a. Spectral partitioning

Spectral partitioning facilitates the division of each directional wave energy spectrum S(f, θ) into distinct subsets that represent individual wind sea and swell wave systems. Each wave system is then conveniently described by a reduced set of statistics. The concept was originally proposed by Gerling (1992); here the approach is based on the ideas of Hasselmann et al. (1994). The WITS partitioning of each input spectrum involves five steps, as outlined below.

2) Step 2: Identify and combine wind sea peaks

Wind seas are identified using a wave age criterion such that wind sea peaks lie within the parabolic boundaries defined by

c_pU₁₀δ,(4)

where c_p is the phase speed of the wind sea, U₁₀ is the 10-m elevation wind speed, and δ is the angle between the wind and the wind sea. In terms of the peak frequency of the wind sea peak in deep water, (4) becomes

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The generous factor of 1.5 ensures that all possible wind sea peaks are included. This relationship defines a parabolic region over the spectral matrix, as depicted in Fig. 2. Any partition whose peak falls within the region described by (5) is considered to be forced by the wind and initially defined as wind sea. All wind sea partitions from a given input spectrum are combined and assigned the partition number 0. All remaining partitions are considered swell.

3) Step 3: Combine mutual swell peaks

Adjacent swell peaks that belong to the same swell system are considered mutual and are combined. This is accomplished with user-supplied threshold factors and helps account for the noise inherent in discrete spectral estimates. To be considered mutual, two neighboring swell peaks must meet one of the following criteria.

Peak separation. Compare the distance between peaks

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with the spread of each individual peak

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where

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with the integrations performed over the measurement domain and the total energy is given by

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Combine the two peaks if the spread of either peak satisfies the peak separation criterion

f²κδf²(9)

where the spread factor κ is adjusted to optimize performance of the spectral partitioning. Optimization procedures are addressed in section 2d.

Minimum between peaks. Combine adjacent peaks if the lowest point between them is greater than the peak minimum factor ζ times the smaller of the two peaks. The peak minimum factor is also optimized for a particular observation set. Newly combined partitions are themselves eligible for further combinations, so all relevant partition statistics are updated whenever a combination is made.

4) Step 4: Energy threshold check

Remove any partitions whose total energy is below an energy threshold

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where A and B are chosen to eliminate noise in low-energy regions of the spectrum (see section 2d). Note that this threshold includes the spectral fall-off proposed by Phillips (1985). Low-energy partitions are represented by negative ones in the template and are ignored for further processing. The remaining partitions are renumbered, and this information is saved to the partition template file. To conserve disk space, the partitioned spectra are not individually saved. The template files can be used at any time to extract spectral partitions from the original “parent” spectra.

b. Swell tracking

Swell tracking in WITS is a two-step process. First, preliminary groups of swell partitions likely to have been generated by the same source are formed. This is accomplished by grouping series of swell partitions with similar or slowly varying f, θ, and a_rms characteristics. As swell arrives from a distant source, these attributes are expected to vary as a result of wave dispersion and storm source evolution. Preliminary wave tracking can be viewed as an initial sorting of swell partitions; groups are formed within which specific subsets, representing swells originating from distinct localized weather events, can be sought.

To determine if a new swell partition belongs with a previously identified preliminary group, the location of the new partition in a weighted frequency–direction–height space is compared to a predicted group location computed from the average properties of the previous five partitions in each existing group. If the group contains fewer than five partitions, then all of the partitions are used in determining the group location. The weighted distance between a new observation and the predicted group value is given by

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where θ is in radians and the coefficients have been introduced to emphasize the relative importance of each term. The new observation is assigned to the group with which it shares the smallest value of d. Values of d greater than an assigned threshold are considered too high for a valid match. When a partition cannot be assigned to an existing group, a new group is created for it. The weighting factors in (11) have been selected through an iterative trial and error process. Although they work well, we anticipate employing more efficient clustering techniques in future upgrades of WITS.

