A strategy to optimize the use of retrievals in data assimilation R. Hoffman, K. Cady-Pereira, J. Eluszkiewicz, D. Gombos, J.-L. Moncet, T. Nehrkorn, S. Greybush 2, K. Ide 2, E. Kalnay 2, M. J. Hoffman 3, and R. J. Wilson 4 AER, 2 U. Maryland, 3 John Hopkins, 4 GFDL Friday, 6 May, 20 Ross N. Hoffman, Vice President, R&D Atmospheric and Environmental Research, www.aer.com 20 Atmospheric and Environmental Research (AER). All rights reserved.
Introduction Is it better to use radiances or retrievals in data assimilation? Using retrievals reduces complexity in the H-operator, reduces data volume, allows arbitrary cloud clearing and retrieval methods and makes the assimilation system more modular. Radiances are available sooner, have uncorrelated errors, and are independent of the prior. Using EOFs from the retrieval scheme can reduce data volume and reduces vertical interpolation errors. The averaging kernel (AK) concept allows removing the influence of the prior (allowing for interactive retrievals) and rotating to a representation where the obs. errors are uncorrelated. 20 Atmospheric and Environmental Research (AER). All rights reserved. 2
Examples Work in progress from on-going Mars data assimilation experiments using GFDL Mars GCM TES and MCS radiances and retrievals LETKF ensemble Kalman filter method Examples from TES retrievals OSS forward model Rodgers type iterative physical retrieval scheme Inverse Methods for Atmospheric Sounding: Theory and Practice Temperature profiles only [for now] 20 Atmospheric and Environmental Research (AER). All rights reserved. 3
Retrievals Retrievals, ˆx (22 of 887 total retrievals) 22 retrievals sampled from 6 hour period Pressure (hpa) 0 0 0 50 60 70 80 90 200 20 220 230 240 250 Temperature (K) 20 Atmospheric and Environmental Research (AER). All rights reserved. 4
Mars Retrieval Locations and Topography 887 retrieval loca7ons 2"#$)$% 6+7$)89% 3"++'$%!"#)0'#)$% Hellas Plani7a 4'++"$% 5"$)0% TES data at the northern fall equinox (Ls=80) of Mars Year 24 (MY24) 20 Atmospheric and Environmental Research (AER). All rights reserved. 5
Forward model vertical grid Retrieval data pressure levels (linear) 0.7 0.39 0.8255.36.7477 2.244 2.884 3.6998 Retrieval data pressure levels (log) 0.7 0.842 0.3037 66.3747 74.859 8.5864 Temperature (x a ) profile (linear) p ln(p) T a 4.7507 6. 7.8326 0.0572 0.5007 0.8255.36 2.244 3.6998 Temperature (K) 88.9546 96.835 204.8438 22.6396 2.937 6. 22.2834 226.927 6.585 3 5 7 9 3 5 7 9 2 0.0572 2 pressure levels 6.585 3 5 7 9 3 5 7 9 2 23.37 238.262 3 5 7 9 3 5 7 9 2 Pressure level index 20 Atmospheric and Environmental Research (AER). All rights reserved. 6
Temperature EOFs Temperature EOFs EOF EOF 2 EOF 3 EOF 4 EOF 5 EOF 6 Pressure level 0 0 N Eigenvalue Variance Sum 0657 89.6% 89.6% 2 654 5.5% 95.% 3 376 3.2% 98.2% 4 25.% 99.3% 5 43 0.4% 99.6% 6 20 0.2% 99.8% 0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 From the prior covariance S a 20 Atmospheric and Environmental Research (AER). All rights reserved. 7
Averaging kernel method (after Rodgers) ˆx = Ax + ( I! A) x a + G y " y. Here ˆx is the retrieval, x is the true state vector, and x a is the prior. A is the averaging kernel.! y includes measurement error and forward model error which is unbiased with covariance matrix, S!. G y is the sensitivity of retrieval to the radiances Here S a is the prior covariance and the retrieval error covariance is determined from K =!F G y =!ˆx!y = ŜK T S # " = S a K T (KS a K T + S " ) # Ŝ! = K T S "! K + S a! And is the Jacobian of the forward problem, F, evaluated at the solution.!x Finally ( ) A = G y K = Ŝ K T S! " K ˆx 20 Atmospheric and Environmental Research (AER). All rights reserved. 8
Interface: required data for assimilation Inputs to the retrieval Prior mean and covariance and radiance error covariance Outputs from the retrieval Retrieval and Jacobian x a, S a, and S! OR, if the channel set is large Prior mean Retrieval, posterior covariance, and averaging kernel ˆx, ˆx and K x a Ŝ, and A 20 Atmospheric and Environmental Research (AER). All rights reserved. 9
Jacobian of the forward problem F 0 Jacobian #374, P 0 =0.hPa K =!F!x 0. 0.08 Pressure (hpa) 0 0 0.06 0.04 0.02 0 0 200 300 400 500 600 700 800 900 000 Wavenumber (cm ) 20 Atmospheric and Environmental Research (AER). All rights reserved. 0
Posterior correlations from Ŝ 0 Correlations for Ŝ #374 0.8 0.6 0.4 Pressure (hpa) 0 0 0.2 0 0.2 0 0.8 0.8 0.4 0.2 0.6 0.2 0.6 0.8 0.6 0.872 0.6 0.872 0.8 0.2 0.2 0.4 0.4 0.6 0.4 0.8 0.4 0.8 0 0 0.8 0.8 0.8 0 0 0 0 Pressure (hpa) 20 Atmospheric and Environmental Research (AER). All rights reserved.
