1. Oceanography 569Oceanographic Data Analysis Laboratory Kathie Kelly Applied Physics Laboratory 515 Ben Hall IR Bldg class web site: faculty.washington.edu/kellyapl/classes/ocean569_2014/

2. Exercise 8: Optimal Interpolation or Objective Mapping • Why compute an objective map? • to get a gridded map from irregularly spaced data • to get an estimate of the map errors at each point Locations of Argo profiles January - June 2003

3. Exercise 8: Estimate Mean to Get Map Anomalies Procedure: Estimate simple form for mean, here M=a*lat+b Remove mean Map anomalies Add mean back

4. Objective Map: Linear Combination of Nearby Data To simplify the derivation assume data are time series at fixed locations (e.g., moorings) to get an ensemble of measurements. Find optimal weighting for nearby data and then generalize to locations that vary in time. Create estimate at each map grid point as a linear combination of nearby data D (time series are columns) where α are the coefficients to be optimized. Want to find the estimate that minimizes the error relative to the “true” value dT : The minimum least-squares error is “orthogonal” to the estimate

5. Objective Map: Coefficients to Minimize Squared Error Substituting the definition of the estimate and the error Defining data-data covariance matrix (times N) covariance of data with true value (times N) Coefficients can be written explicitly as but are solved in Matlab with matrix-left-divide This formulation is general for any distribution of measurements, given C

6. Exercise 8: Coefficients Based on Data Covariances covariance depends only on space and time lags, so generalize to arbitrary data distributions “Numerator” z reflects how close data are to grid point “Denominator” C reflects how close data are to each other o ++ 2 points close together + o + 2 points widely spaced o + single datum coefficients depend on ratio of proximity of data (z) to map grid point and inversely on redundancy of the data (C)

7. Estimating a Covariance Function Space- and time-lagged autocovariance function C: C(ds,dt) = Var exp(-ds2/Ls –dt/Lt) where Var = Total variance minus noise estimate from products of data anomalies function of Var, Ls and Lt space lag (km) time lag (days)

8. Expected Errors for Maps What is the expected squared error for this estimate? which shows that the expected squared error is the data variance minus the covariance between the estimate and the “true” measurement at the map location The squared error approaches the data variance for grid points “far” from the data and gets small for grid points “near” the data

9. Exercise 8: Objective Map & Error Estimate

10. Exercise 8: Series of Objective Maps

11. Exercise 8: Finding Significant DifferencesWhat is the error for a difference?