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Forecast Sensitivity to Observations & Observation Impact Tom Auligné National Center for Atmospheric Research Ackn

Forecast Sensitivity to Observations & Observation Impact Tom Auligné National Center for Atmospheric Research Acknowledgments: Hongli Wang, Qingnong Xiao WRF-Var Tutorial - Feb. 01-03 2010. Outline. Introduction Implementation in WRF Applications Limitations Conclusions. Outline.

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Forecast Sensitivity to Observations & Observation Impact Tom Auligné National Center for Atmospheric Research Ackn

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  1. WRF-Var Tutorial - Feb 01-03 2010 Forecast Sensitivity to Observations & Observation Impact Tom Auligné National Center for Atmospheric Research Acknowledgments: Hongli Wang, Qingnong Xiao WRF-Var Tutorial - Feb. 01-03 2010

  2. Outline WRF-Var Tutorial - Feb 01-03 2010 • Introduction • Implementation in WRF • Applications • Limitations • Conclusions

  3. Outline WRF-Var Tutorial - Feb 01-03 2010 • Introduction • Implementation in WRF • Applications • Limitations • Conclusions

  4. Introduction WRF-Var Tutorial - Feb 01-03 2010 FSO? • What? • Why? • Who? • How? • How much?

  5. Introduction WRF-Var Tutorial - Feb 01-03 2010 • What? • Why? • Who? • How? • How much? • A posteriori, it is possible to evaluate the accuracy of NWP forecasts. • Using an adjoint technique, we can trace it back to the observations used in the analysis. • We can determine quantitatively which observations improved or degraded the forecast. • Forecast Sensitivity to Observations (FSO) is a diagnostic tool that complements traditional denial experiments (OSEs).

  6. Introduction WRF-Var Tutorial - Feb 01-03 2010 • What? • Why? • Who? • How? • How much? • Impact of each observation calculated simultaneously (less tedious than OSEs). • NWP centers use FSO routinely to monitor their Data Assimilation and Global Observing System • Can be used to tune Quality Control, Bias Correction, etc. • Helps assess the impact of specific sensors for data providers.

  7. Introduction WRF-Var Tutorial - Feb 01-03 2010 • What? • Why? • Who? • How? • How much? • Naval Research Laboratory (Monterey, CA) • NASA/GMAO (Washington, DC) • ECMWF (Reading, UK) • Environment Canada (Montreal, Canada) • Meteo-France (Toulouse, France) • NCAR/MMM (Boulder, CO)

  8. Introduction WRF-Var Tutorial - Feb 01-03 2010 • Non-Linear (NL) forecast models can be linearized (with simplifications). • The resulting Tangent-Linear (TL) represents the linear evolution of small perturbations. • The mathematical transpose of the TL code is called the Adjoint (ADJ) and it transports sensitivities back in time. • The ADJ of the Data Assimilation system is needed to compute the sensitivity to observations. It can be computed with various methods: • Ensemble (ETKF, Bishop et al. 2001) • Dual approach (PSAS, Baker and Daley 2000, Pellerin et al. 2007) • Exact ADJ calculation (Zhu and Gelaro 2007) • Hessian approximation (Cardinali 2006) • Lanczos minimization (Fisher 1997, Tremolet 2008) • What? • Why? • Who? • How? • How much?

  9. Introduction WRF-Var Tutorial - Feb 01-03 2010 • What? • Why? • Who? • How? • How much? ---> WRF code and scripts

  10. Introduction WRF-Var Tutorial - Feb 01-03 2010 • What? • Why? • Who? • How? • How much? • 2 runs of non-linear forecast model • 2 runs of adjoint model • 1 run of adjoint of analysis • The computer cost is estimated to 10-15 times the cost of the forecast model.

