## 1. Introduction/background

Propagating mesoscale gravity waves sometimes play a significant role in triggering damaging convection and mesoscale convective systems (Uccellini, 1975; Stobie et al., 1983; Einuadi et al., 1987; Uccellini and Koch, 1987; Koch et al., 1988). Since the gravity wave signal is generally an order of magnitude smaller than the synoptic signal in which it is embedded, many gravity wave studies have concentrated on digital filtering techniques and spectral methods (Koch and Golus, 1988) or numerical simulations (Powers and Reed, 1993; Powers, 1997) to distinguish the underlying gravity wave from the overall environment. This study demonstrates a method to objectively isolate and map the gravity wave signal using an empirical orthogonal function (EOF) analysis. The technique is applied to routinely disseminated 20-min observations taken in the north-central United States during a mesoscale convective event on 30–31 May 1998.

Previous research, by Koch and Golus (Koch and Golus, 1988) and Bosart and Seimon (Bosart and Seimon, 1988) among others, has shown that gravity waves typically have a period between 1 and 5 h. Methods used effectively by Koch and Golus (Koch and Golus, 1988), Powers (Powers, 1997), and others to isolate the period of the gravity wave with minimal aliasing include generating spectra via the fast Fourier transform (FFT) of the raw data, or taking the autospectrum, running the FFT, and then applying a bandpass filter. Inspection of the spectrum can reveal the frequency of the event, and application of an appropriate bandpass filter allows the high- and low-frequency noise to be stripped away. Analysis of the wind-pressure perturbation covariance at gravity wave frequencies can then be undertaken. Unfortunately, this process is often tedious and subjective, and studies of the same event can yield significantly dissimilar results simply due to differences in time series length, choice of bandpass filter, or temporal/spatial density of the input data. For example, Uccellini (Uccellini, 1975) and Stobie et al. (Stobie et al., 1983) chose a 15-min data sampling rate. Koch and Golus (Koch and Golus, 1988) further suggested that the 15-min data (optimally 5-min data as in their case) be gathered over a wide area with station spacing near 20 km to avoid aliasing due to undersampling. The Koch and Golus (Koch and Golus, 1988) data also were divided into two 8-h time series to isolate the two distinct gravity wave pulses during their event. This procedure reduced the effects of the synoptic-scale signal on their datasets and allowed them to study two distinct waves with two different periods.

To demonstrate an EOF filtering technique and its utility in tracking gravity waves with widely available METAR (from the French for aviation routine weather report) observations, this work examines a recent case in the U.S. upper midwest. Section 2 describes the 30–31 May 1998 data used. An analysis detailing the EOF filtering method and its apparent strengths in comparison to more traditional filtering techniques is presented in section 3. Conclusions are outlined in section 4.

## 2. Data

The event studied here is notable as a Derecho mesoscale convective system (MCS) that moved from South Dakota to Lake Ontario in 12 h, causing more than $0.25 billion of damage (NCDC, 1998). The surface low pressure system associated with the MCS (Figure 1) deepened significantly under the influence of the left jet exit region visible over eastern South Dakota and southwestern Minnesota in Figure 2. Observational data analyzed consisted of 35 stations (Figure 3) over a period of 35 h from 1200 UTC 30 May to 2240 UTC 31 May. Stations with automated observing equipment were chosen because the automated sites routinely transmit observations every 20 min. To construct the pressure records, altimeter settings were sampled at every 20-min observation and were then converted from units of inches of mercury to hectopascals (hPa) over the entire domain. All stations had a few missing observations (the average was about 3% missing) and, as a general rule, linear interpolation was used to fill gaps of one to two observations. Stations with three or more consecutive missing data points were not included in the 35 used. Since the time series were long enough to allow the synoptic-scale cyclone to traverse the entire breadth of the data field, an overpowering synoptic signal—representative of the rapid cyclogenesis—dominated the pressure traces of each of the stations.

