Data validation: statistics at Mavericks

Data validation: statistics at Mavericks

The Spoondrift offices are located in Half Moon Bay, about 1 mile east of the legendary Mavericks surf break. In this area, the coastline is fully exposed to energetic swell wave fields radiating in from both the north Pacific and the Southern Ocean. On top of that, local storms produce strong winds driving energetic and steep wind seas. A rough place to be on the water, but ideal testing grounds for Spotter!

Experimental setup

To validate the statistical estimates from Spotter measurements, we deploy a Spotter alongside a Datawell Waverider DWR-G4 in about 40m of water depth at our test site, just north of the Mavericks surf break (see figure on top). During the experiment, the instruments are geographically about 200m apart.

To keep the systems in place, we use identical inverse catenary mooring setups with 1:2 scope for both instruments to eliminate differences in mooring response. The lightweight mooring setup did result in mooring effects (clipping) during heavy winds. These effects were most pronounced in the Datawell observations, and likely due to its larger inertia. Since this has nothing to do with the sensing capabilities and is entirely due to the mooring system, we have excluded such data from quantitative comparisons. Extensive mooring testing is ongoing and will be discussed in a separate blog.

Purely as a measuring device the two instruments are similar in that they both are surface-tracking wave followers (moving with the waves on the surface of the ocean) and they are somewhat similar size (Spotter is slightly smaller). The differences include weight (38lbs for the Waverider vs 12lbs for the Spotter), and sampling rate (1.28Hz vs 2.5Hz). As a result, Spotter can in theory measure higher-frequency waves (Nyquist frequency is 1.25Hz), and due to its lower weight is expected to be more responsive to shorter waves in the wave field. To be consistent, for this comparison we only consider the frequency range 0.03Hz-0.6Hz, which both instruments can resolve.

To be clear, the Datawell Waverider is an established instrument and by many considered the industry standard. The comparisons shown here are not to suggest otherwise, but merely to validate that Spotter produces similar data quality during the period considered.

Time series of statistics

We consider the period from April 18, 2017 through April 30, 2017, during which both instruments were on site and operational most of the time. Over this period, conditions varied from low-energy pure swell, mixed swell-sea, strongly wind-forced conditions, and even a brief spell of fetch-limited wave growth when the wind blew from the east.

Figure 1 shows a time series of bulk statistics, including (from top to bottom): significant wave height (defined here as Hm0 ), peak period, mean direction, and peak direction. The bulk statistics for wave height and mean direction are calculated through integration over the frequency range [0.03Hz - 0.6Hz] (for definitions of these parameters see e.g. Pearman et al., 2014).

From the peak direction and peak period we can deduce that during the first few days of this period there is a mixed see-swell field, followed by low-energy swell, and after April 24 conditions are mostly dominated by locally generated wind seas. The data gaps in in figure 2 are due to either the instruments being out of the water temporarily, or data that was rejected due to mooring effects during high-wind conditions. In general, the agreement between the two instruments is excellent throughout the period.

Figure 2 shows a time series of the distribution of wave variance in frequency space (the 'spectrum'). Here the white bars bars in the color plots indicate times when the instruments were out of the water or otherwise not collecting data. The high-wind conditions that resulted in mooring effects are retained here since in frequency space those effects can be visually somewhat separated (visible as high energy levels at low frequencies). Overall, both instruments provide the very similar spectral structure in the energy carrying wave range (see figure 2).

The variance density plot (see figure 2) considers the variance of the surface elevation, what we typically associate with 'the wave motion'. However, to estimate directional properties of the wave field, a surface-following instrument should measure the three-dimensional displacement vector accurately (displacements in east, north, and vertical direction). To establish that, we compare the lowest-order directional moments from both instruments (see figure 3). These moments involve cross-correlations between the three displacement time series (easting, northing, and vertical), and thus provide a more complete (albeit normalized) check on the complete representation of the wave motion. The evolution of these moments over time (see figure 3) characterize variations in the directional make up of the wave field, due to changes in swell and wind directions. Notable, the directional moments are sensitive and highly variable, the agreement between the two instruments is clearly very good.

