (CIRES)
- University of Colorado at Boulder,
- and
- NOAA Earth System Research Laboratory (NOAA ESRL),
- Physical Sciences Division (PSD)

- This work is supported in part by:
-  NASA Tropical Rainfall Measuring Mission (TRMM),
- Global Precipitation Mission (GPM) Ground Validation (GV) Program,
- NASA Precipitation Measurement Mission (PMM) (Grant NNX07AN32G),
-  NOAA’s contribution to the NASA PMM program, and
- Peter May, CSIRO (formerly BMRC), Bureau of Meteorology, Australia

## Motivation

- In order to improve the parameterization of precipitation processes, we must understand the vertical profile and the time evolution of the raindrop size distribution (DSD) during different rain regimes.
- Profilers are providing estimates of the raindrop size distribution (DSD)
- But before these DSDs can be fully utilized to develop new satellite retrieval algorithms (or to evaluate current algorithms), the error (or uncertainty) of each retrieved parameter of the DSD needs to be quantified
- This study uses Ensemble Modeling to determine the uncertainty of the DSD parameters
- Same input data, but many different models
- The spread in the outputs indicates uncertainty for that retrieval
- Statistical analysis of many retrievals provides a general uncertainty estimate that can propagated through algorithms
- This study describes the uncertainties of DSD parameters retrieved from vertically pointing Doppler radars positioned near Darwin, Australia during TWP-ICE (Nov-Feb 2005-2006).

**Notes:**
Profilers are providing estimates of the raindrop size distribution (DSD). 
But before these DSDs can be fully utilized to develop new satellite retrieval algorithms or to evaluate current algorithms, the error, or uncertainty, of each retrieved parameter of the DSD needs to be quantified. 
As with most remote sensing problems, retrieving the DSD uses inverse modeling techniques. 
As end users of these retrieved DSDs, we need to understand the inverse modeling technique and understand the sources of errors that contribute to the uncertainty of the retrieved DSD parameters.

## Ensemble Modeling

- Basically, we don’t trust one model, so we run multiple models, remove outliers, and aggregate the remaining solutions
- The statistics of the aggregate provides a bulk measure of the uncertainty of the retrieval
- Since retrieving the DSD is an inverse modeling problem (an ill-posed problem), Ensemble Modeling incorporates the retrieval model error of individual models into the aggregate uncertainties
**Ensemble Modeling Method:**
- Radar Inputs – Constant for all model runs:
- 50-MHz vertical air motion
- 50-MHz spectral broadening
- 920-MHz Doppler Velocity spectrum
- Retrieval Model Assumptions:
- Assume a DSD Model (Exponential, Gamma, ...)
- Select a Numerical Method (Forward model vs. deconvolution)
- Select a Cost Function Minimization (χ2 or other best fit score)
- Model Output:
- 	1. Parameterized DSD: Z, Dm, and some shape parameter (μ)
- 	2. Estimate rain rate, R, from the parameterized DSD

## Ensemble Modeling

- A. Assume a DSD Shape
- 1. Modified Gamma DSD: 		N(D) = N0 Dμ exp(-(4+μ) D/Dm)
- 2. Exponential DSD (μ=0): 		N(D) = N0 exp(-(4) D/Dm)
- 3. Marshall-Palmer DSD: 		N(D) = 8000 exp(-ΛMP D), ΛMP=R0.2
- 4. Constant μ=2.5 Gamma DSD: 	N(D) = N0 D2.5 exp(-(6.5) D/Dm)
- 5. Constant μ=5 Gamma DSD: 		N(D) = N0 D5 exp(-(9) D/Dm)
- 6. μ-Λ Constrained Gamma DSD:	N(D) = N0 Dμ exp(-(4+μ) D/Dm), 		      μ-Dm Constrained: Dm = Λ / (4+μ)		     Dm = f(μ)
- 7. Log-Normal DSD: 			N(D) = Nt exp(-ln2(D/Dm) (2ln2 σ))
- B. Select a Numerical Method
- 1. Convolution Method: compare observation with modeled spectrum that includes broadening and air motion effects
- 2. Deconvolution Method: remove broadening and air motion effects from observation and then compare with ideal radar modeled spectrum
- C. Select a Cost Function Minimization
- 1. Spectral Two-Norm: Σ (Sobs(v) – Smodel(v))2 (this is often called χ2)
- 2. Spectral One-Norm: Σ |Sobs(v) – Smodel(v)| (reduces influences of outliers)
- 3. Moment Differences:
- Σ|Zobs – Zmodel|/Zobs + |Vobs – Vmodel|/Vobs + |Wobs – Wmodel|/Wobs
- All models have the same input (one Doppler velocity spectrum), and there are (7*3*2) = 42 different solutions

