Laboratory in Oceanography: Data and Methods

2. Most analysis are designed for long and densely sampled series with equally space measurements in time or space.

6. Polynomial Interpolation To interpolate between more than two points simultaneously. Through three points we can find a unique polynomial of order ? Through four points of order ? Methods to look for are Vandermonde, Lagrange and Newton. f(x) =a0 + a1x^1 + a2x^2 + � + amx^m All coefficients a influence all of x. m needs to be determined by trial and error. Check by comparing the residuals. It oscillates between the data.

7. Vandermonde Matrix p(x) = 3.2 x7 - 4.1 x4 + 9.2 x2 + 1.2 is of order 7. Suppose we have 3 points (2, 5), (3, 6), (7, 4) and we want to fit a quadratic polynomial through these points. The general form is p(x) = c1 x2 + c2 x + c3. Thus, if we were to simply evaluate p(x) at these three points, we get three equations: p(2) = c1 4 + c2 2 + c3 = 5p(3) = c1 9 + c2 3 + c3 = 6p(7) = c1 49 + c2 7 + c3 = 4

8. This, however, is a system of equations. To solve: Writing down the general polynomial of degree n - 1, Evaluating the polynomial at the points x1, ..., xn, and Solving the resulting system of linear equations. Rather than performing all of these operations, simply write down the problem in the form Vc = y where y is the vector of y values, c is the vector of coefficients (�x�), and V is the Vandermonde matrix. See matlab example.

9. Polynomial Interpolation

10. Dense means, that the number of data points in a subregion is more than an order of magnitude larger than the number of inflections points in the fifted curve and there are no aprubt changes in the second derivated. Dense means, that the number of data points in a subregion is more than an order of magnitude larger than the number of inflections points in the fifted curve and there are no aprubt changes in the second derivated.

19. Gridding � oceanographic problems Dense means, that the number of data points in a subregion is more than an order of magnitude larger than the number of inflections points in the fifted curve and there are no aprubt changes in the second derivated. Dense means, that the number of data points in a subregion is more than an order of magnitude larger than the number of inflections points in the fifted curve and there are no aprubt changes in the second derivated.

20. References Data Analysis Methods in Physical Oceanography by W.J. Emery and R.E. Thomson, 1993.

23. Optimal Interpolation Terminology: Optimal Interpolation, Objective mapping, Objective analysis, BLUE (Best Linear Unbiased Estimator) or Gauss-Markov smoothing.

24. Optimal Interpolation Models (approximate dynamics) are imperfect. They are approximations to the truth. Possible errors are: initial conditions, imperfect parameterization, inaccurate forcing. Observations (state variables) are imperfect as well. Errors from instruments, statistical errors, measurement errors. The OI technique can be formulated to simultaneously interpolate different data types (e.g. winds and geopotential heights) provided a linear relationship between the model and data (e,g, as in the case of geostrophic winds (currents) computed from geopotential (sea surface) heights). The OI technique can be formulated to simultaneously interpolate different data types (e.g. winds and geopotential heights) provided a linear relationship between the model and data (e,g, as in the case of geostrophic winds (currents) computed from geopotential (sea surface) heights).

25. One step back � direct insertion Model predictions are replaced with observations available. Assumption: Perfect observations, imperfect model. Model dynamics spread information to nearby gridpoints. Blending uses a weighted average Very Simple ? Leads to dynamical imbalances, creates large noise ? Very Simple ? Leads to dynamical imbalances, creates large noise ?

26. �Nudging� or Newtonian Damping The model is forced over several time steps towards the observation: Equ. of Motion(Xmodel)=- (Xmodel-Xobs)/Tdamp Does not respect smoothness of the fields, or governing physics. ? Simple ? Does not respect smoothness of the fields, or governing physics. ? Simple ?

27. Next step: OI Before: model adjustment only at grid point of observation Now: all points within the de-correlation distance of the observation. OI estimates the fields at an arbitrary location through a linear combination of the available data. Weights are chosen, so that the expected error of the estimate in at a minimum and the estimate itself is unbiased The natural covariance length and time scales of the data and true field enter into the computation of the linear weights.

28. Let�s repeat: Covariance

29. Optimal Interpolation Assumptions: statistics are stationary, homogenous and isotropic For each model variable, only a few observations are important The error covariance is empirically derived and held constant over time If cov=inf, it leads to the direct insertion Fairly Simple, less computational effort than KF ? Spurious noise ? If cov=inf, it leads to the direct insertion Fairly Simple, less computational effort than KF ? Spurious noise ?

30. Let�s go through the math r,s: where the observations are made x: where to interpolate to ?: is the distance from x. T: is the true value, or target value covariance: is represented by a function F(?)

31. The observations are: The measurement error and the observed value is not correlated: Errors at two points are not correlated E is the variance.

32. How to estimate the true value: From the previous slide: F is again the defined covariance function A is the covariance between two data points F is again the defined covariance function A is the covariance between two data points

33. Ars and Cxr are constant for given observation points! The error in the estimation is: it can be used to construct probable error maps in the estimation (derivation follows) Cxx is the natural variation without data present The second term shows data influence Thetax is a linear estimator! Only location r, cov function F and noise level E are needed to be known! Inversion of A can be complicated, if there are lots of observations.Thetax is a linear estimator! Only location r, cov function F and noise level E are needed to be known! Inversion of A can be complicated, if there are lots of observations.

34. How did we derive this? a are some weights still to be determined:

35. The error variance of the estimation: If we minimize this error variance we get the previous equation:

36. Once we know Ars and Cxr We can determine the estimate of the true value: Let�s assume there are M grid locations x and N data locations r:

37. References: Bretherton, 1976: A technique for objective analysis and design of oceanographic experiments applied to MODE-73* Data analysis in physical Oceanography by Emery and Thompson, 2nd edition (watch our for errors in their derivation!)