GV for NASA GSFC’s data assimilation research Arthur Y. Hou NASA Goddard Space Flight Center

1. GV for NASA GSFC’s data assimilation research Arthur Y. Hou NASA Goddard Space Flight Center 2nd International GPM GV Workshop, 27-29 September 2005, Taipei, Taiwan

2. Data assimilation requires knowledge of observation errors • The effect of precipitation observation on the atmospheric analysis is determined by the spatial structures of observation errorsand forecast model errors. • Ground-based observation systems can provide independent measurements for determining the error characteristics of satellite-based precipitation products.

3. 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.

4. Precipitation (Global) Evaporation (Global) Impact of bias uncertainty on water/energy budget analyses • 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. 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.

5. mm/d TMI minus PR RR (mm/d) PR RR 0.1 1 10 102 (mm/d) 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

6. Use of random error information in data assimilation Variational assimilation of precipitation data • Minimizing a cost function that measures the distance between observation yo and model estimate H(x), subject to physical model constraints: • J(x) = (xb – x)TP-1 (xb – x) + ( yo – H(x))T (Ro+Rf )-1 ( yo – 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, Ro, & Rf 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 ofxaffects the form of P. The transformation of yo affects the construction of Ro, and Rf). • For a linear H and Gaussian statistics, the analysis xa is given by: xa = arg{min J(x)} = xb + PHT (HPHT + Ro + Rf )-1( yo – H(xb)) “analysis increment” (correction vector) Structure determined by P, Ro, Rf, and HT

7. Precipitation error covariance models • Ro is the total precipitation observation error including - measurement errors - retrieval errors associated with radiation and cloud-resolving models - error of representativeness (if averaged over space and time) • Ro 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, Ro=<(eo-bo)(eo-bo)T> =so2(RR), wheres =error std. dev., i.e., errors are not correlated in space or time. • Forward model error covariance Rf associated with parameterized precipitation physics is generally not known and consequently ignored.

8. Model level Model level xb: Specific humidity sA(Correct correlation length:+/- 2 levels) sA(Incorrect length = 0) yo: Rain sb sFG so RR(mm/h) and q(g/kg) s (mm/h) Impact of incorrect observation error specification • An idealized example: Assuming xb = xt + eb, yo = yt + eo, where x is the control variable in specific humidity, y is the vertical rain profile, eb and eo are background and observation errors characterized by P and Ro, respectively. • If eo 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

9. Sensitivity of analysis and forecast to precipitation observation error Impact of assigned rain errors on GEOS-3 moisture tendency correction 5-Day Bonnie forecast issued from 12 UTC 8/20/98 Sensitivity of track forecast to weights assigned to rainfall data in the initial condition TMI + SSMI sfc rain so =30% (Estimated so for 1ox1o RR ~ 15-50%) Correlation of precipitation forecast with TMI+SSM/I rain rates (20ox30o moving domain) so =80% so =250% Difference in rain forecasts smaller than TMI+SSM/I sampling errors

10. Modeling error covariances using GV observations J(x) = (xb – x)TP-1 (xb – x) + ( yo – H(x))T (Ro + Rf )-1 ( yo – H(x)) • Assimilation requires knowing the sum of Ro+ Rf , 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> =Ro+ Rf. Note that no explicit knowledge ofRo is needed for evaluating Ro+ Rf • This is possible at the “precipitation process” type of GV sites, which can provide not only precipitation measurements for determining Ro, but also ancillary observations needed for computing H to estimate Ro+ Rf using O-F statistics (in either physical or radiance space depending on the assimilation method). • Knowing Ro will make it possible to estimate Rf , which is needed in order to make better use of precipitation products in data assimilation.

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