## Abstract

In an earlier paper by Wang and Hwang, a wave steepness method was introduced to separate the wind sea and swell of the 1D wave spectrum without relying on external information, such as the wind speed. Later, the method was found to produce the unreasonable result of placing the swell–sea separation frequency higher than the wind sea peak frequency. Here, the following two factors causing the erratic performance are identified: (a) the wave steepness method defines the swell–sea separation frequency to be equal to the wind sea peak frequency with a wave age equal to one, and, (b) for more mature wave conditions, the peak frequency of the wave steepness function may not continue monotonic downshifting in high winds if the high-frequency portion of the wave spectrum has a spectral slope milder than −5. Conceptually, the swell–sea separation frequency should be placed between the swell and wind sea peak frequencies rather than at the wind sea peak frequency. Furthermore the wind sea wave age can vary over a considerable range, thus factor a above can lead to incorrect results. Also, because the slope of the wind sea equilibrium spectrum is typically close to −4, factor b becomes a serious restriction in more mature wave conditions. A spectrum integration method generalized from the wave steepness method is presented here for wind sea and swell separation of the 1D wave spectrum without requiring external information. The new spectrum integration method works very well over a wide range of wind wave development stages in the ocean.

## 1. Introduction

Because of their slow energy decay rate, long surface gravity waves in the ocean may travel vast distances, even across the entire basin. These swells from distant sources then superimpose on the relatively shorter waves generated by the local wind field and produce complex surface wave conditions recorded by either in situ or remote sensors. In many areas of research, such as air–sea interaction and wind wave dynamics, the wind sea portion of the wave spectrum is of interest and the separation of swell and wind sea in the wave spectrum is an important task of the surface wave data analysis.

With directional wave spectra, many methods have been developed to sort out the different wave systems embedded in the wave spectrum (e.g., Gerling 1992; Hasselmann et al. 1994; Hanson and Phillips 2001; Portilla et al. 2009). The procedure usually involves the following two major steps: (a) the identification of local spectral peaks, and (b) the association or combination of neighboring peaks into individual “common” wave systems.

Frequently, in many field campaigns involving large area coverage, such as remote sensing experiments, dedicated in situ wind and wave measurements are not feasible and such data have to be derived from other resources, such as the network of operational weather buoys maintained by the National Data Buoy Center (NDBC). In such cases, most of the buoys provide only the 1D (frequency) wave spectrum. The swell–sea separation algorithms degenerated from the 2D cases also engage in the same two-step procedure of peak search and combination. The first step is relatively straightforward, but for the second step many auxiliary and often subjective criteria have to be designed; examples of these criteria include the frequency spacing between neighboring local peaks, the relative spectral levels between neighboring peaks, and whether the trough between local peaks is sufficiently pronounced (e.g., Portilla et al. 2009).

Deviating from the search-and-combine approach, the NDBC uses a wave steepness method for separating the wind sea and swell (see http://www.ndbc.noaa.gov/algor.shtml; Wang and Gilhousen 1998; Gilhousen and Hervey 2001). As will be further described in section 2, the wave steepness function *ξ*(*f*) is an integrated property of the wave spectrum, and it removes the complication of establishing the criteria of combining local peaks. A set of simple rules, including the use of wind speed, is then developed to relate the swell–sea separation frequency *f _{s}* to the wave steepness function peak frequency

*f*

_{m0}. To remove the dependence on wind speed so that the swell–sea separation can be achieved with the 1D wave spectrum alone, Wang and Gilhousen (1998) define the separation frequency as

*f*

_{sW}= 0.9

*f*

_{m0}. Wang and Hwang (2001) define a different sea–swell separation frequency

*f*

_{s0}, which is related to

*f*

_{m0}through a wind wave spectrum model. The design also eliminates the need to use wind speed in separating wind sea and swell components in the 1D wave spectrum. Portilla et al. (2009, their Fig. 10) apply the method of Wang and Hwang (2001) to a mixed sea dataset collected in the Gulf of Tehuantepec and find that the algorithm produces the rather peculiar result that the separation frequency is higher than the wind sea peak frequency.

