43 Joint probability of tide and rain

High tide levels can restrict discharges from storm overflows on combined sewers and from surface water (storm) sewers and so increase flood risk.?So to model the flood risk of a coastal sewerage system we need to represent the likely effect of tides.

Last year, a client asked me to review the method that they were using for joint probability of rainfall and tide levels affecting an urban drainage system.?They wanted confirmation that their approach was in line with the good practice that I had set out in WaPUG User note 22.?I vaguely remembered writing a user note on selection of tide levels (it was over 30 years ago) but I had no idea what it said.?Was it still valid??Did I still think it was best practice??Had they followed it correctly??

Tide levels

The tide curve for a site is made up of different components. ?The most important ones are:

·????????Regular – independent of weather:

o??Twice daily tidal cycle.

o??Twice monthly spring-neap cycle.?Spring tides give the highest high tides but also the lowest low tides.?Neap tides are more moderate.

o??Apogee-perigee.?A variation several times a year as the moon is closer to or further away from the earth.

·????????Irregular – related to weather

o??Surge due to low atmospheric pressure and wind set due to onshore winds.?The two are often lumped together as “surge”.

Constant or varying

The first decision when modelling tides is whether to use a constant tide level or to use a sinusoidal tide curve.?User Note 22 was based on use of a constant tide level but did suggest use of a tide curve for systems with long drain down times.?I would now favour a tide curve for all cases and I set out here an improvement on the User Note 22 method.

There are four things that we need to consider:

• Where in the daily tidal cycle do we start the event?
• Which part of the spring-neap cycle do we use?
• How do we include surge tides?
• How do we include climate change?

Daily tidal cycle

I propose to generate a sine curve of tide level using a tide cycle rounded to 12 hours.?I recommend starting the tide cycle at mean sea level on a rising tide, which is a slightly pessimistic case.

Spring-neap cycle

Spring tides have both the highest high tides and the lowest low tides.?Neap tides have lower highs and higher lows.?However the worst impact on drainage system outfalls are likely to be from the spring tides where the high levels restrict discharge from the system.?The extra lows of the spring tides are unlikely to give much benefit in allowing additional discharge as there is no extra benefit once the tide level is below the outfall.

We need to consider the joint probability of rainfall and tide.?The regular tidal components are independent of rainfall and the joint probability is relatively easy.?For the weather-related components it gets more complicated.

For the regular components the joint probability of rain and tide happening together is the probability of one multiplied by the probability of the other.?To give any particular overall probability we can have a range of combinations from heavy rain and small tide to light rain and large tide.?Each combination will give a different magnitude of flood risk and the result that we need is the worst case.

For example, an overall 1/50 annual probability of occurrence is given by all of the following combinations and so we would take the worst one.?Adjust the values for the actual overall probability that you want to assess.

The spring tide cycle is the normal maximum tide cycle (but is exceeded occasionally by the Highest Astronomical Tide when the moon is closest to the earth).?We will therefore take mean spring tide as the 10?%ile cycle.?The neap tide cycle is exceeded for all tides (100?%ile).?We can interpolate between these two extremes for other percentiles.

Now we just need values for the high and low water levels for spring and neap tidal cycles.?There are two approaches.

• Published statistical data
• Analysis of long tide records.

Chart datum

There is a gotcha that affects both of the methods, in that tide levels are generally related to a local “chart” datum rather than the national standard datum.?The local datum is often roughly the lowest tide level observed when the tide gauge was set up.?So the calculated levels need to be reduced to change from chart datum to standard datum.

For the UK you can get the corrections from here for each tide gauge.

Using published statistics

For locations in the UK the National Tidal and Sea Level Facility has carried out data cleansing and statistical analysis.?The website lists all of the tide gauges for which data is available.?Select the one closest to your site and click on Site Information (ST) to get the basic data for the site.?

The site gives Mean High and Low Water Spring and Neap tide levels.?These can be used to generate the percentile values that we need.?It also defines Highest Astronomical Tide, which is higher than Mean Spring and occurs when the moon is particularly close to the earth, but we will ignore this for this analysis.

Analysis of tide records

If we do not have published statistical data then we can derive the Mean Spring and Neap tide levels by analysing long records of tide data ideally at 5 minute intervals.

There is another gotcha that there may be bad data in the tide records that needs to be removed before we do the analysis.?How do we do that?

I considered all sorts of clever-clever techniques including machine learning or fast Fourier transforms to identify the underlying patterns.?Then realised that there was a much easier approach as we already knew underlying patterns.?We know that the twice daily tidal cycle is 745.25 minutes.?Therefore we find the first high tide in the record then step forward every 745.25 minutes to retrieve each subsequent high tide.?Similarly for the first low tide.?It is easy to do using the Index function in Excel (other spreadsheet tools are available).?We then plot the sequence of high tides and low tides and look for anomalies.?Below is part of the plot from the project that sparked this blog.

The graph shows two brief anomalies where the gauge jammed for a few tide cycles and one massive anomaly where the coast was struck by a major storm and the gauge appeared to read incorrectly for the next two weeks.?Delete all this anomalous data.?There are some other irregular patterns which probably represent surge conditions.?Retain this data.

We now need to analyse for mean spring and neap high and low tides.

The spring-neap cycle consists of 28.5 tides so in the table that we have created of high tide levels, we number each high tide from 0 to 56 then repeat the sequence.?We then take the average for each sequence number to give us two spring-neap cycles.?Read off the maximum and minimum values to give MHWS to MHWN.?Then repeat the whole exercise for the low tides to give MLWS and MLWN.?

We can then use these values the same as we would the published statistical values.

Surge tides

So far by using mean values rather than extremes we have effectively not included surge in our analysis.?However there can be a correlation between surge and rain.?Low pressure weather systems cause surge tides (like holding a vacuum cleaner over a bucket of water) and also bring heavy rain.?In the UK this is a problem for the west coast where the surge and rain arrive at the same time.?For the east coast the rain generally arrives first as it goes straight across country whereas the surge arrives hours later as it has to go round the north or south.?Similar effects are likely in other countries.?

Research into the correlation for the UK has been published in Defra R&D Technical Reports FD2308/TR1, FD2308/TR2 and FD2308/TR3.?These are masterclasses in how not to explain statistical relationships in an understandable way.?The analysis is too detailed to cover here, but I will come back to it in a future blog.??

Climate change

The dead moose on the table in all of this is of course climate change.?We need to increase all of the calculated tide levels for future conditions to allow for sea level rise due to climate change.?Predictions of this keep being updated.?A recent study suggested that if (when) the West Antarctic Ice Sheet collapses it will lead to a 4m increase in sea level.?So check the latest guidance and take the pessimistic values.

Resilience

We should also test the resilience of the system using an extreme high tide.?For the UK, records of historical extreme tides are given in the National Tidal and Sea Level Facility website.?I suggest taking the second most extreme recorded tide as a constant worst case and combining this with a modest rainfall event, perhaps one with a 1/5 annual probability.

This extreme combination will almost certainly cause flooding and it would not be reasonable to design for this, but check that there is no risk to critical system components or critical infrastructure or community facilities in this situation.