What is Data Assimilation?

1. What is Data Assimilation? Dr. Andrew S. Jones CSU/CIRA

2. Data Assimilation Outline • Why Do Data Assimilation? • Who and What • Important Concepts • Definitions • Brief History • Common System Issues / Challenges

3. The Purpose of Data Assimilation • Why do data assimilation? • I want better model initial conditions for better model forecasts • I want better calibration and validation (cal/val) • I want better acquisition guidance

4. The Purpose of Data Assimilation • Why do data assimilation? • I want better model initial conditions for better model forecasts • I want better calibration and validation (cal/val) • I want better acquisitionguidance • I want better scientificunderstandingof • Model errors (and their probability distributions) • Data errors (and their probability distributions) • Combined Model/Data correlations • DA methodologies (minimization, computational optimizations, representation methods, various method approximations) • Physical process interactions(i.e., sensitivities and feedbacks)Leads toward better future models VIRTUOUS CYCLE

5. The Data Assimilation Community • What skills are needed by each involved group? • NWP Data Assimilation Experts (DA system methodology) • NWP Modelers (Model + Physics + DA system) • Application and Observation Specialists (Instrument capabilities) • Physical Scientists (Instrument + Physics + DA system) • Radiative Transfer Specialists (Instrument config. specifications) • Applied Mathematicians (Control theory methodology) • Computer Scientists (DA system + OPS time requirements) • Science Program Management (Everything + $$ + Good People) • Forecasters (Everything + OPS time reqs. + Easy/fast access) • Users and Customers (Could be a wide variety of responses)e.g., NWS / Army / USAF / Navy / NASA / NSF / DOE / ECMWF

6. The Data Assimilation Community • Are you part of this community? • Yes, you just may not know it yet.

7. Maximum Conditional Probability is given by: P (x | y) ~ P (y | x) P (x) Assuming Gaussian distributions… P (y | x) ~ exp {-1/2 [y – H (x)]T R-1 [y – H (x)]} P (x) ~ exp {-1/2 [x –xb]T B-1 [x – xb]} Bayes Theorem e.g., 3DVAR Lorenc (1986)

8. Minimization Process Jacobian of the Cost Function is used in the minimization procedure Minima is at J/ x = 0 Issues: Is it a global minima? Are we converging rapidor slow? J TRUTH x

9. The Building Blocks of Data Assimilation Start Control Variables are the initial model state variables that are optimized using the new data information as a guide NWP Model Observation Model Minimization Observations They can also include boundary condition information, model parameters for “tuning”, etc. NWPAdjoint Observation ModelAdjoint

10. What Are We Minimizing? Minimize discrepancy between model and observation data over time The Cost Function, J, is the link between the observational data and the model variables Observations are either assumed unbiased, or are “debiased” by some adjustment method

11. How are Data used in Time? Observation model Cloud resolving model forecast time observations Assimilation time window

12. A “Smoother” Uses All Data Availablein the Assimilation Window(a “Simultaneous” Solution) Observation model Cloud resolving model forecast time observations Assimilation time window

13. forecast A “Filter” Sequentially Assimilates Dataas it Becomes Available in each Cycle Observation model Cloud resolving model time observations Assimilation time window

14. Cycle Previous Information forecast A “Filter” Sequentially Assimilates Dataas it Becomes Available in each Cycle Observation model Cloud resolving model time observations Assimilation time window

15. Cycle Previous Information forecast A “Filter” Sequentially Assimilates Dataas it Becomes Available in each Cycle Observation model Cloud resolving model time observations Assimilation time window

16. A “Filter” Sequentially Assimilates Dataas it Becomes Available in each Cycle Observation model Cloud resolving model forecast time Cycle Physics “Barriers” What Can Overcome the Barrier? • Linear Physics Processes and • Propagated Forecast Error Covariances

17. Who are the Candidates for “Truth”? Minimize discrepancy between model and observation data over time Candidate 1: Background Term “x0” is the model state vector at the initial time t0 this is also the “control variable”, the object of the minimization process “xb” is the model background state vector “B” is the background error covariance of the forecast and model errors

18. What Do We Trust for “Truth”? Minimize discrepancy between model and observation data over time Candidate 1: Background Term The default condition for the assimilation when • data are not availableor • the available data have no significant sensitivity to the model stateor • the available data are inaccurate

19. Model Error Impacts our “Trust” Minimize discrepancy between model and observation data over time Candidate 1: Background Term Model error issues are important Model error varies as a function of the model time Model error “grows” with time Therefore the background term should be trusted moreat the initial stagesof the model run and trusted lessat the end of the model run