In this paper, the wave steepness method for swell–sea separation is further examined. The causes leading to the undesirable result as described by Portilla et al. (2009) are identified. The analysis leads to the development of a spectrum integration method, which is a generalization of the wave steepness method for wind sea and swell separation (section 2). The new algorithm is applied to two field datasets. The first dataset has 1-yr hourly wind and wave measurements in deep water taken by an NDBC buoy, and the second dataset is from the Gulf of Tehuantepec air–sea interaction experiment (IntOA) at a location 22 km offshore with 60-m water depth. [The data presented in Portilla et al. (2009) is a subset of IntOA.] The former dataset is characterized by moderate to more mature wave conditions typical of deep-ocean observations, and the latter corresponds to much younger wind-generated waves superimposed on long swells. The spectrum integration method suggested in this paper yields very good results on the swell–sea separation for various stages of wind sea development encompassed in both datasets (section 3). More discussion on the spectrum integration method and related issues is presented in section 4, and a summary is given in section 5.

## 2. Spectrum integration method for swell and sea separation of the 1D wave spectrum

### a. The wave steepness method

The wave steepness method for separating swell and wind sea components of the surface wave spectrum is used operationally at the NDBC (see http://www.ndbc.noaa.gov/algor.shtml; Wang and Gilhousen 1998; Gilhousen and Hervey 2001). The basic idea is that the wave steepness is the product of wavenumber and wave height; thus, it is contributed mainly by short waves. The contribution from the swell components is almost negligible because of their long wavelengths (small wavenumbers). As a result, the peak of the wave steepness function is very close to the peak of the wind sea portion of the spectrum. For the swell–sea separation, the wave steepness function is defined as the ratio of the wave height *H* and wavelength *L* integrated from a given frequency to the maximum frequency of the wave spectrum

Here, the *n*th moment of the wave spectrum is

The equalities and are used in (1). The upper bound of the spectrum integration frequency *f _{u}* is 0.5 Hz because the algorithm development is intended for operational buoys. Extension to a higher spectrum integration frequency is discussed in section 4c. The deep-water dispersion relation

*L = gT*

^{2}/2

*π*is used in (1). The algorithm can also be developed with the more general dispersion relation for all water depths

*L =*(

*gT*

^{2}/2

*π*) tanh(2

*πh*/

*L*), where

*g*is gravitational acceleration,

*T*is wave period, and

*h*is water depth.

In Wang and Gilhousen (1998), the separation frequency is given as *f*_{sW} = 0.90*f*_{m0}, where *f*_{m0} is the peak frequency of *ξ*(*f*); thus, the separation of swell and sea does not require wind data. Gilhousen and Hervey (2001) modify the wave steepness method by defining the separation frequency as

where *f*_{PM} is the peak frequency of the Pierson–Moskowitz fully developed wind sea spectrum (Pierson and Moskowitz 1964), given by *c*_{PM} ≈ 1.25*U*_{10}, where *c*_{PM} is the phase speed of the *f*_{PM} wave component and *U*_{10} is the neutral wind speed at 10-m elevation above mean sea level (http://www.ndbc.noaa.gov/algor.shtml; Gilhousen and Hervey 2001).

Wang and Hwang (2001) seek to improve the method of Wang and Gilhousen (1998) for a wave steepness–based algorithm and maintain the independence on external information. Using a synthetic spectrum similar to the Pierson–Moskowitz model, an empirical relation is found between *U*_{10} and *f*_{m0} as

Assuming that the wind sea components are propagating slower than the wind speed, they define the swell–sea separation frequency as

Substituting (4) into (5), *f*_{s0} is then related to *f*_{m0} for swell–sea separation without the need to use *U*_{10},

Wang and Hwang (2001) apply the algorithm to several datasets obtained by the NDBC buoys in the Gulf of Mexico and off the California coast with very good results. However, as discussed in section 1, Portilla et al. (2009) apply the method to a dataset collected in the Gulf of Tehuantepec. The area is known to have strong mountain gap winds in the winter called Tehuanos. Each Tehuano event may last from 1 day to several days. The south side of the gulf is open to the Pacific Ocean, so under the Tehuanos the wave condition is distinctively bimodal (see Fig. 9 of Portilla et al. 2009, or Fig. 4 of this paper), with swell arriving from the south and young wind sea traveling in the opposite direction. Portilla et al. show that (6) fails for such simple bimodal conditions, with *f*_{s0} frequently higher than the wind sea peak frequency *f*_{pw} (see their Fig. 10, or Fig. 4d of this paper). The cause of the temperamental performance of (6) is investigated next and remedial measures are suggested.

