Kurtosis, what's that? A single value to rule them all!?

Being objective about the impulsiveness of noise - is kurtosis useful?

Warning: This post is speculative and highly nerdy!

We were at the "Effects of noise on aquatic life" conference in July 2019, and one of the topics discussed was an exploration of how we can characterise impulsiveness of sound in an objective way. The silver bullet might be "kurtosis" (Both TNO and JASCO presented good work on this).

Figure 1. Kurtosis definition from Wikipedia.
The definition from Wikipedia does not help us much in terms of getting feel for this metric, but the good crowd from TNO in the Netherlands have a slightly expanded version for us:
Figure 2. ISO definition of kurtosis (ISO 18405

The above equation is fairly easy to implement, even in a spreadsheet, and if you're familiar with digital signal processing you've seen similar expressions before.

With the above equation we can characterise the kurtosis of two signals with the same maximal pressure, but very different appearances:

First a series of impulses from a recording of an impact hammer, striking a  a wood-capped concrete pile. The kurtosis value for this signal will at first vary greatly, but as more impulses are included, the kurtosis value stabilises at 37.5.
Figure 3. Left: Impact pile driving recording. For most of the time it's quiet, but much energy is delivered in short bursts.
Right: Stepwise kurtosis in 0.5 second steps (1st step 0-0.5 sec, 2nd step 0-1 sec, 3rd step 0-1.5 sec etc.)
Kurtosis stabilises at 37.5

Contrasting the impact piling from above a recording of vibration piling takes longer before the kurtosis value stabilises. The signal is less consistent over time, which contributes to the longer stabilisation period. Final kurtosis: 6.3.
Figure 4. Left: Vibration pile driving. Peak not as loud, but noise is continuous.
Right: Stepwise kurtosis in 0.5 second steps (1st step 0-0.5 sec, 2nd step 0-1 sec, 3rd step 0-1.5 sec etc.)
Kurtosis stabilises at 6.3

I have done this for a range of other signals as well (Figure 5):
  • Sperm whale click train from DOSITS archive - Varying kurtosis, not stabilising, probably needs to be characterised in smaller chunks. Kurtosis Max/median: 103/78.
  • Impact piling - kurtosis stabilises after 6-7 impulses at 37.5.
  • Humpback whale call - Like the sperm whale sound, this is quite a variable signal, and probably not well characterised in this way. Kurtosis Max/median: 9.3/6.2.
  • Vibro piling from DOSITS archive  - kurtosis takes a while to stabilise at 6.3 as signal is variable over time. 
  • Suction hopper dredging from DOSITS archive - Kurtosis stabilising at 2.7.
Additionally I had a look at dB metrics of the signals as well at adjusted dB-values as found in literature (you might have to zoom to see all details).
Figure 5. Various signals and their stepwise kurtosis and varying dB-metrics.

We see that for a range of sound types we see a stabilising tendency for the kurtosis value in signals with simple repeating patterns.

An interesting proposal is to incorporate the kurtosis into our established noise metrics so that instead of having multiple metrics (and associated thresholds) dealing with impulses (dB peak-peak and SEL-single-impulse) and continuous noise (dB RMS and SEL-24h) we can incorporate the impulsiveness - and the (often) associated increased risk from it - into a single figure. 

Imagine having a single value that would tell you exactly how hazardous a noise is.

Goley et al. 2011 [1] suggests a modified SEL metric (they use CNE - cumulative noise exposure) where a kurtosis adjusted cumulative exposure (SEL') can be calculated as:

SEL' = SEL + 4.8Log10(B/Br)          {EQ 1}

With "B" being signal kurtosis, and "Br" being a reference kurtosis equal to the kurtosis of Gaussian noise, (it's 3). Note that i have used the figure "4.8" from their Leq metric as multiplier here - this is not in line with what the paper suggests directly, but is derived from data on the additional risk to humans from noise exposure with higher kurtosis.

Applying this to impact and vibro piling, we get two SEL values for the signals: "SEL" and "SEL-Goley". While the "nomal" SEL values are similar for a 10 second exposure (1.4 dB difference), the adjusted value differs by 2.4 dB and thus shows the increased risk associated with the noise from impact piling.
Figure 6. SEL (dashed lines) and adjusted SEL (solid lines) as inspired by Goley et al. 2011 of impact and vibro piling.

In figure 5 there are additional metric suggestions:
  • "SEL-Hong-Wei": Applying method from [2] to normal SEL metric instead of Leq.
  • "SEL + sqrt(kurtosis)": Home-made metric, add square root of kurtosis to SEL.
  • "LEQ-Goley": Method from [1].
  • "RMS (0.5sec)": RMS level with a 0.5 second window.