  11. Outline WRF-Var Tutorial - Feb 01-03 2010 • Introduction • Implementation in WRF • Applications • Limitations • Conclusions

  12. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Task:“Develop adjoint of WRF-Var’s minimization algorithm, additional I/O, and couple with 4D-Var’s adjoint of the ARW forecast model, to create a diagnostic tool for evaluating observation/analysis impact on forecast accuracy”

  13. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Analysis (xa) Forecast (xf) Observation (y) WRF-VAR Data Assimilation WRF-ARW Forecast Model Define Forecast Accuracy Background (xb) Forecast Accuracy (F) Observation Impact <y-H(xb)> (F/ y) Gradient of F (F/ xf) Analysis Sensitivity (F/ xa) Observation Sensitivity (F/ y) Adjoint of WRF-ARW Forecast TL Model (WRF+) Adjoint of WRF-VAR Data Assimilation Derive Forecast Accuracy Background Sensitivity (F/ xb) Obs Error Sensitivity (F/ eob) Bias Correction Sensitivity (F/ k) Figure adapted from Liang Xu (NRL)

  14. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Analysis (xa) Forecast (xf) Observation (y) WRF-VAR Data Assimilation WRF-ARW Forecast Model Define Forecast Accuracy Background (xb) Forecast Accuracy (F) Observation Impact <y-H(xb)> (F/ y) Gradient of F (F/ xf) Analysis Sensitivity (F/ xa) Observation Sensitivity (F/ y) Adjoint of WRF-ARW Forecast TL Model (WRF+) Adjoint of WRF-VAR Data Assimilation Derive Forecast Accuracy Background Sensitivity (F/ xb) Obs Error Sensitivity (F/ eob) Bias Correction Sensitivity (F/ k)

  15. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Analysis (xa) Forecast (xf) Observation (y) WRF-VAR Data Assimilation WRF-ARW Forecast Model Define Forecast Accuracy • Usual WRF-Var 3DVar or 4DVar data assimilation system • Namelist parameter needs to be activated: ORTHONORM_GRADIENT=true Background (xb) Forecast Accuracy (F) Observation Impact <y-H(xb)> (F/ y) Gradient of F (F/ xf) Analysis Sensitivity (F/ xa) Observation Sensitivity (F/ y) Adjoint of WRF-ARW Forecast TL Model (WRF+) Adjoint of WRF-VAR Data Assimilation Derive Forecast Accuracy Background Sensitivity (F/ xb) Obs Error Sensitivity (F/ eob) Bias Correction Sensitivity (F/ k)

  16. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Analysis (xa) Forecast (xf) Observation (y) WRF-VAR Data Assimilation WRF-ARW Forecast Model Define Forecast Accuracy Background (xb) Forecast Accuracy (F) Observation Impact <y-H(xb)> (F/ y) Gradient of F (F/ xf) Analysis Sensitivity (F/ xa) Observation Sensitivity (F/ y) Adjoint of WRF-ARW Forecast TL Model (WRF+) Adjoint of WRF-VAR Data Assimilation Derive Forecast Accuracy Background Sensitivity (F/ xb) Obs Error Sensitivity (F/ eob) Bias Correction Sensitivity (F/ k)

  17. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Analysis (xa) Forecast (xf) Observation (y) WRF-VAR Data Assimilation WRF-ARW Forecast Model Define Forecast Accuracy • WRF ARW forecast • Forecast length is set to reach verification time • Use WRFNL code to write trajectory for adjoint run Background (xb) Forecast Accuracy (F) Observation Impact <y-H(xb)> (F/ y) Gradient of F (F/ xf) Analysis Sensitivity (F/ xa) Observation Sensitivity (F/ y) Adjoint of WRF-ARW Forecast TL Model (WRF+) Adjoint of WRF-VAR Data Assimilation Derive Forecast Accuracy Background Sensitivity (F/ xb) Obs Error Sensitivity (F/ eob) Bias Correction Sensitivity (F/ k)

