1. Small-scale variability and observational error estimates for gridded analyses of SST data Alexey Kaplan Lamont-Doherty Earth Observatory of Columbia University

2. Optimal Interpolation Optimal Interpolation T=T B +e B HT=T o +e o = = = 0 = = = 0 =C =C =R  obs error covariance =R  obs error covariance Solution minimizes the cost function Solution minimizes the cost function S[T]=(HT-T o ) T R -1 (HT-T o )+(T-T B ) T C -1 (T-T B ) S[T]=(HT-T o ) T R -1 (HT-T o )+(T-T B ) T C -1 (T-T B ) T=(H T R -1 H+C -1 ) -1 (H T R -1 T o +C -1 T B ) T=(H T R -1 H+C -1 ) -1 (H T R -1 T o +C -1 T B )

3. Optimal Interpolation: another take Optimal Interpolation: another take T= T B + CH T (HCH T +R) -1 (HT o -T B ) P= C - CH T (HCH T +R) -1 HC Multiscale approach Multiscale approach C = C’ + C’’ C = C’ + C’’ T = T’ + T’’ T = T’ + T’’ P = P’ + P’’ P = P’ + P’’

4. Truncation error autocovariance patterns: Eastern Equatorial Pacific at 110W Western Equatorial Pacific at 140E

6. A.Harris & J.Mittaz, A modeling study of retrieval biases, a poster @ Ocean Sciences Meeting, Orlando, 2008

8. * N obs * * * * * * F(x,y) [or F(x,y,t)] Error variance for the mean of N observ is   2 /N What is the error in the binned obs mean (as estimates of the “true” bin area average)?

9. High spatial and temporal resolution of satellite data can help pinpoint natural SST variability on small scales (below 1 deg) and short terms (within 1 month). A few weeks of background processing of 20 years of daily 4km maps of Pathfinder SST gave us the SST variability inside 1x1 monthly boxes estimated. [http://rainbow.ldeo.columbia.edu/~alexeyk/Satellite_SST.html]

10. Measurement error (or very small-scale variability) has to be taken into account

11. Combining the two estimates to obtain  :

12. Does left look like right?

13. Conclusions: 1.High-resolution data satellite data can be used for estimating data covariance at different scales and then be used in efficient multiscale optimal analysis procedures. 2. Data covariance is spatially nonstationary. 3. We can use variability estimates from satellite data to model data error. This has been verified for MODIS-ICOADS SST comparison. Supported by NASA MODIS Science Team and NOAA CCDD and JCSDA grants