20. How to Adjust for Model Error? Minimize discrepancy between model and observation data over time Candidate 1: Background Term • Add a model error term to the cost function so that the weight at that specific model step is appropriately weighted or • Use other possible adjustments in the methodology, i.e., “make an assumption” about the model error impacts If model error adjustments or controls are used the DA system is said to be “weakly constrained”

21. What About Model Error Errors? Minimize discrepancy between model and observation data over time Candidate 1: Background Term Model error adjustments to the weighting can be “wrong” • In particular, most assume some type of linearity • Non-linear physical processes may break these assumptions and be more complexly interrelated A data assimilation system with no model error control is said to be “strongly constrained” (perfect model assumption)

22. Model What About Model Error Errors? “I just can’t run like I used to.” A Strongly Constrained System? “Little Data People”

23. DA expert What About Model Error Errors? “We’ll… no one’s perfect.” A Strongly Constrained System? Can Data Over Constrain?

24. Who are the Candidates for “Truth”? Minimize discrepancy between model and observation data over time Candidate 2: Observational Term “y” is the observational vector, e.g., the satellite input data (typically radiances), salinity, sounding profiles “M0,i(x0)” is the model state at the observation time “i” “h” is the observational operator, for example the“forward radiative transfer model” “R” is the observational error covariance matrix that specifies the instrumental noise and data representation errors (currently assumed to be diagonal…)

25. What Do We Trust for “Truth”? Minimize discrepancy between model and observation data over time Candidate 2: Observational Term The non-default condition for the assimilation when • data are available and • data are sensitive to the model state and • data are precise (not necessarily “accurate”) and • data are not thrown away by DA “quality control” methods

26. What About other DA Errors? Overlooked Issues? • Data debiasing relative to the DA system “reference”. It is not the “Truth”,however it is self-consistent. • DA Methodology Errors? • Assumptions: Linearization, Gaussianity, Model errors • Representation errors (space and time) • Poorly known background error covariances • Imperfect observational operators • Overly aggressive data “quality control” • Historical emphasis on dynamical impact vs. physical Synoptic vs. Mesoscale?

27. Add DA Bias Here! DA Theory is Still Maturing Lognormal Variables Clouds Precipitation Water vapor Emissivities Many other hydrologic fields Mode  Mean The Future: Lognormal DA (Fletcher and Zupanski, 2006, 2007) Gaussian systems typically force lognormal variables to become Gaussian introducing an avoidabledata assimilation system bias Many important variables are lognormally distributed Gaussian data assimilation system variables are “Gaussian”

28. What “Truth” Do We Have? Minimize discrepancy between model and observation data over time DATA MODEL CENTRIC CENTRIC TRUTH

29. What Approach Should We Use? DATA MODEL CENTRIC CENTRIC TRUTH

30. What Approach Should We Use? DATA MODEL CENTRIC CENTRIC TRUTH

31. My Precious… We Trust the Model! Data hurts us!, Yes… What Approach Should We Use? DATA MODEL CENTRIC CENTRIC TRUTH

32. MODEL CENTRIC FOCUS FOCUS ON “B” Background Error Improvements are Needed “xb” Associated background states and “Cycling” are more heavily emphasized DA method selection tends toward sequential estimators, “filters”, and improved representation of the forecast model error covariances DATA MODEL CENTRIC CENTRIC E.g., Ensemble Kalman Filters, other Ensemble Filter systems

33. What Approach Should We Use? This is not to say that all model-centric improvements are bad… DATA MODEL CENTRIC CENTRIC TRUTH

34. What Approach Should We Use? DATA MODEL CENTRIC CENTRIC TRUTH

35. My Precious… We Trust the Data! Models unfair and hurts us!, Yes… What Approach Should We Use? DATA MODEL CENTRIC CENTRIC TRUTH

36. DATA CENTRIC FOCUS FOCUS ON “h” Observational Operator Improvements are Needed “M0,i(x0)” Model state capabilities and independent experimental validation is more heavily emphasized DA method selection tends toward “smoothers” (less focus on model cycling), more emphasis on data quantity and improvements in the data operator and understanding of data representation errors DATA DATA CENTRIC CENTRIC e.g., 4DVAR systems

37. DUAL-CENTRIC FOCUS Best of both worlds? Solution: Ensemble based forecast covariance estimates combined with 4DVAR smoother for research and 4DVAR filter for operations? Several frameworks to combine the two approaches are in various stages of development now…