### b. Pitfalls of the wave steepness method and recommended modification

In the work of Wang and Hwang (2001), the swell–sea separation frequency *f*_{s0} is defined as (5), which corresponds to the wind sea peak frequency *f*_{pw} for wave age *c*_{pw}/*U*_{10} equal to 1, that is, . In principle, the swell–sea separation frequency should be lower than *f*_{pw} and higher than the swell peak frequency *f*_{ps}. Because the wave age of a wind sea may vary over some range, for example, from about 0.3 in the Gulf of Tehuantepec young sea dataset (Portilla et al. 2009) up to about 1.25 of fully developed sea (Pierson and Moskowitz 1964), (6) may produce unexpected results, such as placing the separation frequency above the wind sea peak frequency when the wind sea wave age deviates from 1, as discovered by Portilla et al. (2009). In retrospect, Fig. 3 of Wang and Hwang (2001), which illustrates the application of (6) to the Joint North Sea Wave Project (JONSWAP) spectrum model (Hasselmann et al. 1973), gives an indication of the problem that *f*_{s0} becomes higher than *f*_{pw} in young sea conditions (the *f*_{pw} greater than about 0.27 Hz in the computational results is illustrated in the figure).

Another limitation of the wave steepness method is related to its dependence on the spectral slope of the high-frequency portion of the wave spectrum. This can be illustrated by studying the shape factor of the wave steepness function (1) with a simple power-law spectrum *S*(*f*) ~ *f ^{a}* for which

*I*

_{0}(

*f*) can be given analytically as

As described later in this paragraph, for the wind sea spectrum the first equality of (7) is most relevant, from which, for *a* < *−*3, *I*_{0}(*f*) is proportional to *f*^{(a+5)/2} asymptotically for *f _{u}* much greater than

*f*; thus, it is a monotonically decreasing function (toward larger

*f*) for

*a*< −5 and a monotonically increasing function for

*a*> −5. For the general case that

*f*may not be very small compared to

*f*, the terms involving

_{u}*f*in the first equality of (7) cannot be discarded. Figure 1a shows the results of

_{u}*I*

_{0}(

*f*) computed with (7) for

*a =*−3, −4, −4.5, −5, and −6. As is clear from the discussion above regarding the asymptotic solutions, the curves for

*a >*−5 have local peaks, as demonstrated in the three curves of

*a =*−3, −4, and −4.5. The curve of

*a*= −4 is of special interest for the wind wave spectrum because the −4 slope is now considered to be an important property of the equilibrium range of the wind-generated wave spectrum (e.g., Toba 1973; Phillips 1985; Donelan et al. 1985; Hwang et al. 2000). For the −4 slope, the local peak is near 0.18 Hz. Thus, for a synthetic spectrum of

*f*

^{−4}dependence, the

*f*

_{m0}of

*ξ*(

*f*) will not be less than 0.18 Hz. As a consequence,

*f*

_{m0}will not respond well to the continuous downshifting of the wind wave spectral peak frequency because the wind wave spectra generally exhibit a prominent equilibrium range. This problem was not discovered by Wang and Hwang (2001) mainly because the algorithm development was based on the Pierson and Moskowitz (1964) spectrum model, which has a −5 high-frequency spectral slope. The −5 slope for the equilibrium range or saturation range in the high-frequency portion of the wind wave spectrum (Phillips 1958, 1977) has long been considered to be out of date (Phillips 1985). More discussion on the subject is deferred to section 4.

As shown in Fig. 1a, to have a monotonic response of the shape function peak frequency, the slope of the spectrum needs to be steeper than or equal to −5. Because the wind sea spectrum most likely has an equilibrium spectrum with a −4 slope, the wave steepness method can be modified by replacing the energy spectrum *S*(*f*) with *S*(*f*)/*f ^{b}*, and setting

*b*≥ 1; that is, the algorithm of the swell and sea separation is modified to be based on the search for the maximum of the spectrum integration function

the peak frequency of which is denoted as *f*_{mb}. Because *I _{b}* no longer carries the meaning of the wave steepness (except for

*b*= 0), the approach is simply called the spectrum integration method. The shape function

*I*

_{0}(

*f*) of the wave steepness function

*ξ*(

*f*) is a special case of

*I*(with

_{b}*b*= 0) and belongs to this general family of the integration functions based on

*S*(

*f*)/

*f*.

