**1. Introduction**

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272 Atmospheric Model Applications

Smith, W.L.; Howell, H.B.; Woolf, H.M. (1979). The use of interferometric radiance measurements for the sounding the atmosphere. *Appl. Opt.*, 36, 566-575.

> Documenting mid-tropospheric global-scale circulation is important to climate modelers. Models are applied to capture the most basic statistics of the flow (e.g., annual and season means and variability). As model physics improve, the goal is to extend the accuracy of the models to shorter, more societally relevant time scales (e.g., monthly average flow). Within a month or season, recurrent modes are present. Comparing such observed flows to modeled counterparts can provide an arbiter of success. There is a long history is isolating such patterns in meteorology using teleconnections (correlation patterns). Some of the initial work by Namias (1980) was used to create atlases of teleconnections at every gridpoint. This approach has the merit of completeness; however, the redundancy in patterns from adjacent gridpoints is inefficient, given present day computational power. An extension of the Namias approach to selected base points provided extensive documentation of Northern Hemisphere winter teleconnection patterns of sea level pressure and 500 mb height (Wallace & Gutzler, 1981).

> A second approach to identifying these variability modes uses eigenvectors to filter the correlation structures into recurrent patterns that are localized. To document the characteristic flow patterns and their time dependent statistics, an objective methodology is applied to portray the localization of the centers of action in each mode. Often, rotated principal component analysis (RPCA) is employed to decompose the aforementioned correlation structure of the flow to obtain information on the morphology of the flow patterns, their associated time behavior and the variability of the total flow associated with each pattern. The eigenvectors are scaled to create principal components that are linearly transformed or rotated to exploit the local structure within the domain, identifying physically meaningful circulation patterns.

> Barnston & Livezey (1987; hereafter referred to as BL87) present an extensive catalogue of mid-tropospheric patterns using this approach. Their work is somewhat limited by use of a specific rotation (Varimax) that enforces an orthogonal transformation matrix from the principal components to their rotated counterparts. Additionally, they did not test rigorously the number of eigenmodes, selecting ten modes for each analysis. Selecting too few eigenmodes can result in multiple patterns being merged on each eigenvector retained. Moreover, if too many patterns are selected, the circulation modes can be fragmented.

Identification of Intraseasonal Modes of Variability Using Rotated Principal Components 275

analysis (Barnes, 1964) to place the data on a Fibonacci grid (Swinbank & Purser, 2006) that, by definition, has equal grid spacing. Such a grid will not result in artificial inflation of the correlations between the 2303 gridpoints (Fig. 1b). To ensure no data extrapolation at lower latitudes, the spacing associated with the Fibonacci grid was kept identical to that at the equator. Interpolation error of the Barnes analysis was calculated at less than 1%. The Fibonacci grid defines the domain and is used in the computation of the correlation matrix.

Variation of geopotential height is a function of latitude. To avoid biasing the analyses and to permit smaller, but equally important, variation in the southern regions of the domain to be represented equally, a correlation matrix (**R**), rather than variance-covariance matrix, was computed. Using the correlation matrix standardizes the data to a mean of zero and standard deviation of one. Hence, all the subsequent analyses are standardized anomalies (**Z**) from the mean. The correlation matrix was formed among the grid points by summing over the 63 year sample for January or July, providing a representative month in the cold or warm season, respectively. The correlation matrix for January (July) was decomposed into a square matrix of eigenvectors (**V**) and associated diagonal matrix of eigenvalues (**D**), given

 **R** = **VDV**T (1) The rank of the eigenvector matrix is equal to the smaller of the number of gridpoints or number of observations minus 1. Because there were 63 observations, only 62 eigenvalues were nonzero and 62 eigenvectors were extracted. The goal of this stage of the analysis is to create a set of basis vectors that compress the original variability in **R** into a new reference frame. It is possible to plot the elements of each vector (**V**) on spatial maps; however, the patterns in **V** do not result in any localization of the spatial variance, nor do they represent well the variability in **R** (Richman, 1986). The eigenvectors were scaled by the square root of the corresponding eigenvalue to create principal component loadings (**A**). Doing so permits

 **Z** = **FA**T (2) where the vectors in **F** represent the new set of basis functions, known as principal component scores and **A** is the matrix of weights that relates the original standardized data (**Z**) to **F**. The vectors in **A** contain elements that are regression coefficients between **Z** and **F**. Many of the 62 dimensions represent small-scale signals (sub-planetary scale) that have variance properties indistinguishable from noise, associated with very small eigenvalues. We truncate the number of principal components to represent only that variance associated with planetary scale wavetrains. To accomplish this goal, a two-step process is applied. First, the magnitudes of the eigenvalues are examined and those with relatively large eigenvalues are retained to yield a subset of *l* principal component loading vectors. The value *l* is selected by implementing the scree test (Wilks, 2011) to provide a visual estimate of the approximate number of non-degenerate eigenvectors to retain. At this stage, a number of roots, *l,* is selected to be liberal, intentionally representing more than the ideal number of roots, *k*, associated with the large-scale signal. To assess the coherent signal, the vectors of **A** are linearly transformed to a new set of vectors, **B**, known as rotated principal

**2.2 RPCA methodology** 

by the equation

the data to be expressed as

In this work, we extend the eigenvector-based approach by relaxing the orthogonality constraint and test each analysis for the optimal number of eigenvectors to retain and rotate. While BL87 used cutting-edge analyses for the 1980's, the limited availability of data (35 years) and computational power (limiting the number of gridpoints to 358) are suboptimal by current standards. Innovation in the analysis procedure in recent years combined with newer data sets, the availability of 63 years of data and much denser grids to document the climate system are strong motivators to re-examine Northern Hemisphere geopotential modes of variability.