## Ensemble Modeling

- A. Assume a DSD Shape (DSD Model)
- 1. Modified Gamma DSD: 		N(D) = N0 Dμ exp(-(4+μ) D/Dm)
- 2. Exponential DSD (μ=0):		N(D) = N0 exp(-(4) D/Dm)
- 3. Marshall-Palmer DSD:	N(D) = 8000 exp(-ΛMP D), ΛMP=R0.2
- 4. Constant μ=2.5 Gamma DSD:	N(D) = N0 D2.5 exp(-(6.5) D/Dm)
- 5. Constant μ=5 Gamma DSD: 		N(D) = N0 D5 exp(-(9) D/Dm)
- 6. μ-Λ Constrained Gamma DSD:	N(D) = N0 Dμ exp(-(4+μ) D/Dm), 		 μ-Dm Constrained: Dm = Λ / (4+μ)			Dm = f(μ)
- 7. Log-Normal DSD:		N(D) = Nt exp(-ln2(D/Dm) (2ln2 σ))

## Ensemble Modeling

- B. Select a Cost Function Minimization
- 1. Spectral Two-Norm: Σ (Sobs(v) – Smodel(v))2 (this is often called χ2)
- 2. Spectral One-Norm: Σ |Sobs(v) – Smodel(v)| (reduces influences of outliers)
- 3. Moment Differences:
- Σ|Zobs – Zmodel|/Zobs + |Vobs – Vmodel|/Vobs + |Wobs – Wmodel|/Wobs
- C. Select a Numerical Method
- 1. Convolution Method: compare observation with modeled spectrum that includes broadening and air motion effects
- 2. Deconvolution Method: remove broadening and air motion effects from observation and then compare with ideal radar modeled spectrum
- All models have the same input and there are (7*3*2) = 42 different solutions

## Ensemble Modeling

- A. Assume a DSD Shape
- 1. Modified Gamma DSD: 		N(D) = N0 Dμ exp(-(4+μ) D/Dm)
- 2. Exponential DSD (μ=0): 		N(D) = N0 exp(-(4) D/Dm)
- 3. Marshall-Palmer DSD: 		N(D) = 8000 exp(-ΛMP D), ΛMP=R0.2
- 4. Constant μ=2.5 Gamma DSD: 	N(D) = N0 D2.5 exp(-(6.5) D/Dm)
- 5. Constant μ=5 Gamma DSD: 		N(D) = N0 D5 exp(-(9) D/Dm)
- 6. μ-Λ Constrained Gamma DSD:	N(D) = N0 Dμ exp(-(4+μ) D/Dm), 		      μ-Dm Constrained: Dm = Λ / (4+μ)		     Dm = f(μ)
- 7. Log-Normal DSD: 			N(D) = Nt exp(-ln2(D/Dm) (2ln2 σ))
- B. Select a Numerical Method
- 1. Convolution Method: compare observation with modeled spectrum that includes broadening and air motion effects
- 2. Deconvolution Method: remove broadening and air motion effects from observation and then compare with ideal radar modeled spectrum
- C. Select a Cost Function Minimization
- 1. Spectral Two-Norm: Σ (Sobs(v) – Smodel(v))2 (this is often called χ2)
- 2. Spectral One-Norm: Σ |Sobs(v) – Smodel(v)| (reduces influences of outliers)
- 3. Moment Differences:
- Σ|Zobs – Zmodel|/Zobs + |Vobs – Vmodel|/Vobs + |Wobs – Wmodel|/Wobs
- All models have the same input (one Doppler velocity spectrum), and there are (7*3*2) = 42 different solutions

## Observations collected near Darwin during TWP-ICE

## Observations collected near Darwin during TWP-ICE

- Step 1. Describe the median and STD of Dm and Rain Rate for each pixel.
- Step 2. Describe the statistics of Dm and Rain Rate for all 26,000 pixels during these two rain events.