  18. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Analysis (xa) Forecast (xf) Observation (y) WRF-VAR Data Assimilation WRF-ARW Forecast Model Define Forecast Accuracy Background (xb) Forecast Accuracy (F) Observation Impact <y-H(xb)> (F/ y) Gradient of F (F/ xf) Analysis Sensitivity (F/ xa) Observation Sensitivity (F/ y) Adjoint of WRF-ARW Forecast TL Model (WRF+) Adjoint of WRF-VAR Data Assimilation Derive Forecast Accuracy Background Sensitivity (F/ xb) Obs Error Sensitivity (F/ eob) Bias Correction Sensitivity (F/ k)

  19. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Analysis (xa) Forecast (xf) Observation (y) WRF-VAR Data Assimilation WRF-ARW Forecast Model Define Forecast Accuracy • Reference state: Namelist ADJ_REF is defined as • 1: Xt = Own (WRFVar) analysis • 2: Xt = NCEP (global GSI) analysis • 3: Xt = Observations • Forecast Aspect:depends on reference state • 1 and 2: Total Dry Energy • 3: WRFVar Observation Cost Function: Jo • Geo. projection: Script option for box(default = whole domain)ADJ_ISTART, ADJ_IEND, ADJ_JSTART, ADJ_END, ADJ_KSTART, ADJ_KEND • Forecast Accuracy Norm: e = (xf-xt)T C (xf-xt) Background (xb) Forecast Accuracy (F) Observation Impact <y-H(xb)> (F/ y) Gradient of F (F/ xf) Analysis Sensitivity (F/ xa) Observation Sensitivity (F/ y) Adjoint of WRF-ARW Forecast TL Model (WRF+) Adjoint of WRF-VAR Data Assimilation Derive Forecast Accuracy Background Sensitivity (F/ xb) Obs Error Sensitivity (F/ eob) Bias Correction Sensitivity (F/ k)

  20. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Analysis (xa) Forecast (xf) Observation (y) WRF-VAR Data Assimilation WRF-ARW Forecast Model Define Forecast Accuracy xt is the true state, estimated by the analysis at the time of the forecast xf is the forecast from analysis xa xg is the forecast from first-guess at the time of the analysis xa Impact of analysis:F = Def,g = ef – eg Products:F/ xaf = C(xaf-xt)F/ xbf = C(xbf-xt) From Langland and Baker (2004) Forecast Error Background (xb) Forecast Accuracy (F) Observation Impact <y-H(xb)> (F/ y) Gradient of F (F/ xf) Analysis Sensitivity (F/ xa) Observation Sensitivity (F/ y) Adjoint of WRF-ARW Forecast TL Model (WRF+) Adjoint of WRF-VAR Data Assimilation Derive Forecast Accuracy Background Sensitivity (F/ xb) Obs Error Sensitivity (F/ eob) Bias Correction Sensitivity (F/ k)

  21. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Analysis (xa) Forecast (xf) Observation (y) WRF-VAR Data Assimilation WRF-ARW Forecast Model Define Forecast Accuracy Background (xb) Forecast Accuracy (F) Observation Impact <y-H(xb)> (F/ y) Gradient of F (F/ xf) Analysis Sensitivity (F/ xa) Observation Sensitivity (F/ y) Adjoint of WRF-ARW Forecast TL Model (WRF+) Adjoint of WRF-VAR Data Assimilation Derive Forecast Accuracy Background Sensitivity (F/ xb) Obs Error Sensitivity (F/ eob) Bias Correction Sensitivity (F/ k)

  22. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Analysis (xa) Forecast (xf) Observation (y) WRF-VAR Data Assimilation WRF-ARW Forecast Model Define Forecast Accuracy First order approximation: Background (xb) Forecast Accuracy (F) Observation Impact <y-H(xb)> (F/ y) Gradient of F (F/ xf) Analysis Sensitivity (F/ xa) Observation Sensitivity (F/ y) Adjoint of WRF-ARW Forecast TL Model (WRF+) Adjoint of WRF-VAR Data Assimilation Derive Forecast Accuracy Background Sensitivity (F/ xb) Obs Error Sensitivity (F/ eob) Bias Correction Sensitivity (F/ k)