## 3. Analysis

Due to the overriding synoptic signal in the data (six example pressure traces are shown in Figure 4), a Fourier analysis of the raw pressure data showed very little spectral amplitude in the 1–5-h range expected for gravity waves (not shown). To remove the masking “red” part of the spectrum, the overall mean and trend were removed from each station's time series, and a bandpass Butterworth filter (Guillemin, 1957) was applied to remove the synoptic signal from the pressure data. The Butterworth filter often is used in geophysical analyses as it passes virtually all the signal in the bandpass range with smooth rolloff into the suppressed ranges (Shanks, 1967; Murakami, 1979). The bandpass Butterworth filter employed in this work is virtually the same as is described in Murakami (Murakami, 1979); its response function is plotted in appendix D (Figure D1). Though the Butterworth filter used has an actual bandpass region of approximately 1.9–5.0 h, it is generally referred to as a 1–5-h bandpass filter in this work for simplicity. Application of the bandpass filter to the pressure data (raw values minus the mean and linear trend) clearly highlighted the gravity wave [Menominee, MI (MNM), example; see Figure 5]. The wave cycles shown in Figure 5 span approximately 9–10 observation points, which is roughly 3.0–3.3 h and is consistent with previously documented gravity waves. In a manner similar to Koch and Golus (Koch and Golus, 1988; see their Figure 2), an examination of the FFT spectrum of the filtered MNM data (Figure 6) found one dominant peak at around 3.8 h (0.004 cycles min^{−1}), which is also consistent with the gravity wave periods in their study and others.

In an effort to find a gravity wave isolation method requiring less a priori knowledge of the gravity wave characteristics, we applied an EOF analysis to the raw pressure records to determine the principal components (PCs) of the total variance (see appendix A for more details of the analysis procedure). The EOF procedure produced a spatial field for each principal component describing the locations where that component expressed the most magnitude. Each spatial field has a PC time series that modulates the spatial expression of magnitude in time. The EOF analysis of the pressure dataset yielded four EOFs explaining slightly more than 99% of the variance. Two dominant EOFs were found that describe approximately 97.4% of the variance (Figure 7 and Figure 8), while the other two EOFs contain 0.8% (Figure 9) and 0.6% (Figure 10) of the variance, respectively. The EOF components 1 and 2 apparently represent two orthogonal modes of the synoptic signal. Evidence of the synoptic signal in EOFs 1 and 2 exist in their PC time series, which are approximately 90° out of phase, thus representing a lag-correlated quadrature pair (Kessler, 2001). A plot of the two time series versus one another reveals an elliptically shaped hodograph about the origin (Figure 11). This behavior is indicative of a propagating signal, which is physically related to the eastward movement of the low pressure trough (see lower-left panel of Movie 2) as it underwent rapid cyclogenesis. Typically, EOFs with relatively low variance are thought to be physically unexplainable and are usually not considered (Legler, 1983; Piexoto and Oort, 1992). However, since EOFs 1 and 2 represent the synoptic signal, the physical signal of interest—the gravity wave—is left within EOFs 3 and 4, and the residual variance. To determine if EOFs 3 and 4 or higher-order EOFs actually represent something truly physical or simply just noise, a reconstitution of the EOF data was undertaken to allow easier comparison with surface pressure data.

While the general convention is to display and analyze spatial and temporal EOF components separately, the principal components can be reconstituted into the units of the original data by multiplying the value at each time in the PC time series by the values in the spatial EOF (Mitchum, 1993). Summing all the resultant reconstituted EOFs yields the original data. Such a reconstitution also could allow direct comparison of EOF components with other analyses in the original units: for example, the filtered pressure field versus an EOF component pressure field. Following a suggestion from J. J. O'Brien (J. J. O'Brien, 2000, personal communication), we created an animated movie of the filtered pressure field for comparison with an animated movie of the residual EOFs—the raw pressure field minus reconstituted EOFs 1 and 2 (see appendixes B and C for more details). This essentially exploited the EOF technique as an objective, synoptic (high pass) filter. As demonstrated in the animations of Movie 1, the residual EOF (upper-left panel) and the 1–5-h Butterworth-filtered (upper-right panel) pressure loops are remarkably similar and capture the wave events well. Note that the colors of the shaded contours in the 1–5-h filtered sequence in Movie 1 represent only half the magnitudes they represent in the residual EOF sequence. Despite excellent qualitative comparison in terms of size, shape, orientation, duration, and propagation speed, the wave extremes in the EOF residual loop consistently exceed the filtered loop values. This difference in magnitudes is attributed to the Butterworth filter design that at many points transmits only 50%–80% of the wave amplitude within the pass band (see Figure D1). Loss of passband signal amplitude is a characteristic to some degree in almost all digital filter designs; in this case it suggests the gravity wave amplitudes isolated by the EOF technique may be more representative of the actual physical values.

