Laboratory in Oceanography: Data and Methods

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Laboratory in Oceanography Data and Methods
Gridding and Interpolation Methods
MAR599, Spring 2009 Anne-Marie E.G. Brunner-Suzuki
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The problem math vs. reality

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

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Data gaps

• Equipment failure
• Ship time is expensive
• Weather conditions (ship, satellite)
• Editing out errors
• Use of historical data, which often had different
  goals (analysing the mean state of the ocean)
• Geographic distribution (moorings, buoys, ships)
  of monitoring stations is usually not uniformely
  spaced
• Resolving smaller subject dynamics

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Interpolation (Theory)

• Linear Interpolation
• Fit a straight line between two data points
  choosing interpolated values at the appropriate
  positions along that line.

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Interpolation (Theory)
Linear Interpolation straight line
first-order polynomial
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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 a1x1 a2x2 amxm
• 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.

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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

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• 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.

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Polynomial Interpolation
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(cubic) Spline Interpolation

• Piecewise polynomial, avoids the Runge
  phenomenon.
• Is applied to a series of segments of the data
  record rather the entire series
• Spline functions can overcome some
  discontinuities or sharp corners, where the
  segments join.
• Good for fitting non-analytical distributions
• No advantage to polynomial interpolation when
  applied to either well-behaved functions or dense
  data

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(cubic) Spline Interpolation

• Approximate the interpolation function y(x) over
  the interval a,b by deviding a,b into
  subregions with continuity at the joints
• a u0 lt u1 lt u2 lt uN b
• For each subinterval y(x) is a polynomial of
  order N or smaller.
• At each joint y(x) and it's N-1 derivatives are
  continuous.
• N3 cubic spline, most common.

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(cubic) Spline Interpolation

• Consider data (xi,yi) i1...N, y'(x), y''(x)
  exist for all x and y'''(x) is constant for all
  x.
• At all joints
• the spline function fi(xi) is continuous
• It's slope y(x) is continuous
• It's curvature y(x) is continuous
• Because y'''(x) const gt y''(x) is also linear.

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(cubic) Spline Interpolation
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(cubic) Spline Interpolation
As a note It can be useful to transform the data
before a spline fit, taking the log of it.
Perform the interpolation, and then convert back
by exponentiation to the original space. This can
ensure positivity.
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FFT Interpolation

• The original vector x is transformed to the
  Fourier domain using fft and then transformed
  back with more points.
• Matlab transforms to the Fourier domain, there
  matlab pads the spectrum with zeros, and then
  transforms the function back with more points.

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More Matlab functions
Interp2 2D interpolation Interp3 3D
interpolation Spline toolbox (not always
available) for other splines but cubic. Delauny
triangulation by finding the natural neighbors.
Voronoi polygon Trimesh mesh with
triangles Dsearch Tsearch
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Gridding

• In many cases in oceanography, we do not have
  evenly spaced observations. We need to grid our
  unevenly spaced data by determining some set of
  evenly spaced estimates that approximate the
  observations.
• Imagesc, pcolor, surf all need equally spaced
  data.

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• Fratantoni Pickart, 2007

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Gridding oceanographic problems

• By plotting 5deg squares spatial coverage
  increases towards lower latitudes
• A mix of historical data and different
  instruments (XBT vs MBT)
• Seasonality in data coverage (winter vs. summer)
• Historical Observations are often along meridians
  or parallel to longitudes
• The main goal was to find the mean state of the
  ocean
• Changes in instrument calibration
• friday objective analysis

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References

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

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Laboratory in Oceanography Data and Methods
Optimal Interpolation
MAR599, Spring 2009 Anne-Marie E.G. Brunner-Suzuki
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Optimal Interpolation

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

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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.

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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

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Nudging or Newtonian Damping

• The model is forced over several time steps
  towards the observation
• Equ. of Motion(Xmodel)- (Xmodel-Xobs)/Tdamp

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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.

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Lets repeat Covariance
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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

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Lets 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(?)

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

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• How to estimate the true value
• From the previous slide

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• 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

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How did we derive this?

• a are some weights still to be determined

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• The error variance of the estimation
• If we minimize this error variance we get the
  previous equation

gt or to 0
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• Once we know Ars and Cxr
• We can determine the estimate of the true value
• Lets assume there are M grid locations x and N
  data locations r

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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!)