  23. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Analysis (xa) Forecast (xf) Observation (y) WRF-VAR Data Assimilation WRF-ARW Forecast Model Define Forecast Accuracy First order approximation: Background (xb) Forecast Accuracy (F) Observation Impact <y-H(xb)> (F/ y) Gradient of F (F/ xf) Analysis Sensitivity (F/ xa) Observation Sensitivity (F/ y) Relative error in WRF (linear vs. non-linear propagation of perturbation) Adjoint of WRF-ARW Forecast TL Model (WRF+) Adjoint of WRF-VAR Data Assimilation Derive Forecast Accuracy -----------------------> 62.25% Background Sensitivity (F/ xb) ---------> 19.68% -----------> 11.45% Obs Error Sensitivity (F/ eob) Bias Correction Sensitivity (F/ k) Results are consistent with Gelaro et al. (2007)

  24. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Analysis (xa) Forecast (xf) Observation (y) WRF-VAR Data Assimilation WRF-ARW Forecast Model Define Forecast Accuracy • Script variable ADJ_MEASURE defined as: • 1: first order • 2: second order • 3: third order • 4: variant of third order • Use WRF+ code to compute WRF-ARW adjoint with Namelist ADJ_SENS=true: • Activate pressure in the adjoint • Switch off intermediate forcing • WRF+ is run for both trajectories from xa and xb • Finally, both sensitivities are added together Background (xb) Forecast Accuracy (F) Observation Impact <y-H(xb)> (F/ y) Gradient of F (F/ xf) Analysis Sensitivity (F/ xa) Observation Sensitivity (F/ y) Adjoint of WRF-ARW Forecast TL Model (WRF+) Adjoint of WRF-VAR Data Assimilation Derive Forecast Accuracy Background Sensitivity (F/ xb) Obs Error Sensitivity (F/ eob) Bias Correction Sensitivity (F/ k)

  25. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Analysis (xa) Forecast (xf) Observation (y) WRF-VAR Data Assimilation WRF-ARW Forecast Model Define Forecast Accuracy Background (xb) Forecast Accuracy (F) Observation Impact <y-H(xb)> (F/ y) Gradient of F (F/ xf) Analysis Sensitivity (F/ xa) Observation Sensitivity (F/ y) Adjoint of WRF-ARW Forecast TL Model (WRF+) Adjoint of WRF-VAR Data Assimilation Derive Forecast Accuracy Background Sensitivity (F/ xb) Obs Error Sensitivity (F/ eob) Bias Correction Sensitivity (F/ k)

  26. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Analysis (xa) Forecast (xf) Observation (y) WRF-VAR Data Assimilation WRF-ARW Forecast Model Define Forecast Accuracy • Analysis increments: x = xa - xb = K [y-H(xb)] = K d • Sensitivity of analysis to observations: xa /y = KT • Adjoint of the variational analysis: F/y = KT F/xa • New minimization package, activated with Namelist USE_LANCZOS=true Background (xb) Forecast Accuracy (F) Observation Impact <y-H(xb)> (F/ y) Gradient of F (F/ xf) Analysis Sensitivity (F/ xa) Observation Sensitivity (F/ y) Adjoint of WRF-ARW Forecast TL Model (WRF+) Adjoint of WRF-VAR Data Assimilation Derive Forecast Accuracy Background Sensitivity (F/ xb) Obs Error Sensitivity (F/ eob) Bias Correction Sensitivity (F/ k)