The animations in Movie 2 allow a comparison of the residual EOF (upper left) with the Butterworth filter loop (upper right) and a 1–5-h Chebyshev filter loop (lower right) where the colors of the shaded contours all represent the same magnitudes. The Chebyshev filter characteristics are described, and its response function is plotted in appendix D (Figure D2). The primary reason the Chebyshev filter loop differs from the Butterworth is not due to differences in filter design, but largely because the linear trend in each of the station time series was not removed prior to filtering. Thus the Chebyshev loop is contaminated with the synoptic trend. This is most apparent at points in the center and eastern parts of the analyzed region, where the surface pressure loop (Movie 2, lower-left panel) shows an almost monotonic pressure decrease from high to lower pressure. The Chebyshev extremes (both maxima and minima) in this region are generally lower than the Butterworth extremes because the Butterworth data were corrected for the downward pressure change. Since in this case the synpotic signal accounts for most of the overall variance and the overall trend in each station's time series, its removal via EOFs 1 and 2 eliminated the need to precondition each time series by detrending prior to application of the EOF filtering method. It is likely the synoptic signal is more cleanly removed using the orthogonal modes of EOFs 1 and 2 rather than crudely adjusting each time series with a linearized trend value.

In the comparison sequences of the residual EOF and the Butterworth/Chebyshev filters, the EOF analysis or filtering was performed on each of the 35 station time series prior to gridding and contouring. Because of the relatively uniform spread of the stations used in this study, gridding the time series data first and then applying EOF analysis and/or filtering on all the grid points (713 in this case) was not necessary. To graphically show the validity of such an assumption, a “first guess grid” was created and an EOF analysis was done on all 713 (23 by 31; see appendix B) grid points. This EOF analysis once again yielded only four EOFs explaining more than 99% of the variance. As with the 35-station analysis there were two EOF pairs: one accounting for 98% of the variance (72% and 26%) and another much smaller pair explaining 1.1% (0.8% and 0.3%). Reconstituting and contouring the values of the smaller two EOFs plus the residual variance over all the time periods of the study produced the “residual EOF with 1st guess grid” animation panel seen in the lower left of Movie 1. The first guess grid animation is little changed from the animation above it where no initial gridding was performed.

Inspection of the first five time steps in both Movie 1 and Movie 2 reveals a significant difference between the animations of the residual EOF and the Butterworth/Chebyshev filtered loops. Despite both filters being low-order digital filters (the Butterworth used here is first order, the Chebyshev is second order), both clearly require several time steps to converge on a stable filtered solution. This instability in the first five time steps of the Butterworth-filtered data is also evident in the Menominee data plotted in Figure 5. Conversely, the reconstituted residual EOF animation—stripped cleanly of the synoptic signal in EOFs 1 and 2—provides a stable solution beginning with the first time step.

Both the residual EOF and 1–5-h Butterworth filter animations isolate the primary convection-inducing gravity wave as a leading trough–trailing ridge couplet in frames 46–67 (*t* = 46 to *t* = 67, approximately 0300-1000 UTC 31 May). For comparison, the Butterworth-filtered (1–5-h) and the residual EOF pressure fields at observation time 50 (0420 UTC 31 May) are shown in Figure 12 and Figure 13. Also displayed, in Figure 14, is a next-generation Doppler radar (NEXRAD) base reflectivity mosaic of the same region at 0420 UTC 31 May. Note that the leading edge of the convection in Figure 14 falls mainly to the west and north of the leading trough of the gravity wave couplet in Figure 12 and Figure 13. This placement of the convection coincides with the location of maximum upward vertical velocity in the conceptualized gravity wave structure described by Bosart and Sanders (Bosart and Sanders, 1986). The similarities of Figure 12 and Figure 13 to one another, and the consistency to the orientation of the convection in Figure 14, leads the authors to conclude that the residual EOF is indeed capturing a gravity wave event. Furthermore, the authors believe the gravity wave in this case is best defined by the reconstitution of all nonsynoptic EOF modes—that is, EOFs 3 and 4 plus the residual variance. This viewpoint is derived from the greater similarity between the residual EOF animation to the Butterworth-filtered animation than the similarity between the EOFs 3 and 4 only animation (Movie 1, lower-right panel) and the Butterworth sequence.