The second step locates specific swell events within preliminary groups. A specific event is an evolving set of swells that were formed at the same time and location. The events are located by searching through the preliminary wave groups for series that obey deep-water wave dispersion characteristics. Each preliminary group is searched as a whole to find embedded specific groups. The slope and variance about the best-fit line through all possible sequences of three or more observations is determined. Both slope and variance thresholds are employed to discriminate against swells that do not exhibit the required dispersion characteristics. The longest possible sequences whose slopes and variance fall within the threshold limits are considered specific swell systems (Kline and Hanson 1995).

c. Swell source identification

The resulting wave partitions allow automated identification and tracking of multiple storm systems. A distinct swell source time and location is calculated from each specific group of swell partitions. The slope and intercept of the best-fit line through the mean frequency and time data pairs (f, t) are used in the linear gravity wave theory simplifications of Eqs. (2) and (3) to determine the distance to the swell source and the swell origination time, respectively. A specific group mean wave direction ϒ, obtained by averaging the θ_s values from all group members, is used to indicate the great-circle route over which the waves have traveled. The source location is then obtained by spherical geometry. As depicted in Fig. 3, the wave observation site, the North Pole, and the swell source location all form a spherical triangle on the surface of the earth. The source latitude (ϕ) and longitude (ϕ) are then given by (Bartsch 1974):

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where ϕ₀ = observation latitude, ϕ₀ = observation longitude, θ_d = (d/R_E) = angular distance to source, and R_E = radius of the earth. Once the swell generation times and locations are determined, they are located on regional weather maps to determine the credibility of each prediction.

d. Parameter selection

The success of the WITS partitioning process is dependent upon proper selection of the various adjustable parameters. These parameters are chosen to optimize the performance of the partitioning, tracking, and storm source identification algorithms on a given observation set. The need for adjustable parameters arises primarily from variations in regional wave climatology, although differing frequency and direction resolutions and instrument noise levels can contribute. The described methods have been applied to the datasets listed in Table 1. Included are data from several different instruments as well as a wave model; these observations and predictions cover a wide variety of climatic environments. The parameters listed in Table 1 were chosen through iteration to optimize the wave system identifications and storm source predictions of each set.

The minimum energy threshold parameters A and B from Eq. (10) are the most crucial, as the spectral noise floor is a function of the wave climate at a given site. This is demonstrated by the computed threshold levels from the Datawell WAVEC buoy observations in the North Pacific Ocean and the Gulf of Mexico. The smaller values from the Gulf of Mexico are a result of a much calmer wave environment, which effectively lowered the instrument noise floor. The WITS toolkit includes a utility for optimizing the selection of A and B. As demonstrated in Fig. 4, the utility places the energy threshold curve (10) over a plot of the total energy of each isolated partition as a function of peak frequency. The values of A and B are then easily adjusted based on output sensitivity requirements and the noise floor of the dataset.

The peak combination parameters κ and ζ are selected through iteration to optimize both the selection of distinct wave systems and the prediction of storm source locations. One measure of performance is the mean regression coefficient from the specific swell group linear correlations of mean frequency versus time (see section 2c). Another is the percentage of storm source predictions that can be verified on surface weather charts. Note that the same values of κ and ζ were selected in all of the field datasets in Table 1; only the WAM model predictions required slightly adjusted parameters to optimize peak combinations.