^{b}Following an analysis with simulated wave spectra (the detail of which is given in section 4), we recommend *b* = 1 for developing the swell–sea separation method. That is, the algorithm is operated on the momentum spectrum *S*(*f*)/*f*. The corresponding spectrum integration function is

and the peak frequency of which is *f*_{m1}.

Here we show the result of the application of the method to simulated spectra computed with the Donelan et al. (1985) spectrum model, which has a −4 slope spectral tail, for *U*_{10}/*c*_{pw} = 0.8, 1.6, and 2.4 and wind speeds between 5 and 30 m s^{−1}. The correlation between *f*_{pw} (the wind sea spectral peak frequency) and *f*_{m1} (the peak frequency of *I*_{1}) is shown in Fig. 1b. The next step is to define the swell–sea separation frequency *f*_{s1}. For design purpose, we can write

where *X* < 1. If *X* is close to 1 (*f*_{s1} is close to *f*_{pw}), as in the algorithm of Wang and Hwang (2001), then it may produce the undesirable result of placing the swell–sea separation frequency above the wind sea peak frequency when applied to the actual field data, as has been pointed out by Portilla et al. (2009). Even if such mishap does not occur, the definition of *X* close to 1 will lead to an underestimation of the variance of the wind sea. On the other hand, a small *X* may place *f*_{s1} inside the swell portion of the wave spectrum. From experimenting with the Donelan et al. (1985) spectrum model, it is found that about 95% (the exact percentage is wave age dependent) of the wave energy is contained in the frequency components higher than 0.75*f*_{pw}, and *X* = 0.75 is used in this paper. More quantitative discussions of the multiplication factor *X* are presented in section 4d.

With *f*_{s1} = 0.75*f*_{pw}, least squares fitting applied to the data in Fig. 1b produces the following polynomial function for *f*_{s1}(*f*_{m1}):

## 3. Application to field data

### a. Open ocean deep-water measurements

The spectrum integration method of swell and sea separation is applied to two field datasets. The first dataset contains 1-yr-long (2006) hourly measurements of NDBC station 41001 at 150 n mi east of Cape Hatteras (34°40′30″N, 72°41′54″W). The local water depth is 4462 m. Figure 2a shows a 10-day segment of the wave spectrum. The *f*_{pw}, *f*_{s1}, and *f*_{ps} are shown with dashed, solid, and dashed–dotted lines, respectively. Of the 240 cases in this example, the method failed to identify the wind sea peak for 24 cases: 8 of them are at day-of-year (YD) close to 12.7; 8 are near YD 16, 1 is near YD 18.1, 3 are near YD 19.3, and 4 are near YD 19.7. These failed cases are under low wind conditions, such that either the spectral density at the actual wind sea peak is less than the spectral density at the separation frequency or no clear wind sea peaks can be found in the spectrum as a consequence of swell dominance. There are also 12 cases where the swell peak frequencies are assigned to the lowest frequency bin. These are high wind cases with single-peaked (wind sea) spectra. A second step to identify single-peaked spectra can be designed relatively easily, because in such cases *f*_{s1} would be very close to either *f*_{pw} or *f*_{ps}. This second step can use wave age to determine whether the single-peaked spectrum belongs to either the wind sea (*U*_{10}/*c*_{pw} ≥ 0.8) or swell (*U*_{10}/*c*_{pw} < 0.8). In the absence of wind speed information, the temporal variation of the spectrum evolution may provide useful indication of whether the single-peaked spectrum belongs to either wind sea or swell. More sophisticated designs of identifying single-peaked spectrum are not pursued further in this paper. At this point, we clarify the following conventions used in this paper: (a) YD starts at 0 from 0000 UTC 1 January [the Julian day starts at Greenwich noontime (4713 BC, which is frequently adjusted to the year of interest), thus it is 12 h off from the timing system used here], and (b) NDBC 42001 wind speed *U*_{5} is measured at 5-m elevation. It is converted to *U*_{10} by multiplying with 1.06. For a logarithmic wind profile, the range of *U*_{10}/*U*_{5} is from 1.05 to 1.08 for dynamic roughness *z*_{0}, ranging from 10^{−5} to 10^{−3} m.

The significant wave heights of the wind sea and swell portions of the wave spectrum are shown in Fig. 2b. The time series of wind speed is also displayed for reference. In terms of the wave height, the wind sea generally lags the wind event during the developing phase, sometimes by more than 10 h. Interestingly, during the decaying phase of a wind event the lag may be significantly reduced in some cases (e.g., YD 15–16 and 18–19), or remain very large in other cases (e.g., YD 11.5–13 and 16.3–17). These features may be influenced by the conditions of background swell and wind steadiness.

