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Optimum
Interpretation |
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Composite Analysis |
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Observations are often available only at a few
stations that are unevenly spaced in the domain of interests. |
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In order to compute derivatives of the field
variables, as would be required in diagnostic studies or in the
initialization of a numerical model, or simply to perform a sensible
averaging process, one often requires values of the variables at points on
a regular grid. |
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Assigning the best values at the grid points,
given data at arbitrarily located stations and perhaps a first guess at
regular grid points, is what has traditionally been called objective
analysis. |
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The methods described are applicable to any
problem where the data you are given do not fill the domain of interest
fully, and/or where the data must be interpolated to a regular grid. |
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The regridding can be in space, in time, or
both. |
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We can use a polynomial function to fit the
observational data and then use this function to generate data on regular
grids. |
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In meteorological applications, up to thrid
order polynomial have been used. But usually, quadratic equation is
sufficient for most purpose: |
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The problem now it to determine the values of
the coefficients: a1, a2, …., and a6. |
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If we have 6 observations of h, then we can
determine the 6 coefficients in the quadratic polynomial function. |
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We can then use the polynomial function for
gridding purpose. |
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The instability of the polynomial fit is such
that when one key data point is removed, the polynomial fit in that region
may change radically. |
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Polynomial fits are unstable in the sense that
the values the polynomials give at points between the stations vary greatly
for small changes in the data at the station points, and especially so when
data are missing. |
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The problem gets worse as the order of the
polynomial is increased. The method is nearly useless where the data are
sparse. |
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The optimum interpretation (OI) is a linear
interpretation which requires its root-mean square error to be minimum. |
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The discussion of this method focuses on the
“deviations from a normal state”: |
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Let’s say we have a variable f and its normal
state f norm. This normal state can be the climatological value
of f or a first guess of f. |
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Then we can define the deviation of f from its
normal state as f’: |
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Now, we want to approximate the value of f at a
grid point (fg), in terms of a linear combination of the values
of f at neighboring station points (fi): |
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We want to determine the coefficients pi
by minimizing the mean squared error: |
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We can normalized the error E: |
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where |
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Now we can determine the weightings (pi) by asking: |
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We then solve the N linear equations for the N p’s. |
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It can be shown that the error obtained after
fitting the coefficients is: |
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We need to know rij and rgi
in order to solve the N linear equation for the N Ps. |
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don’t data at grid points (can not calculated rgi !). |
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It is typical to assume that correlations
between points depend only on the distance between them and not on location
or direction. |
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So we calculate the rij from station
data and obtain rgi from rij based on the distance
between the grid points and the stations. |
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Compositing analysis (also called superposed
epoch analysis) is to sort time series into different categories (or
phases) and to compare means in these categories. |
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For example, how do Pacific SST evolve before,
during, and after an El Nino event? In this case, there are three
categories (before, during, and after) in the composite analysis. |
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Compositing is useful when you have many
observations of some event and you are looking for responses to that event
that are combined with noise from a lot of other influences. |
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The basic idea of the compositing analysis is
that the averaging process will remove noise and keep the signals of
interest. |
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Often compositing will reveal periodic phenomena
with fixed phase that cannot be extracted from spectral analysis if the
signal is small compared to the noise. |
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Select the basis for compositing and define the
categories |
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The
categories might be related to the phase of some cyclic phenomenon or
forcing, or to time or distance from some event. For example, we can use
NINO3 index as the basis for compositing the ENSO cycle. |
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Compute the means and statistics for each
category |
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We
calculate the mean SST, wind stress, or heat flux for the onset, growing,
and mature phases of the ENSO cycle. |
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Organize and display the results |
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Validate the results |
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Validation
of the results can be achieved in many ways. Statistical significance tests
are only one of these. |
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