Month: April 2023

Finding maximums part 3: How a repeating sea surface hides extreme mooring loads

This is part 3 of an article series on maximum dynamic loads in moorings. If you haven’t seen it yet, start from part 1 here first.

Mooring design is often driven by the maximum load in a severe sea state. It’s common to use a dynamic analysis process and a numerical model of a mooring system to resolve the extreme peak loads. In a time-domain analysis, the mooring loads are calculated in a specific realization of a sea state. A critical mistake to avoid is a repeating sea surface in the sea state realization. But what does this mean exactly and how does a repeating sea surface happen?

Southern Ocean Flux Station (SOFS) full ocean depth buoy recovered after over a year at sea in 4km deep water near Tasmania. With an average sea state of 4m significant wave height, often reaching 10m significant wave height and higher, peak loads are important to understand and anticipate. Picture credit: Peter Jansen from CSIRO / IMOS / Marine National Facility

A repeating sea surface what happens when there isn’t enough detail in the sea state realization

In a time domain analysis, this looks like a pattern of water surface that repeats itself over time. How often it repeats depends on how much detail is in the sea state realization. A sea state realization is formed by superimposing a finite number of sinusoidal ocean waves. Each sinusoidal wave segment has its own period with an associated height governed by the representative wave spectrum. The problem with a repeating wave surface is when there aren’t enough wave segments to represent the sea surface. Consider, in the absolute worst case, a sea state realization of a wave spectrum using only a single ocean wave segment: it will have a repeating sea surface after one wave period! The effect of this repetition can mean a disaster for mooring analysis.

The problem is that a repeating sea surface fools you into a false sense of security

A repeating sea surface causes a repeating mooring load response. No matter how long you run the analysis, you still get the same repeating tension. A consequence is that you are missing an accurate representation of the extreme values of the mooring response. You might think you’ve captured a reasonable maximum dynamic mooring history by checking it with a long simulation time and multiple sea state realizations. But in reality, you’ve just been looking at the results of a repeating pattern over and over. You can miss a lot of detail in the system’s dynamics, and there’s a risk you can underestimate the maximum loads. In this way, the repeating sea surface hides the extreme values from you.

How can you tell you have a repeating sea surface?

It’s difficult to tell in a systematic way, but often looking at a time history of the sea surface and the mooring tensions, it can become evident with some experience. If you see a repeating pattern of groups of peaks, there’s a good chance that you may have a repeating sea surface. A repeating pattern may be evident in the motion or mooring tension, too. But how can you avoid the problem of a repeating sea surface?

The key is in having enough detail in your sea state realization. You need enough individual wave segments to have a healthy representation of the sea state spectrum without repetition. Building up experience and visually checking your sea surface and mooring response is essential to look for telltale signs of a repeating signal.

But why not use thousands of wave segments in all sea state realizations?

The practical consideration is that the computational burden increases with wave segments. The sea state surface and water particle velocities and accelerations are all the result of contributions from every wave segment that’s included in the sea state. In turn, these physical parameters are all factors in the forcing of a floating system and its mooring. So with the number of wave segments used, the computational burden goes up, and the analysis speed goes down. A careful balance is needed to get enough detail for a realistic design process without slowing the entire process down too much.

It’s time for an example

Let’s look at more data generated by ProteusDS of the Southern Ocean Flux Station (SOFS) full ocean-depth mooring. This mooring response was calculated in a 9m, 13s wave spectrum. The sea state realization was constructed using a custom sea state with only 10 wave segments with an anticipated energy repetition of 36 seconds.

Looking at a time series of the sea state realization, you can see a repeating wave pattern, but it looks like it repeats around every 100 seconds.

However, the dynamic mooring tension shows something slightly different. The largest peaks show a repeating pattern and similar shape approximately every 36 seconds. While the sea surface looks like it takes longer to repeat, the most dominant energetic wave pattern is driving the mooring tensions with a repetitive tension at this rate.

The big problem is that these repetitive peaks are not representative of what to expect in a realistic sea state – just like you wouldn’t expect the peak loads from a single sinusoidal ocean wave to realistically reconstruct the dynamic loads of a mooring in an irregular sea state. You can’t rely on results from any statistical extrapolations to estimate maximum tensions over a time period longer than your simulation either. The consequence is that you may drastically overestimate or underestimate the maximum tensions in the mooring because of this uncertainty. This is an effect that can show up in any dynamic analysis tool that generates a time series of sea state realizations.

Note that ProteusDS uses a few tricks to avoid a repeating sea surface even when only a few wave segments are specified. While you may not see a repeating sea surface with only a few wave segments, there still may not be enough detail in the sea state realization to get meaningful maximum values – and again you may significantly under- or overestimate maximum tensions. When you’re looking at a new kind of mooring in a particular sea state, sensitivity studies on the detail in the sea state realization and the resulting mooring and motion peaks may help give you an idea of what works for future reference.


A sea state realization is formed from the superposition of many sinusoidal ocean waves. But beware that you can get a repeating sea surface if you don’t use enough of these wave segments. The problem is that once the sea surface starts repeating, you don’t get any new information from the dynamic response of a floating system and its mooring. You could even be significantly underpredicting the maximum loads. You might think you are capturing a lot of detail in a long simulation or in multiple sea state realizations, but you’re missing important information on the true extreme loads in the system.