Figure 3 shows several examples of the swell–sea separation results applied to cases of different wind wave development stages. In the NDBC dataset, the maximum value of the inverse wave age *U*_{10}/*c*_{pw} is 1.99, and for active wind wave development, *U*_{10}/*c*_{pw} ≥ 0.8 (Pierson and Moskowitz 1964). The three cases on the top row represent young seas, and those on the bottom row are mature seas. The three numbers in each panel are time (YD), *U*_{10}/*c*_{pw}, and *U*_{10} for the displayed case. Dashed and dashed–dotted lines identify *f*_{pw} and *f*_{ps}, respectively, and the dotted line with circles represents *f*_{s1.} For reference, *f*_{s0} of Wang and Hwang (2001) is also shown with the dotted line with triangles: the results produced by *f*_{s0} are not very satisfactory. The spectrum integration method with *f*_{s1} produces very good results for separating the wind sea and swell in both young and mature wave development. For wave spectra that are clearly bimodal, such as Figs. 3b–d, or those that are less prominently bimodal cases, such as Fig. 3e, *f*_{s1} demarcates the swell–sea separation properly. With a second step follow-up, the cases in Figs. 3a,f, with *f*_{s1}/*f*_{ps} close to unity, would be classified as single peaked.

### b. Bimodal wave spectrum

The second dataset is from the IntOA Experiment conducted from 22 February to 24 April 2005 (García-Nava et al. 2009; Ocampo-Torres et al. 2011). The analyzed data are restricted to wind sea generated by strong mountain gap winds from the north, and swell from the south is a constant presence. As mentioned earlier, the cases presented by Portilla et al. (2009) belong to a subset of this dataset. Figure 4 shows the result of swell and sea separation using the spectrum integration method described in section 2b. The results of *f*_{ps}, *f*_{s1}, and *f*_{pw} are shown, respectively, with dashed–dotted, dotted, and dashed lines in Figs. 4a,c for the mountain gap wind events throughout the full period of the experiment. The detailed spectrum of a strong wind case is shown in Fig. 4b and a weak wind case is shown in Fig. 4d, with the resulting *f*_{ps}, *f*_{s1}, and *f*_{pw} marked with the same line styles as those in Figs. 4a,c. For comparison, *f*_{s0} of Wang and Hwang (2001) is also shown with dotted line with triangles. The problem of *f*_{s0} > *f*_{pw} occurs frequently in low winds when the signal (of surface waves)-to-noise ratio is generally low, but even when *f*_{s0} < *f*_{pw}, the demarcation of sea and swell is too close to the wind sea peak frequency, and the algorithm will result in an underestimation of the wind sea variance. A more quantitative discussion is presented in section 4d.

Using the recommended spectrum integration method, 4 out of the total of 494 cases (with 3 near YD 55.5 and 1 near YD 72, all of which are in low winds) produce questionable results, which represent a better than 99% accuracy for these almost ideal bimodal wave spectra. For reference, the wind speed is also plotted in Figs. 4a,c.

## 4. Discussion

### a. Wave steepness method and spectrum integration method

In this section, a more detailed analysis of the spectrum integration method is presented using simulated data, and the results are compared with the wave steepness method. Figure 5 shows examples of the simulated mixed sea spectra *S*(*f*) and the corresponding *I*_{1}(*f*) and *I*_{0}(*f*) at the following two different inverse wave ages: *U*_{10}/*c _{p}* = 0.8 and 2.4. The wind sea components of the simulated spectrum are calculated with the Donelan et al. (1985) spectrum model for

*U*

_{10}= 5, 10, 15, and 20 m s

^{−1}. The swell components are Gaussian distributed over a narrow frequency band surrounding the swell period of 10 s and significant swell height of 3 m. Noise is introduced to each spectral component by multiplying with a factor 1 +

*N*, where

*N*is a random number uniformly distributed between 0 and 1. The integration process of

*I*

_{1}(

*f*) and

*I*

_{0}(

*f*) removes the spikiness of

*S*(

*f*), making the task of peak searching considerably simpler. This figure illustrates one of the foremost advantages of the integration operation, that is, the elimination of the peak combination step and the associated combination criteria of the conventional swell–sea partitioning methods, as described in section 1. While both

*I*

_{1}(

*f*) and

*I*

_{0}(

*f*) produce smoothed spectral products for processing swell–sea separation, the decreased sensitivity of the peak frequency downshift with the increasing wind of

*I*

_{0}(

*f*) in more mature wave conditions (lower panels) is obvious, and will be further discussed next.