Data assimilation interface Within the data assimilation, define the new observation as and the new observation operator by We will now be comparing observed and simulated quantities with the same degree of smoothing. Now the observation increments are unbiased if the ŷ A = ˆx! ( I! A) x a, y A = Ax. ŷ A! y A = G y " y,! y S m = G y S! G y T = AŜ are unbiased, and have covariance 20 Atmospheric and Environmental Research (AER). All rights reserved. 2
Averaging kernel, A Rows of A R (eq ) for # 374, P 0 =0.hPa Pressure (hpa) 0 0 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 20 2 0 0. 0.05 0 0.05 0. 20 Atmospheric and Environmental Research (AER). All rights reserved. 3
Old vs. New observations ˆx ŷ A = ˆx! I! A Retrievals, ˆx ŷ a eq(5) ( ) x a Pressure (hpa) 0 0 Pressure (hpa) 0 0 0 0 50 60 70 80 90 200 20 220 230 240 250 Temperature (K) 00 20 40 60 80 200 220 240 260 20 Atmospheric and Environmental Research (AER). All rights reserved. 4
Forecast as prior (interactive retrievals) Since the observation increment ŷ A! y A = G y " y, does not depend on the prior, we may use the ensemble background mean as and the ensemble sample covariance as S a. x a But G y does depend on S a, the prior covariance. 20 Atmospheric and Environmental Research (AER). All rights reserved. 5
Rotated averaging kernel method S m!/2 If we scale by we effectively rotate to a space where the observation errors are unbiased, uncorrelated, and have unit variance. The new observation is ŷ A = S m!/2 and the new observation operator by ( ˆx! ( I! A) x a ), y A = S m!/2 Ax = A R x, S m is covariance of the errors of the new observakons where A R = S m!/2 A. R is for rotated 20 Atmospheric and Environmental Research (AER). All rights reserved. 6
Rotated averaging kernel A R Rows of A R (eq ) for # 374, P 0 =0.hPa Rows of A R (eq ) for # 374, P 0 =0.hPa A Pressure (hpa) 0 0 A R = S m!/2 A 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 0.05 0 0.05 0 0 0.05 0. 0.5 0.2 0.25 0.3 0.35 0.4 0.45 20 Atmospheric and Environmental Research (AER). All rights reserved. 7
New vs. rotated observations!/2 ŷ A = ˆx! I! A ŷ A = S m ( ) x a ŷ a eq(5) ( ˆx! I! A ) ŷ a2 eq(9) ( ) x a Pressure (hpa) 0 0 Pressure (hpa) 0 0 0 0 00 20 40 60 80 200 220 240 260 00 200 300 400 500 600 20 Atmospheric and Environmental Research (AER). All rights reserved. 8
EOF analysis of the retrieval method Subtract from to get a version in terms of anomalies, We now project this version into EOF coefficient space. For justification, the filtering step is normally done within the retrieval algorithm initially and at each step of iteration. If we define then we have with x = Ax + I! A ( ) x ˆx = Ax + ( I! A) x a + G y " y x ˆ! = A x! + ( I " A) x a! + G y # y.!a = E T AE ˆ! =! A! + ( I " A! )! a + #!! " = E T G y! y ˆ! = E T ( ˆx " x )! a = E T ( x a " x ) 20 Atmospheric and Environmental Research (AER). All rights reserved. 9
Rotated EOF averaging kernel method! a S! = E T S m E S! "/2 has covariance given by. We scale by to rotate to a space where the observation errors are unbiased, uncorrelated, and have unit variance. The new observation as ŷ A = S ( "/2! ˆ! " ( I "!A )! ) a The new observation operator by y A =! A R where we have defined the rotated EOF-space AK, ( x! x ), "/2!A R = S!!AE T. Or work in EOF coefficient space:!a R = S! "/2!A y A =! A R! 20 Atmospheric and Environmental Research (AER). All rights reserved. 20
EOF projected (rotated) averaging kernel ÃE T for #374, P 0 =0.hPa Rows of à R (eq 28) for #374, P 0 =0.hPa!AE T 2 3 4 5 6!A R = S! "/2!AE T 2 3 4 5 6 0 0 Pressure (hpa) 0 0 0 0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.80.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 20 Atmospheric and Environmental Research (AER). All rights reserved. 2
Old vs. EOF-projected observations ˆx Retrievals, ˆx ŷa = S ( "/2! ˆ! " ( I "!A )! ) a EOF ŷ a3 eq(26) 2 EOF 2 Pressure (hpa) 0 0 EOF # 3 4 EOF 3 EOF 4 5 EOF 5 0 50 60 70 80 90 200 20 220 230 240 250 Temperature (K) 6 EOF 6 20 00 80 60 40 20 0 20 40 20 Atmospheric and Environmental Research (AER). All rights reserved. 22
Summary Convert standard retrievals into "observations" with expected errors that should be zero mean, uncorrelated, and unit variance, and independent of the background or prior. Define a corresponding obs-function (or H-operator) that is a weighted sum of the temperatures on the radiative transfer model vertical grid. No changes to the assimilation method are needed, except to interpolate to the radiative transfer model vertical grid and to calculate the weighted sum. Projecting onto EOFs used by the retrieval can reduce the number of observations Based on ideas in Rodgers' book, "Inverse Methods for Atmospheric Sounding: Theory and Practice. Next step: Try it out in our Mars LETKF data assimilation system. For details visit h<p://arxiv.org/abs/009.56 20 Atmospheric and Environmental Research (AER). All rights reserved. 23
Vertical interpolation We want to determine the EOF coefficients that best fit the temperatures from the dynamical model. Define an interpolation V that transforms x from the retrieval grid to the dynamical model grid. Then the best fit solution is E T V T VE! = E T V T (x b " V x) 20 Atmospheric and Environmental Research (AER). All rights reserved. 24