# Presentation: 113032

## Accelerating the spin-up of EnKF

- Eugenia Kalnay and Shu-Chih Yang
-  Motivation and theory
-  4D-LETKF
-  No-cost smoother
-  “Running in place”
-  Results
-  Conclusions

## Motivation

-  4D-Var is known to spin-up faster than EnKF because it is a smoother: it keeps iterating until it fits the observations within the assimilation window as well as possible
-  EnKF spins-up fast if starting from a “good” initial state, e.g., 3D-Var
-  In a severe storm where radar observations start with the storm, there is little real time to spin-up
-  Caya et al. (2005): “EnKF is eventually better than 4D-Var” (but it is too late to be useful, it misses the storm). Also Jidong Gao, (pers. comm.)

## Theory (1)

-  Hunt et al., 2007: The background term represents the evolution of the maximum likelihood trajectory given all the observations in the past
-  After the analysis a similar relationship is valid:
-  From here one can derive the linear KF equations
-  Also the rule: “Use the data once and then discard it”

## Theory (2)

-  The rule: “Use the data once and then discard it” makes sense when the analysis/forecasts are the most likely given all the past data, not when we start from scratch. Hence we propose “Running in place” until we extract the maximum information form the observations.
-  We need
-  4D-LETKF (Hunt et al, 2004) to use all the observations within an assimilation window at their right time
-  A No-Cost Smoother (Kalnay et al., 2007b)
-  An appropriate iterative scheme

## Theory (3)

-  4D-LETKF: observational increments are linear combinations of the ensemble forecasts at the observation time. We can use the same linear combination at analysis time. Equivalent to 4D-Var.
-  No-Cost Smoother (Kalnay et al., 2007b):
-  At the end of the assimilation window, the analysis ensemble is a linear combination of the forecast ensemble.
-  We can use the same linear combination at the beginning of the assimilation window.
-  This no-cost smoother will give an improved initial analysis (but the same final analysis, Yang et al, 2008b)

## Running in Place (1)

-  EnKF is a sequential data assimilation system where, after the new data is used at the analysis time, it should be discarded...
-  **only** if the previous analysis and the new background are the most likely states given the past observations.
-  **If the system has converged after the initial spin-up all the information from past observations is already included in the background. **
-  During the spin-up we should use the observations repeatedly if we can extract extra information
-  But we should avoid overfitting the observations

## Running in Place algorithm (1)

-  Cold-start the EnKF from any initial ensemble mean and random perturbations at *t**0*, and integrate the initial ensemble to *t**1*. The “running in place” loop with *n=1*, is:
- a) Perform a standard EnKF analysis and obtain the analysis weights at *t**n*, saving the mean square observations minus forecast (OMF) computed by the EnKF.
- b) Apply the no-cost smoother to obtain the smoothed analysis ensemble at *t**n-1* by using the same weights obtained at *t**n*.
- c) The smoothed analysis ensemble is perturbed globally with a small amount of random Gaussian perturbations, a method similar to additive inflation.

## Running in Place algorithm (2)

- a) Perform a standard EnKF analysis and obtain the analysis weights at *t**n*, saving the mean square observations minus forecast (OMF) computed by the EnKF.
- b) Apply the no-cost smoother to obtain the smoothed analysis ensemble at *t**n-1* by using the same weights obtained at *t**n*.
- c) Perturb the smoothed analysis ensemble with a small amount of random Gaussian perturbations, a method similar to additive inflation.
- d) Integrate the perturbed smoothed ensemble to *t**n*. If the forecast fit to the observations is smaller than in the previous iteration according to a criterion, go to a) and perform another iteration. If not, let                and proceed to the next assimilation window.

## Running in Place algorithm (notes)

- Notes:
- c) *Perturb the smoothed analysis ensemble with a small amount of random Gaussian perturbations, a method similar to additive inflation.*
- This perturbation has two purposes:
- Avoid reaching the same analysis as before, and
- Encourage the ensemble to explore new unstable directions
- d) *Convergence criterion*: if
- with                  do another iteration. Otherwise go to the next assimilation window. (If you only have done one iteration for a few windows, stop running in place: it’s already spun-up).

## Results with a QG model

- Notes:
- With a poor initial ensemble the LETKF takes 120 cycles to get good errors of the day. Then it converges quickly (~ 180 cycles)
- 4D-Var converges in about 90 cycles
- With eps~1%, too many iterations
- With eps~5%, 90 cycles
- Faster with smooth perturbations (Zupanski)
- Fixing 10 iterations overuses the observations

## Number of iterations

- Notes:
- With eps~1%, too many iterations
- With eps~5%, 2-8 iterations were enough
- Faster (in real time) with smooth (3D-Var) perturbations (Zupanski et al., 2006) but at the end small scale perturbations are more effective.

## Discussion

- Number of iterations during spin-up: 2-8, computationally acceptable
- We could use the weights interpolation of Yang et al. (2008b) and run in place only where “the action is”.
- There are many applications where a fast spin-up is important.
- (Junjie Liu has developed LETKF “adjoint sensitivity” without the adjoint.)
- It seems like there is nothing that 4D-Var can do that EnKF cannot do as well, usually better.