### b. Frequency weighting of the spectrum integration method

In section 2b, we have recommended the modification of the spectrum integration method for swell–sea separation by substituting *S*(*f*)/*f ^{b}* (with

*b*≥ 1) for

*S*(

*f*). Examples of setting

*b=*1 for field data analyses have been presented in section 3. Here we also analyze the case with

*b*= 2 using simulated data. For

*b*= 2, the spectrum integration function is

and the corresponding peak frequency is *f*_{m2}. Figure 6 shows *f*_{m0}, *f*_{m1}, and *f*_{m2} as functions of *U*_{10} between 5 and 30 m s^{−1} for *U*_{10}/*c _{p}* = 2.4 and 0.8. The swell wave height is also varied in this part of the simulation, and the cases of

*H*

_{ss}= 0, 1, and 3 m are illustrated (the swell period remains 10 s, as in Fig. 5).

Let us first examine the three curves *f*_{m0}, *f*_{m1}, and *f*_{m2} for *H*_{ss} = 3 m. In a young sea, for example, Fig. 6a showing *U*_{10}/*c _{p}* = 2.4, all of the integration functions display expected monotonic downshifting of the peak frequencies as wind speed increases. The wind speed sensitivity of the frequency downshift increases as

*b*increases. In low winds, the dominant wind sea components may be much higher than 0.5 Hz, as shown in Fig. 5a for

*U*

_{10}= 5 m s

^{−1}, and they do not contribute to the evaluation of the integration function; thus, the resulting

*I*are overwhelmed by the swell components, and the impacted range of low wind speed expands with increasing value of

_{b}*b*(from

*U*

_{10}≤ 5 m s

^{−1}for

*b*= 0,

*U*

_{10}≤ 6 m s

^{−1}for

*b*= 1, to

*U*

_{10}≤ 8 m s

^{−1}for

*b*= 2 in the examples illustrated here). In a more mature sea, for example, Fig. 6b showing

*U*

_{10}/

*c*= 0.8, the stagnation problem of peak frequency downshift for the steepness function starts to emerge, as reflected in the low sensitivity with respect to wind speed of the

_{p}*f*

_{m0}curve in moderate to high winds. For the other two curves

*f*

_{m1}and

*f*

_{m2}, the wind speed sensitivity of the former (

*b*= 1) is somewhat better, with a more consistent monotonic decrease.

The impact of swell height is generally not strong on the wind speed sensitivity of the *f*_{mb} downshift, as shown with the three *f*_{m1} curves for *H*_{ss} = 0, 1, and 3 m in both Figs. 6a,b. For a young sea (Fig. 6a), the lower swell does improve the wind speed range of steady frequency downshift, from *U*_{10} > 6 m s^{−1} for *H*_{ss} = 3 m, *U*_{10} > 5 m s^{−1} for *H*_{ss} = 1 m to *U*_{10} > 3 m s^{−1} for *H*_{ss} = 0 m.

Summarizing the results from the numerical simulations, while both *b* = 1 and 2 remove the stagnation problem of frequency downshift in the wave steepness method, based on overall considerations *b* = 1 appears to be a better choice than *b* = 2.

### c. Extending to high-frequency spectral measurements

The frequency resolution of operational buoys rarely exceeds about 0.5 Hz, but many wave sensors for research applications are designed to achieve much higher-frequency wave measurements. It is natural that one would like to be able to process the data with a frequency resolution that is as high as is practical. Increasing the upper-bound integration frequency *f _{u}* of

*I*

_{1}in (9) changes the relation between

*f*

_{pw}and

*f*

_{m1}and the design curve of

*f*

_{s}_{1}(

*f*

_{m1}) (11) for swell–sea separation. The results of

*f*

_{pw}and

*f*

_{m1}are obtained for the simulated spectra (as described in sections 4a and 4b, with the swell period fixed at 10 s and swell heights of 0, 1, and 3 m), as displayed in Fig. 7 for

*f*= 0.5, 1.5, 3, and 5 Hz. It is clear that with increasing

_{u}*f*,

_{u}*f*

_{m1}becomes an almost ideal proxy of

*f*

_{pw}, such that a simple

*f*

_{s1 }

*= Xf*

_{m1}can be used for the algorithm of setting the swell–sea separation frequency when applying the spectrum integration method to wave measurements with high-frequency resolution (

*f*is greater than about 1.5 Hz).

