In 1995, astronomers wanted to point the multi-billion dollar Hubble Space Telescope at a small portion of a completely empty sky. It was risky because they couldn’t tell in advance anything about what they might get: there was truly no evidence to expect anything but total emptiness. Yet they still wanted to follow a process and see what they got out of it. Because Hubble has limits of light sensitivity, it needed to point at precisely the same spot for 100 hours to collect enough light to have a chance at detecting something. The result was a huge success that showed the small portion of the sky wasn’t so empty. Thousands of new galaxies were discovered by following their process to reveal the data.

Similarly, the design of floating systems in a marine environment involves using a process to reveal data. The maximum load in a severe sea state often drives mooring design. But mooring systems can behave in a complex way, and you can’t tell in advance anything about what a mooring might do in a sea state just by looking at it. So how do you calculate the maximum loads? It’s common to use a process and see what you get out of it. In this case, the process consists of a dynamic analysis of a numerical model of the mooring in different sea states. The ISO mooring standard recommends a process that determines stable statistical values of mooring response. Following an approach based on this reveals data on the maximum loads. In this article, we’re going to cover three elements that are key to the process of finding these maximum loads:

- Checking the simulation length
- Investigating multiple sea state realizations
- Avoiding a repeating sea surface

First, we’re going to cover the simulation length.

**An ocean wave spectrum causes an irregular sea surface in space and time **

The general characteristics of the sea surface can be predictable, but what’s not so predictable is the resulting mooring response in such conditions. A numerical model of the mooring that accounts for the components’ weight, buoyancy, drag, and elastic stretch is necessary, with the forcing from ocean waves moving the system around. A mooring will usually have some irregular dynamic tension proportional to the intensity of the sea state. However, what’s of most interest is the maximum load. In a well-designed mooring, these peak loads only occur sometimes and appear unpredictably. What’s crucial to understand is that to capture some reasonable expected value of the maximum tension, you need a time series of the calculated mooring tension that covers a suitable amount of time.

The ISO offshore mooring standard recommends running simulations in a storm duration scenario but leaves it to the mooring designer to determine a specific time to use. Another standard for offshore moorings by DNVGL suggests a typical storm duration is three hours. But ultimately, what’s important is finding a stable statistical indication of the maximum loads. Running longer simulations always has a chance of producing higher loads. But pay attention to the maximum values you find in a mooring tension – if you find multiple examples of peaks that are reasonably close to each other, it’s a good chance you have a simulation that’s long enough to capture an expected value of the maximums. Simulation length is crucial, but other aspects to the analysis process are essential to keep track of. This brings us to the second point on investigating multiple sea state realizations.

**Not all sea states with the same spectrum are created equal**

An ocean wave spectrum is one form of description of the state of the ocean. It shows you how much energy is present at different frequencies. A sea state realization is a physical manifestation of that wave spectrum. It’s a specific instance with a time history of water surface elevation as the waves propagate through the sea. What’s vital about sea state realizations is how they can drive the extreme values of a mooring analysis. It’s not simple to anticipate the characteristics of each sea state and how the mooring will respond to it in advance. Each sea state realization will have different attributes like the maximum trough-to-crest wave height. But even with a severe trough-to-crest wave event, the maximum mooring loads depend on the mooring’s position and orientation at the time. The result can be a significant difference in peak mooring loads between different sea state realizations. Even in a single sea state realization, a fairly long span of time may not show the same peak load as another sea state realization.

Both ISO and DNVGL offshore mooring standards recommend completing dynamic mooring analysis in multiple sea state realizations to check the characteristics of the mooring response. By studying several sea state realizations, you increase your chances of capturing extreme mooring responses to severe wave loading. Sea state realizations are an important concept to grasp and leverage in the mooring design process, but there’s also a crucial mistake to avoid when working with them. This is the hazard of the repeating sea surface and is the third and final point we’re going to cover.

**A sea state realization is formed from a finite number of sinusoidal ocean waves**

As the mooring analyst, it’s in your power to control how many sinusoidal ocean waves are used to construct the realization of the wave spectrum. The problem of a repeating sea surface comes up when there aren’t enough of these waves used to represent the spectrum. For example, in the worst-case scenario, using only a single sinusoidal wave to construct a wave spectrum produces a sea surface that repeats once every wave period! While a repeating sea surface does give an initial picture of the dynamics of a mooring system, it causes a serious problem in the analysis process because it hides an accurate representation of the extreme values.

It doesn’t matter how long your analysis runs or how many sea state realizations you check – the repeating sea surface will keep producing the same repeating mooring load response. The consequence is that you may be significantly underpredicting the maximum mooring loads.

**Let’s look at an example**

Maintained and designed by CSIRO, the Southern Ocean Flux Station (SOFS) includes a full ocean-depth mooring. Located in extremely harsh conditions, with an average sea state of 4m significant wave height, it often sees much higher and intense waves. Here’s a time series of the top tension of the SOFS mooring in a severe wave spectrum with 9m significant wave height and 13 second period calculated by ProteusDS. How long should you run an analysis to find the maximum tension? The answer depends on what the peaks look like in the results.

Here’s a plot of the tension history from the first 200 seconds. There’s quite a few peaks in the mooring tension around 22kN. But how do we know these are the maximums?

A longer simulation time to 500 seconds reveals a jump up in peak loading to 33kN.

Extending to 1000 seconds shows another peak at 34kN. Extending even further to 2000 seconds reveals another lower peak around 31kN.

** **

This doesn’t mean we always need at least 2000-second simulations to find the peak loads for oceanographic moorings like the SOFS system. The takeaway is that if you don’t have an expectation of the mooring response, an analysis that is too short in time can miss critical information on the maximum load. If you miss these maximum peaks and select components with lower load ratings, there’s a higher chance the mooring may fail in a severe sea state. Your general analysis needs to have some kind of stable statistical representation of the values. Generally, from this one example, the standard deviation of the mooring tension stabilizes around 500-750 seconds, and there tend to be a few of these large peaks in the 500-750 second range.

**It’s summary time**

We covered a few topics on finding the hidden extreme in mooring design, and it’s time to summarize. The most important concept to understand is that you need to have stable statistical values of your dynamic mooring loads. You can do this by evaluating the dynamic mooring response in an ocean wave spectrum with a few careful considerations. The first consideration of the analysis process is the simulation length. The analysis should be long enough that you see a grouping of maximum peak values that are reasonably close to each other. The second element is checking multiple sea state realizations. This is because it’s not easy to anticipate uncommon characteristics of the water surface – like a maximum individual wave – and how it will interact with the floating system. Finally, watch for the hazard of a repeating sea surface. You need to avoid a repeating sea surface in your analysis because it will obscure the true extremes in the system – no matter how long your simulation or how many realizations you investigate. All of these concepts are part of a process in finding important information. Much like Hubble starting at a single spot for 100 hours, you also need to follow a process to find new information on maximums in dynamic analysis.

**Next step**

This is the first article in a series of three on finding extremes in mooring analysis. Check out the second article that focuses on sea state realizations and why they are important in finding maximums here.

**PS**

Check out the result of the Hubble Deep Field image here.