Snapshots of wave statistics

For a more quantitative comparison between the two instruments, we further consider statistical estimates of spectral variables from observations collected on three times during the experiment, namely:

  • Case 1 (April 19, 1500 UTC): remnants of swell from west, and fetch-limited wave growth driving wind waves from east (from shore).
  • Case 2 (April 23, 0400 UTC): low-energy swell from south and west, light winds
  • Case 3 (April 28, 1430 UTC): energetic. locally-generated sea state

In figure 4 we show spectral variables as a function of frequency for Case 1. In what follows we will have more of these types of figures (with the same variables), so we will step through them here in more detail to help interpretation.

For Case 1 (figure 4), we see a double swell peak in the variance spectrum (top left panel), one peak below and one just above 0.1Hz. The variance density spectrum provides a measure of wave energy at each frequency. It is derived from the measured surface elevation alone (although it can equally well be derived from horizontal displacements or velocities if the water depth is known).

The mean directions (top right panel, figure 4) show that the lower-frequency swell peak comes from slightly south of west (slight less than 270o), whereas the higher-frequency swell peak comes from west (270o). The mean direction (top right) abruptly transitions from west (270o) to east (90o), at around 0.35 Hz. This is due to the wind blowing from the east (from land), which results in a fetch-limited condition and causes the higher-frequency (wind-forced) waves to come from the east (from land). This is also reflected in the directional spreading function (figure 4, bottom left panel). The swell peaks have relatively low directional spreading, with increasing spreading in the higher-frequency parts of the spectrum, consistent with a transition to fetch-limited conditions in that range. Finally, the cross-coherence Gxz , see bottom right, measures the cross-coherence of the easting and vertical displacements. This value is close to unity if the waves are narrow-band in directional space and traveling along a east-west line. This is consistent with the large coherence values we see both below 0.35Hz (swell peaks from west) and above 0.35Hz (wind sea from east) is consistent.

In summary, this is a fairly complex wave field, consisting of at least three distinct wave fields that are readily identified by measurements of both instruments. Both instruments are also in excellent agreement for all these parameters. The only systematic difference is that in the high-frequency tail of the spectrum, Spotter measures slightly more energy. This is most likely due to the fact that Spotter is lighter and slightly smaller, making it a bit more responsive to very short and steep waves. Note that the vertical scale is logarithmic and these high-frequency differences are extremely small.

The directional moments for Case 1 are shown in figure 5. The agreement is excellent throughout the frequency range.

From the directional moments and the variance spectrum, we can estimate the two-dimensional frequency-direction distribution, which is shown in figure 6. Note that the two-dimensional spectrum as shown in figure 6 is insightful but technically cannot be fully resolved by a single wave buoy (it can be resolved with an array). The spectra here are estimated using the Maximum Entropy Method proposed (Lygre & Krogstad, 1986), which introduces additional constraints to provide a reasonable (and smooth) spectral estimate. That being said, both estimates are very similar, which is of course also implied by the excellent agreement seen in the variance spectrum (figure 4, top left) and the directional moments (figure 5).

The same figures are shown for Case 2 (4/23/17, 0400 UTC), and Case 3 (4/28/17, 1400 UTC) below, to shows the excellent agreement between Spotter and Waverider for these different conditions. 

Case 2 (below, figure 7, 8, and 9) is characterized by a long-period (17s) swell field from south-west and a locally generated wind peak (around 0.25Hz), driven by 20 mph winds from the northwest. This is fairly wide-band wave field with wind waves from the northwest at large angles with the south swell. 

Case 3 (below, figures 10, 11, and 12) is characterized by strong wind forcing, resulting in short-period swell with a peak period of about 8.5 s and significant wave height of 2.2m. For these wind-forced conditions, the wave spectrum takes on a Jonswap-like shape, with an f -5 energy roll-off in the tail of the spectrum (see figure 10, top left panel). Again, the statistics obtained from both instruments are in excellent agreement everywhere.



Lygre, A. and H.E. Krogstad, 1986; 'Maximum entropy estimation of the directional distribution in ocean wave spectra', J. of Phys. Ocean.16, 2052-2060. 

Pearman, D.W., T.H.C. Herbers, T.T. Janssen, H.D. van Ettinger, S.A. McIntyre, and P.F. Jessen, 2014; 'Drifter observations of the effect of shoals and tidal-currents on wave evolution in San Francisco bight', Cont. Shelf Research91, 109-119.