  27. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Analysis (xa) Forecast (xf) Observation (y) WRF-VAR Data Assimilation WRF-ARW Forecast Model Define Forecast Accuracy • Analysis increments: x = xa - xb = K [y-H(xb)] = K d • Sensitivity of analysis to observations: xa /y = KT • Adjoint of the variational analysis: F/y = KT F/xa • New minimization package activated with Namelist USE_LANCZOS=true Background (xb) Forecast Accuracy (F) Observation Impact <y-H(xb)> (F/ y) Gradient of F (F/ xf) Analysis Sensitivity (F/ xa) • Cost Function and Gradient are IDENTICAL to Conjugate Gradient • Lanczos estimates the Hessian = Inverse of Analysis error A-1 • KT = R-1 H A-1 • We calculate the EXACT adjoint of analysis gain: KT < x, x > = < x, K d> compared to <KTx, d> -----> 10-13 relative error Observation Sensitivity (F/ y) Adjoint of WRF-ARW Forecast TL Model (WRF+) Adjoint of WRF-VAR Data Assimilation Derive Forecast Accuracy Background Sensitivity (F/ xb) Obs Error Sensitivity (F/ eob) Bias Correction Sensitivity (F/ k)

  28. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Analysis (xa) Forecast (xf) Observation (y) WRF-VAR Data Assimilation WRF-ARW Forecast Model Define Forecast Accuracy Background (xb) Forecast Accuracy (F) Observation Impact <y-H(xb)> (F/ y) Gradient of F (F/ xf) Analysis Sensitivity (F/ xa) Observation Sensitivity (F/ y) Adjoint of WRF-ARW Forecast TL Model (WRF+) Adjoint of WRF-VAR Data Assimilation Derive Forecast Accuracy Background Sensitivity (F/ xb) Obs Error Sensitivity (F/ eob) Bias Correction Sensitivity (F/ k)

  29. Implementation in WRF WRF-Var Tutorial - Feb 01-03 2010 Scripts: • Analysis Experiment • WRF-Var with Namelist ORTHONORM_GRADIENT=true • Trajectories • WRFNL from Xa and from Xb • Forecast Accuracy • ADJ_REF to choose reference for forecast accuracy • ADJ_ISTART, ADJ_IEND, etc to define a box • Adjoint of Model • ADJ_MEASURE to select order of Taylor expansion • WRF+ (Adjoint mode) with Namelist ADJ_SENS=true • Adjoint of Analysis • RUN_OBS_IMPACT=true launches WRF-Var with Lanczos

  30. Outline WRF-Var Tutorial - Feb 01-03 2010 • Introduction • Implementation in WRF • Applications • Limitations • Conclusions

  31. Applications WRF-Var Tutorial - Feb 01-03 2010 Impact of METAR data on forecast error

  32. Applications WRF-Var Tutorial - Feb 01-03 2010 ConventionalObservations SatelliteRadiances Total per obs

  33. Applications WRF-Var Tutorial - Feb 01-03 2010 from Gelaro et al. 2009

  34. Applications WRF-Var Tutorial - Feb 01-03 2010 from Langland 2009

  35. Applications WRF-Var Tutorial - Feb 01-03 2010 from Gelaro 2009

  36. Outline WRF-Var Tutorial - Feb 01-03 2010 • Introduction • Implementation in WRF • Applications • Limitations • Conclusions

  37. Limitations WRF-Var Tutorial - Feb 01-03 2010 • Uncertainties are difficult to estimate and represent. • The reference for the calculation of forecast accuracy is NOT perfect and often correlated with the initial analysis. • The adjoint model is not an accurate representation of the NL model behavior (linearization, simplification, dry physics). Langland (2009) proposes a method to mitigate these errors. • For higher than first-order approximation of de, nonlinear dependence on dy, which complicates the separation of observation impact (Errico 2007). These errors are small for the calculation of average impact (Gelaro et al. 2007).

  38. Limitations WRF-Var Tutorial - Feb 01-03 2010 • Results are strongly dependent on the norm chosen to define forecast accuracy. • The interpretation of information and application is not always straightforward.

  39. Outline WRF-Var Tutorial - Feb 01-03 2010 • Introduction • Implementation in WRF • Applications • Limitations • Conclusions

  40. Conclusions WRF-Var Tutorial - Feb 01-03 2010 • All code and scripts for FSO will be available in next WRF public release. • A small User’s Guide will be provided. Due to lack of funding, no support to be expected ;-( • Have fun!

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