In combining swells with preliminary groups, the weighted distance, d in amplitude–direction–frequency space is determined by (11). A threshold for d allows for discrimination between different swell sources. For the observations reported here, a value of d = 0.6 provided the best performance in forming preliminary groups. The performance of this threshold is evaluated by plotting the evolution of wave vectors or wave vector history. An example is provided in section 4. Discontinuities in evolving swell systems would indicate that d needs to be increased; the inclusion of disparate wave systems in the same group would indicate the opposite.

a. Measurements

Atmospheric and sea surface measurements were obtained by a MINIMET meteorological buoy (Coastal Climate Company) deployed from R/V John P. Tully. The instrument was outfitted with a drogue at a depth of 165 m to minimize advection by near-surface currents and was allowed to drift within the test site. Observations obtained at 1-Hz resolution included wind speed and direction at 3-m height, air temperature, sea surface temperature, and barometric pressure. MINIMET data were processed using a marine atmospheric boundary layer model (Smith 1988) to generate surface layer profiles, adjust meteorological measurements to common reference elevations, and compute atmospheric stability parameters and wind stress (Hanson and White 1991). Output quantities were averaged to form 30-min values that coincide with the surface wave observations. Vector-averaged wind speed estimates at 10-m height (U₁₀) are used for the results reported here. These observations compared well with those obtained by a nearby shipboard meteorological station (Hanson et al. 1993).

Continuous measurements of wave height and direction over the experimental period were taken by a Datawell WAVEC heave, pitch, and roll buoy deployed from the John P. Tully (Farmer 1992). The instrument was drogued at a 100-m depth and allowed to drift with the surface currents. Although not required for this analysis, this deployment approach reduced wave Doppler contamination by surface currents and also stabilized the buoy against capsizing. Surface displacement and vector slope were sampled at 1.28 Hz and transmitted to a receiver aboard the John P. Tully. A few record gaps, of several hours duration, occurred when the buoy drifted out of telemetry range from the ship.

Wave statistics and a total of 236 half-hourly directional spectra were computed by Juszko et al. (1995) using a data-adaptive eigenvector (EV) technique (Marsden and Juszko 1987). Spectra containing 36 degrees of freedom were computed over the frequency band 0.05 ⩽ f ⩽ 0.5 Hz to obtain estimates over the range that the wave buoy is considered most accurate. A particularly attractive aspect of the EV method for this investigation is an exceptional resolution of individual peaks within multimodal spectra such as those obtained from the Gulf of Alaska.

Typical directional (2D) and one-dimensional (1D) wave spectra from a single 30-min wave record from this study appear in Figs. 5a and 5b, respectively. The directions reported are the directions from which the waves are propagating. The directional spectrum has two large peaks, indicating a wind sea from the northwest coexisting with a lower-frequency swell from the west. These two events also appear as isolated peaks on the 1D spectrum, calculated using

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The 95% confidence intervals in Fig. 5b indicate that the wind sea and swell wave peaks are distinct wave events (Shumway 1988).

b. Experimental conditions

The winter weather we experienced was influenced by four separate atmospheric low pressure systems moving through the North Pacific region. Selected meteorological and sea surface records from the experiment appear in Fig. 6. The first two lows, occurring on 24 and 26 February, passed almost directly over the test site and had the most significant impact on meteorological conditions. Note in particular the rapid air temperature changes due to the atmospheric fronts associated with these two lows. A notable amount of local variability in wind speed and direction existed (Fig. 6), with high-wind events associated with each passing storm. The most significant of these were the two approximately 12-h wind episodes associated with the 25–26 February winter storm. The structure of this storm as it passed northward over the test site at a speed of approximately 12 m s⁻¹ is depicted by the National Oceanic and Atmospheric Administration (NOAA) surface pressure map series of Fig. 7. This single low pressure cell dominated the weather in nearly the entire gulf for a period of approximately 36 h.

Of particular importance for this study was the effect these meteorological events had on the wave field. The significant wave height record of Fig. 6 indicates that surface waves peaked at 5.3 m as a result of the 25–26 February storm. Wave heights did not develop as dramatically during the other high-wind events because the winds were either too low, too unsteady, or too variable in direction. However, even during times of low winds, significant wave heights were at least 2 m because of ocean swells propagating through the gulf. In fact, the continual arrival of long-period swell from remote weather events, often at considerable angles to the wind waves, led to a “confused sea” visual appearance during much of the test.

a. Swell systems

The results of the Gulf of Alaska preliminary wave tracking appear in the wave vector history plot of Fig. 8. Swell mean frequency f, rms height, and propagation direction are given by vector origin, length, and azimuth, respectively. Preliminary groups are color coded and numbered. Note that wind seas, isolated using the wave age criterion [Eq. (5)], do not appear on these figures; they are discussed briefly at the end of this section.