_{u}In practice, the signal-to-noise ratio in the high-frequency portion of the wave spectrum is generally not too good, especially in low wind conditions, so the processing of swell–sea separation with increasing *f _{u}* may not necessarily generate a better outcome. For example, the wave spectrum frequency resolution of the IntOA data is nominally 10 Hz (the capacitance wave gauges for surface displacement measurements are sampled at 20 Hz). Figure 8 shows the results of swell–sea separation using

*f*= 0.5, 1.5, and 3 Hz. The separation frequency derived using higher

_{u}*f*becomes increasingly more erratic. This is caused mainly by the strong contribution of the high-frequency components, which are more likely to have low signal-to-noise ratios, to the spectrum integration function

_{u}*I*

_{1}. Figures 9a,b show the two detailed spectra (the same ones illustrated in Figs. 4b,d for frequencies up to 0.5 Hz) for high and low winds, respectively. Here, the swell–sea separation results of

*f*

_{ps},

*f*

_{s1}, and

*f*

_{pw}for

*f*= 3 Hz are illustrated. For the high wind case, the results are the same as using

_{u}*f*= 0.5 Hz (cf. Figs. 9a and 4b). For the low wind case, the obvious wind sea spectrum with a peak near 0.4 Hz is missed in the

_{u}*f*= 3 Hz processing. However, the wind speed at the time of measurement is 1.9 m s

_{u}^{−1}and it is not likely to be the driving force of the 0.4-Hz wave component, the phase speed of which is about 4 m s

^{−1}. A close inspection of the wind speed time series (e.g., Fig. 8d) indicates that the wind speed dropped from about 5 to 1.9 m s

^{−1}within less than 2 h before YD 71.93. Thus, the apparent wind sea components near 0.4 Hz, while they belong to the same wind event, are technically the remnant of the earlier wind forcing unrelated to the instantaneous wind speed of 1.9 m s

^{−1}. There are many similar cases in low winds, especially following a sharp decrease in wind speed. These cases exemplify the difficulty of wind wave analysis in low wind conditions.

The spectrum integration functions *I*_{1}, corresponding to the wave spectra shown in Figs. 9a,b, are illustrated in Figs. 9c,d. The three curves in each panel are calculated with *f _{u}* = 0.5, 1.5, and 3 Hz. For the high wind case (Fig. 9c), the three peaks of the integration functions remain relatively unchanged because the wind sea signal is strong enough to overcome the noise contribution from the high-frequency portion of the spectrum; thus, the swell–sea separation is not impacted by the choice of

*f*. For the low wind case (Fig. 9d), the peaks of the three integration functions differ considerably as a result of the low signal-to-noise ratio of the wind sea components. To reduce the noise contamination to the evaluation of

_{u}*I*

_{1}, it is recommended that

*f*does not deviate too much from 0.5 Hz for processing swell–sea separation.

_{u}### d. Swell–sea separation frequency and wind sea peak frequency

The basic concept behind the spectrum integration method is the use of the peak frequency *f*_{mb} of the spectrum integration function *I _{b}*(

*f*) as the proxy for the wind sea peak frequency

*f*

_{pw}. The wave steepness method is a special case of the wave spectrum integration method for

*b*= 0. After

*f*

_{mb}is derived from the spectrum integration, the swell–sea separation frequency can be calculated with the analytical formulation established from the wind wave spectrum model simulation (11). In the design of Wang and Hwang (2001), the swell–sea separation frequency (5) is effectively and the empirical relation of

*f*

_{m0}and

*f*

_{pw}is based on the Pierson–Moskowitz wave spectrum. As described earlier, when applied to the field data,

*f*

_{s0}from such a design may become higher than

*f*

_{pw}, and it is no longer capable of separating the sea and swell successfully. Even when the problem of

*f*

_{s0}>

*f*

_{pw}does not occur, placing

*f*

_{s0}so close to

*f*

_{pw}will severely underestimate the wind sea variance. In this paper, our design curve of