Next step

Comparison of measured mooring tensions to dynamic analysis results is always an important validation exercise. The comparison of SOFS mooring measured tensions with ProteusDS results factors in multiple sea state realizations, long-duration simulations, and many wave segments to ensure reliable results. Read more on the detailed comparison of measured peak mooring loads on the SOFS mooring compared to ProteusDS calculations in the oceanographic validation report here.

Finding maximums part 2: Why using multiple sea state realizations quickly reveals extreme mooring loads

This is part 2 of an article series on maximum dynamic loads in moorings. Start from part 1 here first.

The maximum load in a severe sea state usually drives a mooring design. It’s common to use a dynamic analysis process and a numerical model of a mooring system to resolve the extreme peak loads. In a time-domain analysis, the mooring loads are calculated in a specific realization of a sea state. But what exactly is a sea state realization?

It’s not always flat calm in the ocean

Most places in the ocean at any given time have many different sizes and frequencies of ocean waves. A compact way to express the sea state is with a wave spectrum. A wave spectrum is like a convenient short-hand that describes how much wave height there is at different frequencies. But a wave spectrum isn’t the same thing as an actual sea state. This is where a sea state realization comes in.

A sea state realization is a fully detailed description of the state of the ocean surface. This is in the form of a complete picture of the water surface height in space and time. Thousands of specific sea state realizations have the same wave spectrum, but each realization is unique in that it has its own differences in surface elevation through space and time. The subtle differences in the water surface are vital to revealing extremes in the analysis of floating systems, like the maximum peak load in a mooring system.

Sea state realizations all have the same spectrum, but they are individually different from each other.

How do sea state realizations reveal extreme mooring loads?

The key is in the subtle differences in the time variation in the water surface. A wave spectrum can give an idea of the general characteristics of the sea surface. But it’s not always clear what rare events in the realization will look like. For example, the maximum distance from a wave trough to the next wave crest is one indication of the worst-case wave height in a specific sea state realization. You can’t know what this worst-case wave will be for a sea state realization by looking at the wave spectrum alone. You also won’t know exactly how a floating system will respond to each worst-case wave. A systematic approach is needed to check how a floating system will react in specific sea state realizations and confirm the resulting dynamic mooring loads. But rather than just checking one very long sea state realization, there are good reasons to use several.

One reason to check multiple sea state realizations is because of variations in the intensity of the water surface

In theory, a well-constructed sea state realization will have a perfect match to the characteristics of its matching wave spectrum. Attributes like the significant wave height and peak period should be identical. But the practical reality is that sea state realizations tend to have some variation in these parameters. One sea state realization might have a slightly higher significant wave height, and another might have a slightly lower one.

Significant wave height is an essential driver of motions and mooring loads, so checking multiple sea state realizations ensures you are aware of a single sea state realization that is less severe than you intend. This is a critical facet in using multiple sea state realizations, but there’s also a practical reason to evaluate multiple sea state realizations.

Another key reason to check multiple sea state realizations is that calculations can be computed in parallel

Calculating one long time series of mooring dynamics can take a while, slowing down the design process while you need to wait for the results to get design feedback. Yet it’s trivial to use multicore computers to calculate the mooring response in different sea state realizations in parallel. Often, shorter simulations in various sea state realizations can be computed much faster than one extended analysis, giving more robust design feedback than a single simulation.

While each simulation may be shorter than a storm duration, it still needs to be long enough individually to capture a meaningful number of extreme peak loads. Comparing the extreme peak loads produced by a mooring in each sea state realization gives you confidence that you have a reasonable peak load to base the design on the mooring.

But how much variation can you have between extreme peak loads?

There isn’t a specific number to use and you need to rely on engineering judgment. The extreme loads from each sea state realization will be different every time. However, the spread between the extreme values should not be too severe. If they are too far apart in magnitude, it’s an indication that your simulations are too short or that there may be severe tension shocks in the mooring response, and some changes may be needed in the design. Typically, for oceanographic mooring design, it makes sense to start by looking at least three sea state realizations to get a good picture of the extreme peak loads.

Example time

Let’s look at more data generated by ProteusDS of the Southern Ocean Flux Station full ocean-depth mooring. This mooring response was calculated in a 9m, 13s wave spectrum. The first plot shows the mooring tension from the first sea state realization. The maximum tension is around 32kN.

Another simulation in ProteusDS with a new realization of the same sea state reveals a different time series of dynamic tension. The peak is just under 30kN.

The third realization shows a peak tension of about 28kN. These load cases were all calculated in parallel on a desktop computer much faster than one long individual simulation. It’s encouraging to see similar maximum peak values across the three realizations, building confidence that the maximum is likely in this range of 30kN. A shorter duration of 500 seconds may even be justified but this is driven by the characteristic peak values that have already come out of the analysis.


A sea state realization is a fully detailed representation of the ocean surface. It’s derived from a wave spectrum and includes a time series of the water surface elevation everywhere in space and time. Thousands of sea state realizations may have the same wave spectrum and statistical characteristics, like the significant wave height. But it’s the specific details and time variation that drive floating systems, and sometimes rare events like a sequence of wave troughs and crests, that affect extreme peak loads. While in theory, sea state realizations should be characteristically the same, they often are slightly more or slightly less intense in terms of significant wave height. With low-cost multicore computing readily available, it’s also easy to run numerical models of moorings in different sea state realizations in parallel. These are both good reasons to check the extreme loads in multiple sea states to find the extreme values faster.

Next step

This is the second article in a series of three on finding extremes in mooring analysis. Read the next and final article in the series that focuses on the effect of a repeating sea surface here.