## Profiler Observations- Darwin, Australia

- 920-MHz Profiler

- 50-MHz Profiler

**Precipitation Motion**

**Air Motion**

- 22 January 2006

## Sample Reflectivity spectral density

- The reflectivity weighted Doppler velocity spectral density [(mm6/m3)/(m/s)] is recorded at each range gate:

- Area under the curve is the total reflectivity [mm6/m3]

## Sample Reflectivity spectral density

- Marshall-Palmer DSDs

- A. 7 DSD models
- B. Convolution
- C. 3 Cost Functions
-      = 21 estimates

## Sample Reflectivity spectral density

- The reflectivity weighted Doppler velocity spectral density [(mm6/m3)/(m/s)] is recorded at each range gate:

- Area under the curve is the total reflectivity [mm6/m3]

- Original spectrum has been shifted upwards by 0.67 m/s
- Original spectrum has been deconvolved to remove the spectral broadening effects.

## Sample Reflectivity spectral density

- A. 7 DSD models
- B. Deconvolution
- C. 3 Cost Functions
-      = 21 estimates

- Marshall-Palmer DSDs

## Sample Reflectivity spectral density

- Represent the ensemble Dm and Rain Rate with the median and Standard Deviation (STD)

- A. 7 DSD models
- B. Convolution +      	Deconvolution
- C. 3 Cost Functions
- 	= 42 estimates
- Remove the Marshall-Palmer DSD model (6)
- Remove outliers (4)
- 	= 32 estimates

## Observations collected near Darwin during TWP-ICE

- Step 1. Describe the median and STD of Dm and Rain Rate for each pixel.
- Step 2. Describe the statistics of Dm and Rain Rate for all 26,000 pixels during these two rain events.

## Ensemble Model Solutions for all Observations (2 Rain Events)

- Histogram of Ensemble mean Dm for all observations
- Histogram of Ensemble σ{Dm} for a narrow range of
- 1.45 < Dm < 1.55 mm
- ~Symmetric spread

- cnt / %

- Histogram and Percent Accumulation of Dm: ALL OBSERVATIONS

- Histogram and % Accum. of σ{Dm}  for 1.45 < Dm < 1.55 mm interval

## Ensemble Model Solutions for all Observations (2 Rain Events)

- Histogram of Ensemble mean Dm for all observations
- Histogram of Ensemble σ{Dm} for a narrow range of
- 1.45 < Dm < 1.55 mm
- 10-to-90 percentile spreads as a function of Dm

- cnt / %

- Histogram and Percent Accumulation of Dm ALL OBSERVATIONS

- 10-to-90 percentile spread as a function of Dm

- 1.5 mm

## Ensemble Model Solutions for all Observations (2 Rain Events)

- Histogram of Ensemble mean R for all observations
- ~Log-Normal distribution of rain rate occurance
- Histogram of Ensemble σ{R} for a narrow range of
- 1.45 < R < 1.55 mm
- Spread is Not Symmetric

- cnt / %

- Histogram and Percent Accumulation of R: ALL OBSERVATIONS

- Histogram and % Accum. of σ{R}  for 1.45 < R < 1.55 mm hr-1 interval

## Ensemble Model Solutions for all Observations (2 Rain Events)

- Histogram of Ensemble mean R for all observations
- Histogram of Ensemble σ{R} for a narrow range of
- 1.45 < R < 1.55 mm
- 10-to-90 percentile spreads as a function of R

- cnt / %

- Histogram and Percent Accumulation of R :ALL OBSERVATIONS

- 10-to-90 percentile spreads as a function of R

## Ensemble Model Solutions for all Observations (2 Rain Events)

- 10-to-90 percentile spread of σ{Dm} for small intervals of <Dm>
- 10-to-90 percentile spread of σ{R} for small intervals of <R>
- Horizontal lines are Median values

## Concluding Remarks

- Uncertainty estimates of profiler derived DSD parameters (Dm and R) were estimated using Ensemble Modeling.
- For each observed Doppler velocity spectrum: was used as the input
- 42 different DSD retrieval models produced 42 Dm and R estimates.
- Calculated Mean (<Dm> and <R>) and STD( sigma{Dm} and sigma{R})
- The spread around small intervals of <Dm> and <R> were studied to quantify the uncertainty of the retrieved Dm and R.
- Spread of sigma{Dm} and sigma{R} were not symmetric
- Used 10-90 percentiles to describe uncertainties
- The uncertainty of Dm was approximately constant for Dm < 2 mm
- Median uncertainty was about 0.1 mm with the 90% approaching 0.2mm
- The uncertainty of R increased with rain rate
- Median uncertainty was about 8% with 90% approaching 30% relative error