## 4. Conclusions

This study has demonstrated that a gravity wave pressure signal can be efficiently isolated using EOF filtering techniques. The resulting signal produced by the EOF filtering was in this case remarkably similar to the processed signal obtained via more traditional means with a digital bandpass filter. The EOF filtering method appears to offer at least three advantages over digital bandpass filtering: 1) more complete capture of the gravity wave magnitude; 2) less preconditioning of the initial time series in the manner of trend removal; and 3) stable, filtered solutions at the first time step.

This work also documented an EOF decomposition where even the smallest components (the gravity wave composed less than 3% of the total variance in this case) can be considered physically significant. The robustness of the technique is that only the clearly discernable synoptic EOF components need to be removed to expose underlying signals such as atmospheric gravity oscillations. Evidence presented here suggests future studies could distinguish between synoptic EOF and residual EOF components by simply identifying where the EOF variance drops by an order of magnitude or more.

This research additionally showed that mesoscale gravity wave signals could possibly be identified and tracked in near–real time using routinely disseminated 20-min interval observation data and existing automated surface observing site (ASOS) density. Such availability of data could allow severe weather forecasters to use either EOF filtering or bandpass filtering to monitor convection-induced or possibly convection-triggering gravity waves.

The EOF filtering technique clearly has potential uses in other areas of mesoscale dynamics such as distilling sea-breeze or mountain–valley wind circulations from the overall synoptic wind field. Furthermore, this filtering method could be used with other timescales as well—such as identifying Madden–Julian oscillations in long–term climatic time series (see Kessler, 2001). Further research should include demonstrating the repeatability of this work on other gravity wave events with different observation time intervals, data network densities, and digital filter designs.

## Acknowledgments

This research was a team project completed in partial fulfillment of the Time Series Analysis course requirements in the Department of Meteorology at The Florida State University. We wish to thank Dr. J. J. O'Brien for his many suggestions and guidance on this project. We would also like to thank Dr. K.-Y. Kim for his comments and use of his EOF analysis algorithms (http://cyclo.met.fsu.edu/software.html), and Dr. T. N. Krishnamurti for providing the code for the Butterworth filter. Dr. H. E. Fuelberg provided a thorough review of an early version of the manuscript and greatly improved its clarity. The authors are also grateful to the reviewers, who aided in the correction of several errors and whose comments enhanced the quality of the paper. Data for this project were obtained from the COMET Case Study Library. The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. government.

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## Appendix A: EOF Analysis Procedure

- Make a 35 column by 105 row station versus time matrix of raw pressure data.
- Input station versus time matrix into EOF algorithms to obtain spatial and temporal components.
- Put each spatial EOF onto a 27 by 26 grid using MATLAB.
- Contour plot each EOF using Grid Analysis and Display System (GrADS), plot principal component time series (PCTs) with spreadsheet software.

## Appendix B: Movie Loops

- Make five additional 35 by 105 station versus time matrices: one for the Butterworth-filtered pressure, one for the Chebyshev-filtered pressure, one for the residual EOF pressure (see appendix C), one for the EOFs 3 and 4 only pressure (see appendix C), and one for the residual EOF with first guess grid pressure.
- At each time step, use MATLAB to put the pressure observations of the 35 stations into a 23 by 31 grid, which makes a 713 by 105 matrix.
- Use GrADS to contour plot each grid at each time, then export each contoured grid as a GIF image, and run in sequence using Quicktime.

## Appendix C: Residual EOF and EOFs 3 and 4

For the residual EOF,

- Multiply EOFs 1 and 2 by their associated PCTs, which produces two 35 by 105 matrices of pressure-unit data, and
- Subtract the matrices of EOFs 1 and 2 from the raw pressure matrix, which results in a 35 by 105 residual EOF matrix.

For EOFs 3 and 4,

- Use 713-point first guess EOF grid as data source and
- Add reconstructed EOFs 3 and 4 together, which results in a 23 by 31 by 105 EOFs 3 and 4 only matrix.

## Appendix D: Butterworth and Chebyshev Filters

For the Butterworth filter,

- Construct a first-order bandpass filter following that of Murakami (Murikami, 1979).
- Choose a passband of 1.87–5.00 h to maximize signal passage at peak gravity wave periods.
- Plot Butterworth response (Figure D1), note passband amplitude loss regions.

For the Chebyshev filter,

- Construct a second-order 1.00–5.00-h bandpass filter using MATLAB.
- Use passband ripple loss of 5 dB.
- Plot Chebyshev response (Figure D2).