38. What Have We Learned? Your Research Objective is CRITICAL to making the right choices… • Operational choices may supercedegood research objectives • Computational speed is always critical for operational purposes • Accuracy is critical for research purposes DATA MODEL CENTRIC CENTRIC TRUTH

39. DA Theory is Still Maturing A Brief History of DA Hand Interpolation Local polynomial interpolation schemes(e.g., Cressman) Use of “first guess”, i.e., a background Use of an “analysis cycle” to regeneratea new first guess Empirical schemes, e.g., nudging Least squares methods Variational DA (VAR) Sequential DA (KF) Monte Carlo Approx. to Seq. DA (EnsKF)

40. Eliassen (1954), Bengtsson et al. (1981), Gandin (1963) Became the operational scheme in early 1980s and early 1990s Optimal Interpolation (OI) OI merely means finding the “optimal” Weights, W A better name would have been “statistical interpolation”

41. Major Flavors: 1DVAR (Z), 3DVAR (X,Y,Z), 4DVAR (X,Y,Z,T) Lorenc (1986) and others… Became the operational scheme in early 1990s to the present day Variational Techniques Finds the maximum likelihood (if Gaussian, etc.) (actually it is a minimum variance method) Comes from setting the gradient of the cost function equal to zero Control variable is xa

42. What is a Hessian? A Rank-2 Square Matrix Containing the Partial Derivatives of the Jacobian G(f)ij(x) = DiDj f(x) The Hessian is used in some minimization methods,e.g., quasi-Newton…

43. The Role of the Adjoint, etc. Adjoints are used in the cost function minimization procedure But first… Tangent Linear Models are used to approximate the non-linear model behaviors L x’ = [M(x1) – M(x2)] /  L is the linear operator of the perturbation model M is the non-linear forward model is the perturbation scaling-factorx2 = x1 + x’

44. Useful Properties of the Adjoint <Lx’, Lx’> <LTLx’, x’> LT is the adjoint operator of the perturbation model Typically the adjoint and the tangent linear operator can be automatically created using automated compilers y =  (x1, …, xn, y) *xi = *xi + *y /xi *y = *y /y where *xi and *y are the “adjoint” variables

45. Useful Properties of the Adjoint <Lx’, Lx’> <LTLx’, x’> LT is the adjoint operator of the perturbation model Typically the adjoint and the tangent linear operator can be automatically created using automated compilers Of course, automated methods fail for complex variable types (See Jones et al., 2004) E.g., how can the compiler know when the variable is complex, when codes are decomposed into real and imaginary parts as common practice? (It can’t.)

46. Kalman (1960) and many others… These techniques can evolve the forecast error covariance fields similar in concept to OI Sequential Techniques B is no longer static, B => Pf = forecast error covariance Pa (ti) is estimated at future times using the model K = “Kalman Gain” (in blue boxes) Extended KF, Pa is found by linearizing the model about the nonlinear trajectory of the model betweenti-1 and ti

47. Ensembles can be used in KF-based sequential DA systems Ensembles are used to estimate Pf through Gaussian “sampling” theory Sequential Techniques f is a particular forecast instance l is the reference state forecast Pf is estimated at future times using the model K number model runs are required (Q: How to populate the seed perturbations?) Sampling allows for use of approximate solutions Eliminates the need to linearize the model (as inExtended KF) No tangent linear or adjoint models are needed

48. Notes on EnsKF-based sequential DA systems EnsKFs are an approximation Underlying theory is the KF (circa 1960) Assumes Gaussian distributions Many ensemble samples are required Can significantly improve Pf Where does H fit in? Is it fully “resolved”? What about the “Filter” aspects? Future Directions Research using Hybrid EnsKF-Var techniques Sequential Techniques

49. Zupanski (2005): Maximum Likelihood Ensemble Filter (MLEF) Structure function version of Ensemble-based DA (Note: Does not use sampling theory, and is more similar to a variational DA scheme using principle component analysis (PCA) Sequential Techniques NEis the number of ensembles S is the state-space dimension Each ensemble is carefully selected to represent thedegrees of freedom of the system Square-root filter is built-in to the algorithm assumptions

50. No M used 3DDA Techniques have no explicit model time tendency information, it is all done implicitly with cycling techniques, typically focusing only on the Pf term 4DDA uses Mexplicitly via the model sensitivities, L, and model adjoints, LT,as a function of time Kalman Smoothers (e.g., also 4DEnsKS) would likewise also need to estimate L and LT Where is “M” in all of this? M used