I am eagerly looking for a viable answer, I don't have an answer myself presently. 

A freaque wave is not characterized by its size, many storm waves are quite large, but they are not freaque because they are to be expected during a storm. Therefore I don't regard extreme waves can be freaque waves in general.

I have read that a large wave greater than 2*significant wave height has been classified as a freak wave. In working with realistic simulations, I categorized such waves according to steepness, where some of them had very small wavelengths, so large steepness, while others were high in wave length and in wave height and near or greater than  the 2*significant wave height  metric. see my MIT/WHOI thesis here: https://www.google.tt/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CBwQFjAA&url=https%3A%2F%2Fdarchive.mblwhoilibrary.org%2Fbitstream%2Fhandle%2F1912%2F3230%2FHenry_thesis.pdf%3Fsequence%3D1&ei=PKB3VMeuHIOcgwSZgIMg&usg=AFQjCNE-W1uIKyhByrH78EYGgQOvopIazQ&sig2=LE16KkwOGMYTO3gGd_WCeQ&bvm=bv.80642063,d.eXY

Dear Paul C. Liu,

            I just got a initial  idea to try to  define a freaque wave based on K-line diagram technique .Since you have strong  expertisement in this field . can you tell me all the  freaque waves in the past 30 years or so? 

I dare to answer the question of Paul( by the way one of the personthat knows more on this topic). Of course his own answer gives a insight of the situation and I can't agree more. In my opinion the unexpectedness of freque waves has to be considered in the frame of the mechanisme of generation as so many researches do at present. I have the conjecture that the "old" Lyapunov instability could be involved.

Thanks, Antinio, for your discussion. I am fascinated by your mention of Lyapunov instability as possibly connected to unexpectedness. I am not familiar with Lyapunov instability, but any new idea is exciting. I hope you would look into it and further develop your conjecture. That would be something really inspiring!

Dear Paul,

I really appriciate on how you point your finger on the points which hurts a wave counter.

As I am with an institution which is responsible for the delivery of online wave measurement data in the German waters (which can be quite 'freaquey' my perspective on this question is biased by the possibilities of in situ instruments.

We yet do not know how good the instruments are in regard to measure extreme or freque wave so we still have to put a lot of effort in data quality control ans the structured data collection to target on the day where we will be able to detect dangerous waves.

Our data recipients come from a bunch of branches:

* Shipping

* Offshore wind industry

* Coastal engineering

...

The potential answer probably shall be given individually for a specific data recipient.

As you already mentioned, dangerous waves not only appear during strong storm events. Here I would like to give some illustrative examples:

* A video cam is installed at the research platform FINO1. On the 4th of Oktober 2009 at 00:44 UTC a video with a wave of about 16 m to 18 m has been aquired. At 08:43 another wave video with a wave height in the same order has been acquired too. The ADCP wave data has measured Hs in the order of 6 m. The 100 yr. design wave height of the platform FINO1 is 18 m. So this may would be comparable to a lottery winning twice a day: two extreme wave in the order of H = 3 * Hs events very close to the design wave in one day.

* On the 7th of March 2011 sea state was calm with Hs in the order of 0,9 m with peak direction 340° (NNW). On the catamaran Windforce I (http://www.frisia-offshore.de/index.php/Wind-Force-I/518/0/?&L=3) we were on the way to an offshore wind power station in the wind farm 'alpha ventus' 60 km north of the island of Borkum in the German Bight. The safety regulations of offshore wind farm allow the ascent on a station up to a treshold of 1.3 m. As the catamaran had to change its direction in the right angle to the wave fields main direction the vessel started to suffer from a kind of resonant phenomenon with high rolling angles. Therefore the ascent operation on the station was skipped although everybody thought that we had the best conditions. This example may show how a wave field 'suprisingly' becomes 'frequey'.

Please do not hesitate to contact me for a copy of the videos.

So the following tasks could be adressed to come closer to an answer of Pauls question:

* Use all available data from all available instruments

* Data archive with sea state raw data

* highly automated data quality control with extreme (freaque: which features? height? steepness? ...) wave detector which works with all instruments (not an easy task)

The above topics would help to find extreme wave statistics to prove model based extreme wave statistics.