The swell events depicted in Fig. 8 are essentially tracers of local and distant storms traveling through the North Pacific. Perhaps the most striking feature is that at any given time, there were at least two, and often three, simultaneous swell systems evolving through the Gulf of Alaska. Swells within the lower-frequency groups (shown in red) had presumably traveled the farthest distance; their directions suggest they propagated into the Gulf of Alaska from the wide-open Pacific basin. These groups exhibit a linear frequency increase with time, as predicted by the wave–dispersion relationship [Eq. (1)]. It is shown later that the higher-frequency groups contain subsets that exhibit this same behavior. Two midfrequency groups (shown in magenta and cyan) persisted for nearly 3.5 days from the start of observations on 24 February until a record gap just before 1200 UTC on 27 February. The swell events in these groups nearly always opposed each other in direction and shifted forward and back within a narrow frequency range of approximately 0.09–0.13 Hz. The green groups have some members at fairly high frequency for swell. These events are correlated with abrupt changes in the wind forcing; very young, local swells persist for a short time after the wind shifts.

The swell tracking algorithm identified 19 original preliminary groups. The seven groups that do not appear in Fig. 8 were eliminated because they contained only three partitions, each with extremely low energy levels. It could not be determined if these observations were poorly resolved swell events or just sampling noise. These discarded partitions represent less than 1% of the total amount of partitioned swell energy.

b. Storm source predictions

Specific wave tracking was performed on each preliminary group from the Gulf of Alaska observations. A total of 44 specific swell events were identified in the 6-day wave record. These events are shown on the frequency evolution plot in Fig. 9. Here, the mean frequency of each swell partition is plotted as a function of observation time. Specific swell groups are numbered and depicted with enlarged symbols. A least squares regression line is drawn through each specific group. A very good correlation exists for each group, with r = 0.9 being the average correlation coefficient for the entire set.

Note that in Fig. 9, not all members of each preliminary group fall into a specific swell group. The unselected partitions (identified by small dots) contain about 35% of the total swell energy that passed by the observation site during the 6-day measurement period. These are valid swell events that do not exhibit the dispersive characteristics predicted by linear wave theory. They may have originated with events that are not well represented as point sources, such as fast-moving storms or steady, continuous wind. These events will result in a less focused dispersive signal at a distant observation point. Furthermore, interactions between frequency components from the same storm, interactions with wind and sea from other meteorological events along the route, and refraction by major ocean currents may all contribute to the attenuation and scattering of the original swell. Thus across ocean basins only selected windows of opportunity may allow swell systems to pass unaltered; these are represented by the specific groups in Fig. 9.

As wave dispersion dictates, the specific groups in Fig. 9 all display a steady increase of wave frequency with time. Only the slope and intercept of each regression line and the mean direction of each series are required to predict the swell generation times and locations using Eqs. (2), (3), and (12). The storm parameters of all 44 specific swell events are listed in Table 2. This information was directly obtained from the automated processing of the directional wave spectra. The (f, t) regression slope m_ft and intercept b are listed along with the mean wave propagation direction ϒ for each specific group. The calculated swell origination time t₀, distance traveled d, and source geographic coordinates (ϕ, ϕ) are also provided in Table 2.

