Part 6: Objective Analysis

Optimum Interpretation

Composite Analysis

Purpose of Objective Analysis

Observations are often available only at a few stations that are unevenly spaced in the domain of interests.

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.

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.

When to Do the Gridding?

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.

The regridding can be in space, in time, or both.

Polynomial Fitting Method

We can use a polynomial function to fit the observational data and then use this function to generate data on regular grids.

In meteorological applications, up to thrid order polynomial have been used. But usually, quadratic equation is sufficient for most purpose:

The problem now it to determine the values of the coefficients: a₁, a₂, …., and a₆.

If we have 6 observations of h, then we can determine the 6 coefficients in the quadratic polynomial function.

We can then use the polynomial function for gridding purpose.

Problems With the Polynomial Method

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.

Problems With the Polynomial Method

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.

The problem gets worse as the order of the polynomial is increased. The method is nearly useless where the data are sparse.

Optimum Interpretation

The optimum interpretation (OI) is a linear interpretation which requires its root-mean square error to be minimum.

The discussion of this method focuses on the “deviations from a normal state”:

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.

Then we can define the deviation of f from its normal state as f’:

Interpretation

Now, we want to approximate the value of f at a grid point (f_g), in terms of a linear combination of the values of f at neighboring station points (f_i):

We want to determine the coefficients p_i by minimizing the mean squared error:

Determine the Weightings

We can normalized the error E:

where

Now we can determine the weightings (p_i) by asking:

We then solve the N linear equations for the N p’s.

It can be shown that the error obtained after fitting the coefficients is:

What Do We Need to Get P_i?

We need to know r_ij and r_gi in order to solve the N linear equation for the N Ps.

è But we don’t data at grid points (can not calculated r_gi !).

It is typical to assume that correlations between points depend only on the distance between them and not on location or direction.

So we calculate the r_ij from station data and obtain r_gi from r_ij based on the distance between the grid points and the stations.

Composite Analysis

Compositing analysis (also called superposed epoch analysis) is to sort time series into different categories (or phases) and to compare means in these categories.

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.

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.

The basic idea of the compositing analysis is that the averaging process will remove noise and keep the signals of interest.

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.

Steps in Compositing Analysis

Select the basis for compositing and define the categories

     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.

Compute the means and statistics for each category

      We calculate the mean SST, wind stress, or heat flux for the onset, growing, and mature phases of the ENSO cycle.

Organize and display the results

Validate the results

      Validation of the results can be achieved in many ways. Statistical significance tests are only one of these.

An Example -the MJO Cycle

Step 1: Define Index

Time Series of the Indices

Step 2: Compute the Means

Step 3: Display the MJO Life Cycle


	Observations are often available only at a few stations that are unevenly spaced in the domain of interests.
	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.
	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.


	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.
	The regridding can be in space, in time, or both.


	We can use a polynomial function to fit the observational data and then use this function to generate data on regular grids.
	In meteorological applications, up to thrid order polynomial have been used. But usually, quadratic equation is sufficient for most purpose:

	The problem now it to determine the values of the coefficients: a₁, a₂, …., and a₆.
	If we have 6 observations of h, then we can determine the 6 coefficients in the quadratic polynomial function.
	We can then use the polynomial function for gridding purpose.


	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.


	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.
	The problem gets worse as the order of the polynomial is increased. The method is nearly useless where the data are sparse.


	The optimum interpretation (OI) is a linear interpretation which requires its root-mean square error to be minimum.
	The discussion of this method focuses on the “deviations from a normal state”:
	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.
	Then we can define the deviation of f from its normal state as f’:


	Now, we want to approximate the value of f at a grid point (f_g), in terms of a linear combination of the values of f at neighboring station points (f_i):


	We want to determine the coefficients p_i by minimizing the mean squared error:


	We can normalized the error E:

	where

	Now we can determine the weightings (p_i) by asking:



	We then solve the N linear equations for the N p’s.
	It can be shown that the error obtained after fitting the coefficients is:


	We need to know r_ij and r_gi in order to solve the N linear equation for the N Ps.
	è But we don’t data at grid points (can not calculated r_gi !).
	It is typical to assume that correlations between points depend only on the distance between them and not on location or direction.
	So we calculate the r_ij from station data and obtain r_gi from r_ij based on the distance between the grid points and the stations.


	Compositing analysis (also called superposed epoch analysis) is to sort time series into different categories (or phases) and to compare means in these categories.
	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.
	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.
	The basic idea of the compositing analysis is that the averaging process will remove noise and keep the signals of interest.
	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.


	Select the basis for compositing and define the categories
	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.
	Compute the means and statistics for each category
	We calculate the mean SST, wind stress, or heat flux for the onset, growing, and mature phases of the ENSO cycle.
	Organize and display the results
	Validate the results
	Validation of the results can be achieved in many ways. Statistical significance tests are only one of these.