*f*

_{s1}(

*f*

_{m1}) (11) effectively places

*f*

_{s1}at

*Xf*

_{pw}with

*X*= 0.75. The procedure is to integrate the wave spectrum to obtain

*I*

_{1}(

*f*), search for the maximum of

*I*

_{1}(

*f*) to obtain

*f*

_{m1}, and then apply (11) to obtain

*f*

_{s}_{1}. Once the separation frequency

*f*

_{s1}is identified, the peak frequencies

*f*

_{pw}and

*f*

_{ps}can be easily obtained by searching for the spectral maxima in the regions of

*f*≥

*f*

_{s1}and

*f*≤

*f*

_{s1}, respectively. After processing the swell–sea separation, it is a simple procedure to obtain the swell and wind sea variance by spectrum integration from the minimum frequency bin to

*f*

_{s1}for the former, and from

*f*

_{s1}to the maximum frequency bin for the latter. Figure 10 plots the cumulated wind sea variance (normalized to have a maximum equal to one)

of the simulated wave spectrum as a function of *f*_{s1}/*f*_{pw} for *f _{u}* = 0.5 Hz (upper panels) and

*f*= 2 Hz (lower panels) and three different wave ages. For young sea and low wind speed, the wind sea spectrum peak may be higher than 0.5 Hz (see Fig. 5a) and the separation algorithm may produce unreliable results in those cases. In general, setting

_{u}*X*=

*f*

_{s1}/

*f*

_{pw}at 0.75 provides a very good estimation of the total wind sea variance, while

*X*= 1 clearly underestimates the wind sea variance. Varying

*X*in the neighborhood of 0.75 only results in very small differences in the resulting wind sea variance computation.

## 5. Summary

The wave steepness method for the swell and sea separation of the 1D wave spectrum (Wang and Gilhousen 1998; Gilhousen and Hervey 2001; Wang and Hwang 2001) employs spectral integration to yield a smooth wave steepness function *ξ*(*f*). The peak frequency of *ξ*(*f*) is then related to the separation frequency of wind sea and swell. The method may be designed to use only the wave spectrum without external input, such as wind speed (Wang and Gilhousen 1998; Wang and Hwang 2001), but unexpected result of placing the separation frequency above the wind sea peak frequency using the published algorithms may occur (Portilla et al. 2009).

In this paper, the wave steepness method is analyzed further. Two major shortcomings of the wave steepness method described in Wang and Hwang (2001) are noticed: (a) the method is designed on *U*_{10}/*c*_{pw} = 1 and the swell–sea separation frequency is placed at the wind sea spectral peak frequency; and (b) for an analytical power-law spectrum with a −4 slope high-frequency tail, the peak of the wave steepness function *ξ*(*f*) may not downshift monotonically in response to the wave spectral development as wind speed increases. Because the equilibrium spectrum of wind-generated waves is likely to have a −4 spectral slope, and the wave age of a wind sea can vary over some range, the wave steepness method of Wang and Hwang (2001) may not perform well in high winds and conditions with wave ages much different from 1.

To preserve the advantage of the spectrum integration approach in smoothing out the spikiness of the wave spectrum, a generalization of the wave steepness method is suggested. The procedure is applied to the weighted spectrum *S*(*f*)/*f ^{b}*, with

*b*≥ 1, such that the peak of the corresponding spectrum integration function

*I*(

_{b}*f*) may advance monotonically as the peak of the wave spectrum continues downshifting in increasing winds. Based on the analysis on simulated spectra,

*b*= 1 is recommend (section 4). The resulting algorithm (11) relates the swell–sea separation frequency

*f*

_{s1}and the peak frequency

*f*

_{m1}of the integration function

*I*

_{1}(

*f*). This equation places the separation frequency below the expected wind sea peak frequency instead of at the expected wind sea peak frequency, as in the algorithm of Wang and Hwang (2001). The revised method is applied to two field datasets to cover a wide range of the wind sea wave age and yields very good result on swell–sea separation.

## Acknowledgments

This work is sponsored by the Office of Naval Research (NRL Program Element 61153N) and CONACYT (Project 62520, DirocIOA). The IntOA field experiment was supported by CONACYT (SEP-2003-C02-44718). The deep-ocean field data are provided online by the U.S. Department of Commerce, National Oceanic and Atmospheric Administration/National Weather Service/National Data Buoy Center (http://www.ndbc.noaa.gov/hmd.shtml). We are grateful for the constructive comments from three anonymous reviewers.

## REFERENCES

## Footnotes

U.S. Naval Research Laboratory Contribution Number JA/7260-11-0114.