The last two columns of Table 2 provide comments pertaining to the identification of the predicted source storms on National Weather Service (NWS) regional surface pressure charts. A direct detection indicates that a storm was found at the predicted time and location with winds oriented toward the observation site. Wind orientations are assumed to be parallel to the isobars, with cyclonic rotation about low pressure centers in the Northern Hemisphere. Inferred detections suggest that a storm occurred in the general region of the predicted site within ±12 h of the predicted observation time. This makes allowances for 1) slight modifications to the propagating swell by the processes discussed above; 2) inaccuracies in the weather charts, which are based on sparsely distributed observations; and 3) noise inherent in the collection and processing of the wave data. As Table 2 indicates, most of the inferred sources were either stationary features or quickly moving storms. In a few instances, no detections were made, suggesting that either an erroneous specific group was selected or the storm was missed on the NWS chart. Of the total swell energy in the 44 specific groups, 70% arrived from directly detected sources, 16% was from inferred sources, and 14% was of unidentifiable origin. The process of identifying swell sources using predicted parameters and NWS charts is illustrated below, with a few examples from Table 2.

Specific groups 1, 2, 3, 4, and 11 have t₀ that fall in the latter part of 23 February 1992. The corresponding source locations (ϕ, ϕ) are plotted on the North Pacific surface pressure chart for 1800 UTC 23 February 1992 in Fig. 10. The great circle paths that connect the source locations with the wave observation site are also shown on the chart. Note that a single low pressure storm centered directly in the Gulf of Alaska is responsible for swell groups 1–4. The predicted source positions are located precisely where the cyclonically rotating winds would be expected to generate waves directed toward the wave observation site. These are essentially young swell events that have traveled distances of 243–509 km (Table 2).

The predicted source for swell group 11 is 724 km south of the observation area in a wide swath of increasing pressure to the east (Fig. 10). As with the previous groups, the source is located precisely in a region where wave generation would be directed toward the observation site. As this pressure “wall” was a semistationary feature on the weather charts from 22 February 1992 to 26 February 1992, many of the specific groups evolving from this direction did not exhibit a clear dispersive trend.

In contrast to the swell from meteorological events in the general region of the Gulf of Alaska, wave groups 27 and 32–36 all appear to be a result of distant storms moving across the Pacific basin (Table 2). The swell from these events have predicted travel distances of 1000–5000 km or more. Specific group 27 is a particularly clear example of these waves. The predicted source location for specific group 27 appears on the 22 February 1992 (1200 UTC) weather chart reconstruction of Fig. 11. The source is located to the southeast of a large low-pressure system in a region of significant wind fetch well away from the storm center.

Specific swell groups 8, 21, and 44 are all predicted to have been generated on 25 February 1992, but at widely separated locations (Table 2). As depicted in Fig. 9, these swell events cover a wide range of frequencies and actually arrived at the observation site over 4 days. This is due to the travel distance of each example swell. Values of d were 380, 918, and 4995 km for groups 8, 21, and 44, respectively. Predicted source locations for these three groups appear on the 25 February 1992 (0600 UTC) weather chart reconstruction in Fig. 12. The predicted regions correspond to the locations of three distinct atmospheric low-pressure storms scattered throughout the North Pacific Ocean. The identified source positions are precisely located where the cyclonically rotating winds would be expected to generate waves: in areas of sufficient fetch with winds directed toward the wave observation site.

Specific group 44 is especially interesting, with swell that traveled almost 5000 km over 4 days before reaching the observation site with a strong dispersive signal. The f, θ, and h_s records from group 44 appear in Fig. 13. Unlike some systems, which exhibited varying directions, the directions of group 44 observations have little variance. This is attributed to the source storm’s comparatively slow translation speed of approximately 45 km h⁻¹ to the east. Group 44 h_s exhibited a rapid increase over the first 9 h and then leveled off at approximately 2.5 m until the end of our observation period. It should be noted that the system 44 observations end because our experiment was over and the wave buoy was retrieved. Examination of NOAA buoy data (#46003) at 51°51′N, 155°54′W indicates the swell of system 44 persisted through the following day. System 44 waves were of such low period that they were not visually detected at the time of the observation. Their existence went unsuspected until post-test processing of the buoy data was completed and the results were validated with the NWS weather charts.