@Paul: A more detailed definition of 'unexpectedness' especially in regard to the 'probability' may would help me to better understand your question.

I guess we all agree that the answer to Pauls question for freaque waves would more complex than freak waves.

Here the spectral and directioanl parameters should be considered too (see second example).

@Ji Qing Tang: Dear Ji, can you give me the rough idea of the K-line diagram technique?

I think Christian's answer covers a lot of ground.  Is the wave a freak because it is different from the wave field it is in, is it an extreme wave (which is the public idea) or is it a wave that produces unusual forces/motions/effects.  All are 'freaks' in some way and size does not really matter :)

A real 'freak' would be defined as inexplicable in terms of existing theoretical paradigms.

If the question is about defining a probability in philosphical terms then maybe we are into the idea of 'surprise' - I think there's a very old book by Spanner - it was new when I was young.

Or there is a field of study that deals with the probability of events that never happen where the boundary limit is estimated by the duration within which nothing has been observed.

In my opinion, a freaque wave can be defined by its non-uniform continuity property,which  is not characterized by its size,in a previous of my study, I had developed a non-uniform continuity index to identify this kind of characteristics,

and many arrogant leading scientists in China had no patience to sit down to listen to my opinion simply because they know I don't agree with the greenhouse effect can cause the global climate change.When I send an application for a funding of China National Naturual Science, they didn't read what I said in the application documents,

simply write down some negative words prevent me to obtain fundings.

Dear Paul,

I like to talk about probabilities in a different sense and finally relate this to the question raised. 1st we need to define fully specified and non-fully specified random samples, where the latter would relate e.g. to water waves, storm surge levels, maximum water levels in rivers, rain intensities etc., the fully specified once are e.g. the shoe sizes of all kids in the school, suppose nobody is missing then u have the full sample of all shoes which is then a fully specified sample.

When we talk about freak waves or max. wave heights in general we must assume that these are not known to us in a fully specified way for a random sample. E.g. for some rivers we should measure the water levels for the existence of the river but this is also not enough because the river was changed by humans.  So why is it a problem? It is a big problems since we try to fit probability density functions (PDF) on discrete datasets and then we are doing extrapolation and look at the tail of the integrated PDF in order to derive extreme return periods.

When u decide for a certain PDF that fits well to your sample u need to do some TEST to check if u can accept your 0-hypthesis (your PDF fits the data) or u cannot deny the alternative-hypthesis (your data does not fit), following Gliwenko-Cantelli, the PDF should collapse with more data to the knows distribution but this is only valid for "FULLY SPECIFIED SAMPLES", so in conclusion we can say that if we measure extreme waves and fit some PDF that makes us happy to the given dataset we must prove that the data set is a full specification of the phenomena we want to describe (we know that this is not the case, especially not for freak waves not even for any kind of nonlinear waves) or we need to develop some testing method that takes into account that the samples are NOT FULLY SPECIFIED, I know only very little work that handle this topic properly.

Again there are two ways and a purely theoretical 3rd way. 1st assume some theory and do Monte Carlo in a way that u get all possible representation of the phenomena and then assume u have a FULLY SPECIFIED SAMPLE, 2nd u do some monte carlo on the PDF you intend to use and the you derive KS-TEST (Kolmogorov-Smirnoff Test) tables that incorporate the influence of the NON-SPECIFIED SAMPLE and then do the KS-TEST using this tables, it is quite some work that has to be done and it was done in Germany (Dr.-Ing. Detlev Belke, DWA Communications) for the PDF like Gumble, Pierson-III, Normal distribution and but not too many not e.g. for Rayleigh PDF, the 3rd one is a bit esoteric you need to convince yourself that you know everything e.g. that you are god, well there is all kind of folks out there, so all of the options I think are feasible but the latter one would be pathological.

In conclusion i am sceptic about the PDF derived and used for extremes as long they are not tested as defined above and I am not aware that this is done, in a sense it can been summarized that I question most of the work done on extreme PDF's. People often tend to show their PDF fits their data very well. I am sorry to say that it is not enough. The KS-TEST would tend to accept the 0-hypthesis even if it is not true! Therefore it is a long work but I am trying to do something in this directions but maybe it will take many years since I am so busy in my life with more trivial topics.

