# Presentation: 484645

## GV for NASA GSFC’s data assimilation research

**Arthur Y. Hou**
***NASA Goddard Space Flight Center***

**2****nd**** International GPM GV Workshop, 27-29 September 2005, Taipei, Taiwan**

## The effect of precipitation observation on the atmospheric analysis is determined by the spatial structures of observation errors and forecast model errors.

- The effect of precipitation observation on the atmospheric analysis is determined by *the spatial structures of observation errors* *and forecast model errors*.
- Ground-based observation systems can provide independent measurements for determining the error characteristics of satellite-based precipitation products.

**Data assimilation requires knowledge of observation errors**

## The type of error matters

-      In data assimilation it is crucial to differentiate systematic errors from random errors because:
- The analysis equation is based on the following assumptions:
- The underlying statistics are random, unbiased, stationary, and normally distributed
- Observation and background errors are uncorrelated.
- This means that any systematic error in observations be removed.
- The random components of observation or background errors are typically *parameterized* based on known error statistics - often with simplifying assumptions (e.g., homogeneous, isotropic, separable horizontal and vertical structure functions).
- This requires that the observables being assimilated have well-defined error statistics to *guide* the construction of error covariance models.

## Impact of bias uncertainty on water/energy budget analyses

-   Impact of rainfall assimilation on the global P-E budget

*    **Understanding and removal of biases in precipitation observations are essential for obtaining unbiased analysis for climate research and NWP.*

- Difference between GEOS-3 analyses with and without TMI+SSM/I rainfall data is        ~ 0.5 mm/d for the tropics and    ~ 0.3 mm/d for the global mean.
- Constraining rainfall but not evaporation with observations leads to an imbalance in the P-E budget, which could be used to diagnose model errors and guide improvements in evaporation and land surface modeling – but only if bias uncertainty in observed global rain is << 0.3 mm/d.
- In terms of energy fluxes, a global-mean bias of 0.13 mm/d in rain rate is comparable to 4 Wm-2 signal from CO2 doubling.

## Bias uncertainty in current rainfall observations

- PR and TMI V6 monthly global mean     rain rates differ by 7%, or ~ 0.2 mm/d.
- Uncertainties in rain algorithms may account for this. But undetected low rain rates by PR/TMI could add up to 0.07 (?) mm/d (assuming 1% of all raining grids are light rain).
- GPM will have the ability to measure light rain rates to reduce this uncertainty. Ground measurements form an important part of the validation strategy.

- 0.5o x 0.5o        January 1998

## Use of random error information in data assimilation

-  Minimizing a cost function that measures the distance between observation *y**o* and model estimate ***H****(x)*, subject to physical model constraints:
*            **J(x) = (x**b** – x)**T** ****P****-1** (x**b** – x) + ( y**o** – ****H****(x))**T** (****R******o**** **+** ****R******f ****)**-1** ( y**o** – ****H****(x))*
-  The observation operator ***H*** (forward model) is a physical model of precipitation, which may be a function of space and time (e.g., 4DVAR).
-  ***P***, ***R******o***, & ***R******f*** are error covariances characterizing the forecast model, observations, and the forward model. *They define both the problem and the solution.*
-  Minimization is done w.r.t. a control variable *x*, which may be the initial condition, a model tendency correction, or a model physics parameter. (*The choice of** * *x*  *affects the form of P. The transformation of **y**o** affects the construction of ****R******o***, and ***R******f*** *).*
-  For a linear ***H*** and Gaussian statistics, the analysis *x**a* is given by:
*             **x**a** = arg{min J(x)} = x**b** + ****P****H**T** (H****P****H**T** + ****R******o**** + ****R******f**** )**-1**( y**o** – H(x**b**))*

**Use of random error information in data assimilation**

- “analysis increment” (correction vector)

- Structure determined by *P, R**o**, R**f**,* and *H**T*

## Precipitation error covariance models

*- measurement errors *
*- retrieval errors associated with radiation and cloud-resolving models*
*- error of representativeness (if averaged over space and time)*
-  ***R******o*** is *user-defined* – depending on data usage and possible transformations of the observation variable to render errors more consistent with underlying statistical assumptions (e.g., *rain rate vs. log of rain rate*).
-  In practice, observation error covariances are *parameterized*. The current error models for precipitation are exceedingly simple: typically, ***R******o*** =** **<*(****o**-b**o**)(****o**-b**o**)**T*> =** *****o**** *(RR), where ** = error std. dev., i.e., errors are not correlated in space or time.
-  Forward model error covariance ***R******f*** associated with parameterized precipitation physics is generally not known and consequently ignored.

**Precipitation error covariance models**

## Impact of incorrect observation error specification

- An idealized example: Assuming *x**b* = *x**t* + ***b*, *y**o* = *y**t* + ***o*, where *x *is the control variable in specific humidity, *y *is the vertical rain profile, ***b* and ***o* are background and observation errors characterized by ***P*** and ***R******o***, respectively.
- If ***o* is vertically correlated, assimilating the rain profile assuming no correlation leads to suboptimal use of data and less accurate analysis.

- Independent observations provide more information

## Sensitivity of analysis and forecast to         precipitation observation error

**Sensitivity of analysis and forecast to         precipitation observation error**

- 5-Day Bonnie forecast issued from 12 UTC 8/20/98

**   **Impact of assigned rain errors on GEOS-3 moisture tendency correction

***o *= 30%

***o *= 80%

***o *= 250%

- TMI + SSMI sfc rain

-       Sensitivity of track forecast to weights assigned to rainfall data in the initial condition

-       Correlation of precipitation forecast with TMI+SSM/I rain rates (20ox30o moving domain)

- Difference in rain forecasts smaller than TMI+SSM/I sampling errors

**(Estimated ***********o *****for 1****o****x1****o**** RR ~ 15-50%)**

## Modeling error covariances using GV observations

*J(x) = (x**b** – x)**T** ****P****-1** (x**b** – x) + ( y**o** – ****H****(x))**T** (****R******o**** + ****R******f ****)**-1** ( y**o** – ****H****(x))*

- Assimilation requires knowing the sum of ***R******o******+ R******f*** , which may be estimated from observation minus forecast (O-F) statistics if the inputs (*T,q,..*) for *H* are highly accurate: <*(O-F)(O-F)**T*> =** *****R******o******+ R******f*** .* *
*	**	**Note that no explicit knowledge of** ****R******o**** is needed for evaluating ****R******o******+ R******f***
- This is possible at the “precipitation process” type of GV sites, which can provide not only precipitation measurements for determining ***R******o***, but also ancillary observations needed for computing *H* to estimate ***R******o******+ R******f***  using *O-F* statistics (in either physical or radiance space depending on the assimilation method).
- Knowing ***R******o*** will make it possible to estimate ***R******f*** , which is needed in order to make better use of precipitation products in data assimilation.

## Summary

-  Precipitation measurements at GV sites may be used to estimate the spatial structure of ***R******o*** associated with satellite precipitation estimates.
-  Precipitation assimilation requires knowing the sum of ***R******o******+ R******f*** . But ***R******f***  is generally not known. Ancillary measurements at GV sites can be used to directly estimate the combined error ***R******o******+ R******f***  (in either physical or radiance space).
-  Knowing both ***R******o*** and ***R******o****** + R******f***  makes it possible to estimate ***R******f ***. The knowledge of ***R******f***  at multiple GV sites may allow ***R******f***  to be parameterized in terms of local variances and correlation functions.
-  A parameterized model of ***R******f***  could ultimately lead to the more effective use of precipitation information in data assimilation.