A few comments:
  • I choose to apply the methods from [1] & [2] to SEL because most limits for underwater noise are in SEL. I have no idea if the methods are suitable for SEL as they were developed for use with the Leq metric and tested on humans.
  • I need time to evaluate the NOAA/NMFS criteria for impulsive/non-impulsive against a range of realistic noises to estimate sensible constants for these methods, e.g. EQ 1 above.
  • We need information on the dependence on kurtosis for hearing injuries in aquatic animals (if a dependence exists), for this to be useful.
  • Less impulsive noise can have very high indirect impact via their continuous masking (short impulses don't do much masking...!).
  • When determining kurtosis for a signal with the view to use it for determining risk ranges, the signal should probably be filtered according to the receivers hearing, as removing low frequency energy from a signal tends to increase its kurtosis (B. Martin demonstrated this nicely at the AN 2019 conference).
UPDATE 2019/08/22

I decided to do a comparison of ranges to TTS and PTS limits, using limits from the NOAA/NMFS guidelines. 
Method as follows:
  • Generate several signals of equal SEL with varying kurtosis:
    I made ten signals of SEL 220 dB, with kurtosis from 0.1 to 45.
  • Classify signals as either "impulsive" or "non-impulsive":
    Gaussian noise has a kurtosis of "3", so this served as my limit of impulsiveness - a kurtosis over "3" means a signal is "impulsive".
  • Calculate the ranges to TTS and PTS limits of the "Low Frequency group" using simple semi-spherical spreading and using the appropriate "impulsive" or "non-impulsive" threshold (LF Group: TTS-non-impulsive: 179 dB; TTS-impulsive: 168 dB; PTS-non-impulsive: 199 dB; PTS-impulsive: 183 dB).
  • Calculate ranges to TTS and PTS limits by using the non-impulsive threshold and two different adjusted SEL values - "SEL-Goley" as above and "SEL+B^0.65" (my own).

Figure 7. Black bars: Range to TTS limits using NMFS thresholds of LF group.
Blue bars: Using "SEL-Goley" and the non-impulsive TTS threshold from NMFS.
Red bars: Using my own "SEL+B^0.65" and the non-impulsive TTS threshold from NMFS.
Signals are sketched at the bottom of the figure (amplitude at correct scale).

Figure 8. Black bars: Range to PTS limits using NMFS thresholds of LF group.
Blue bars: Using "SEL-Goley" and the non-impulsive PTS threshold from NMFS.
Red bars: Using my own "SEL+B^0.65" and the non-impulsive PTS threshold from NMFS.
Signals are sketched at the bottom of the figure (amplitude at correct scale).

  • Using any kurtosis-adjusted value for assessments removes the sharp change in predicted range to limit seen in Figure 7 & 8, where the signal geos from being classified as "impulsive" to being classified as "non-impulsive" (notice "jump" in black bars). In my opinion removing this jump is highly desirable, as we can remove the discussion about whether or not a sound is impulsive - very important for determining risk ranges as there is over 11 dB difference in the two thresholds.
  • Scaling does not have to be complicated to produce reasonable results. "SEL-Goley" and "SEL+B^0.65" are very simple.
  • Scaling by using the "SEL+B^x" pattern seems to mimic the risk ranges produced by the NMFS thresholds better than the "SEL+x*ln(B/Bref)" pattern like the approach of Goley et al. 2011.

Now I just need to go to all authors of TTS and PTS literature and ask for their stimulus-signals so we can get the kurtosis-corrected threshold shifts!

I found that the good guys from Ocean Conservation Research have some good writing on topic as well:
  • https://ocr.org/projects/noise-criteria-kurtosis-thresholds/
  • https://ocean-noise.com/2019/08/what-is-this-thing-called-kurtosis/?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+OceanNoise+%28Ocean+Noise+Blog%29
  • https://ocr.org/ocr/wp-content/uploads/Poster-Noise-Exposure-Damage-Potential-Compatibility-Mode.pdf

  • [1] G. Steven Goley, Won Joon Song, and Jay H. Kim. (2011) Kurtosis corrected sound pressure level as a noise metric for risk assessment of occupational noises. J. Acoust. Soc. Am. 129 (3).
  • [2] Hong-wei Xie, Wei Qiu, Nicholas J. Heyer, Mei-bian Zhang, Peng Zhang, Yi-ming Zhao, and Roger P. Hamernik. (2016). The Use of the Kurtosis-Adjusted Cumulative Noise Exposure Metric in Evaluating the Hearing Loss Risk for Complex Noise. EAR & HEARING, VOL. 37, NO. 3, 312–323