It would be interesting to do a Monte Carlo on the Non-linear Schrödinger Equations and  investigate the all realisations and then take a part of it and define in this way KS-TEST tables for non-specified random samples.

I did not a very deep literature review on this topic with respect to water waves. So maybe Paul and others will give clear arguments literature or whatever that will make my knowledge of what was done fully specified as by now it is only a non-specified random sample of the existing literature.

In terms of unexpectedness it is the gap between specified and unspecified since it would not be unexpected any more, but this is somehow trivially deduced from above and basically just a little more than nothing but one can do something with it.

Paul, have nice free days and let me know if this makes any sense at all in context to your knowledge which is like a big ocean in contrast to my little lake.

All the best hope to meet you soon again!

Aron

The main used idea, is  the definition "greater, than twice of Hsign". It is too weak. In order the wave became freak it has to obtain some other important features.( E.g. see fig 4 of attached paper).

Hi Leonid, thank you so much for your discussion and sharing your paper with the schematic figure.That's a very thoughtful way of determining a freaque wave. My only reservation is that for a mariner or sailor in the open ocean when encountering an unexpectedly large wave coming, there may not have enough time to check through the criteria to determine. If it's some large wave unexpected, it must be a freaque wave. We just need to be able to define the unexpectedness!

Dear Paul,

Some addtions about our favorite freaks :-)

There exist  two different problems.

1. The probability of a freak wave in specific place. We encounter with this problem for the probability of freak waves  in the Barents Sea oil field. This is one of the problem of wave climate investigation, and yesterday paper is about this.

2.Problem, you are speaking about is a sailor at a stormy sea. Nobody never will know at what moment and at what specific place a freak wave will arise. Such wave arises suddenly (the reasons are different, and this is another problem), exist for a short time and then disappear. At sea we can imagine only, that the probability of freak wave is increasing (e.g., due to opposite current intensification or veering of the wind, etc). This is our measure of unexpectedness.

VERY SIMPLE example from everyday life:

During thunderstorm you will never exactly guess at what moment and at what limited place the lighting will strike.

So goes the world,

Regards,       Leonid L.

 .

As usual, the answer of Leonid is precise and clearly formulated. I can't agree more. Excepting the shown cases of opposite currents and also bathymetric caustics the generating mechanism of rogue waves is linked to instability, here the unexpectedness.

Hi  Leonid and Antonio, Interesting we live in different parts of the world and we see waves in different parts of the Oceans. Most of all we have different perception of freaque waves also. Unexpectedness in freaque waves, while I think it's an important element, it is too abstract to define or measure. Leonid made the interesting analogy between a freaque wave and lightning strike. All are basically unknown physical phenomena we face. Answers to them are just all elusive!

The captain of a ship can only anticipate on a statistical basis until we have rapid models of the full 3D hydrodynamics and detailed remote data acquistion and evaluation. I would guess 20 years from now it will be routine. In the meantime the probability of such encounters can be reduced. A relatively energetic and narrow spectrum leads to strong non-linear interactions and is one cause of freak waves. Factors that enhance the energy in a narrow spectral band  include opposing current and locally rising bed elevation, so keep away.

I had an interesting example with transient outflow from a nearshore zone which had an offshore barrier pierced by a few narrow inlets. During a morning of persistent low ocean swells, resonant oscillations of the inshore water were excited by the external wave group period and resulted in pulsed flow through one of the inlets. At one time during the efflux, several incoming waves were trapped and crest elevations were increased fourfold relative to the approach elevation. A fishing boat was overturned and several people suffered serious injuries while negotiating the inlet at this time. It was predictable in hindsight only.

Hi Jon, thank you so much for the discussion and answering my question. It's been some years since last we conversed. You told an interesting case of near-shore freaque wave occurrence which we can only find it predictable in hindsight. But it's happening can be regarded as nature and expect-able even it's in hindsight. Of course that's easy for a desk-sailor like me to say. For real sea-going mariners that's their daily life. Science seems still unable to help them in the foresight!