Tide predictions based on tide-gauge observations are not just
the astronomical tides; they also contain radiational tides –
periodic sea-level changes due to atmospheric conditions and solar forcing. This poses a
problem of double-counting for operational forecasts of total water level
during storm surges. In some surge forecasting, a regional model is run in
two modes: tide only, with astronomic forcing alone; and tide and surge,
forced additionally by surface winds and pressure. The surge residual is
defined to be the difference between these configurations and is added to the
local harmonic predictions from gauges. Here we use the Global Tide and Surge
Model (GTSM) based on Delft-FM to investigate this in the UK and elsewhere,
quantifying the weather-related tides that may be double-counted in
operational forecasts. We show that the global

The operational forecast in several countries of storm surge still-water
levels is based on a combination of a harmonic tidal prediction and a
model-derived forecast of the meteorologically induced storm surge component.
The forecast is based on the “non-tidal residual”, the difference of two
model runs with and without weather effects. This is linearly added to the
“astronomical prediction” derived from local tide-gauge harmonics

There are several possible sources of error in this procedure. The purpose of
the combined tide-and-surge model is to capture the well-documented
non-linear interactions of the tide and surge.

In Sect. 2, we show that the double-counting of radiational tides has a
potential contribution to forecasting error not just on long timescales
(through

Specific radiational tides have been studied using response analysis, for
example the solar-diurnal

The Highest and Lowest Astronomical Tide (HAT and LAT) are important datums used for navigation and are calculated from tidal predictions. In Sect. 4 we use the model predictions to quantify to what extent HAT and LAT are influenced by weather-related tides and show that in many places several centimetres of what is reported as HAT is attributable to periodic weather patterns.

There are other contributors to water level, including steric effects and river flow, that will also create differences between the tide gauge and the forecast water levels, particularly seasonally, and which may be out of phase with the atmospheric contribution. The problem of double-counting of periodic changes does not arise if they are omitted from the surge model entirely, but they may contribute to HAT and LAT calculations. These effects are not included in this study.

The current procedure for forecasting total water level in the UK is as follows.

Run a barotropic shelf model (CS3X, currently transitioning to NEMO surge

At individual tide-gauge locations, derive a tide harmonic prediction

Forecast the total water level

Finally, it has been proposed

Similar procedures are implemented elsewhere in the world, so in this paper
we replace the regional models with GTSM. This is the forward Global Tide and
Surge Model developed at Deltares on the basis of Delft-FM (Flexible Mesh)

Harmonic analysis

The choice and number of tidal constituents determined by harmonic analysis
are typically chosen according to the length and frequency of data available.
In this paper we use 62 harmonics for which there is 1 year of data, as listed
in Table

Time series (2013) of
error (

A significant source of error for this method is that a tide gauge is
measuring the total water level, and hence the harmonic prediction

To estimate

If it were possible to avoid the double-counting and provide astronomical
tidal harmonics for the observations, the prediction would instead be
equivalent to

Fortnightly cycle of prediction change
(metres) due to small changes in constituents

The forecasting approach of the linear addition of a non-linear model residual to
a harmonic prediction,

Consider a simplified example in which the tide can be modelled by a single
constituent,

Suppose the harmonic prediction at the gauge agrees in amplitude to the
tide-only model, but has slightly different phase:

The skew surge is defined as the difference between the maximum water level,
here

This is illustrated in Fig.

Although in practice there are more constituents, a similar relationship will
still hold in a small window about each high tide. Where there are frequent
surges with a consistent effect on the tidal phase we would expect

Suppose a surge adds a constant
amplitude of 20

Figure

The diurnal constituents

Vector difference (

It may come as a surprise that constituents such as

The effect on higher-order constituents is everywhere less than 5

We tested the stability of these results to the number of constituents fitted using the list of 115 harmonics usually associated with 18.6 years of data (see the Supplement) and found that the changes remain within 0.2 cm.

Some of the difference between the harmonics of surge and tide-only models is
directly attributable to the atmospheric tides. The global atmospheric
pressure field contains

Amplitude (

The Highest Astronomical Tide (HAT) is used internationally for
flood-forecasting reference levels and in navigation for clearance under
bridges. HAT can be used in structural design alongside skew surge as an
independent variable for determining return-period water levels. The Lowest
Astronomical Tide (LAT) is also an important parameter recommended for use
as the datum on navigation charts

An approximate calculation of
range as

Figure

LAT tends to move the opposite way, so in most places the maximum tidal range
is increased by using the tide-and-surge model. That is, the true
astronomical-only tidal range is slightly less than that quoted from
harmonics based on predictions. In Scotland (just above Liverpool in
Fig.

The most extreme changes shown in Figure

In places with small tide, seasonal signals may be dominant and they may be important to include for practical purposes. For example along the French–Italian coast from Mallorca to Sicily there is about a 7 cm increase in HAT and 3 cm decrease in LAT using the surge rather than tide-only model, so a highest “astronomical” tide based on predicted tide from observations actually contains about 7 cm due to seasonal winds.

There are substantial changes in tidal constituents fitted to
tide-only and tide-and-surge model results. Even constituents with purely
lunar frequencies, including

Some effects of the weather on tides are double-counted in the forecast
procedure used in the UK, in which model residuals are added to gauge-based tide
predictions. Even if the model were perfect, the minimum error from the
current forecast procedure would be at least the error in the harmonic
prediction including surge at estimating the tide-only model. If
62 constituents are fitted, this has a standard deviation of 20 cm at
Avonmouth and 4–10 cm at most other UK gauges. 5–8 cm of the error at
Avonmouth is due simply to a small change in phase of the

Understanding and quantifying these errors is extremely important for
forecasters, who will often need to advise or intervene on the expected surge
risk, often based on a direct comparison between observed residuals and the
forecast non-tidal residual. Where, for example, such a comparison may lead
to the observed residual falling outside the bounds of an ensemble of
forecast non-tidal residuals, forecasters may significantly (and
potentially incorrectly) reduce their confidence in the model's estimate of
surge if they are unaware of the additional errors associated with the
harmonic tide and whether or not they have been addressed within the ensemble
forecast's post-processing system. For comparison, across the UK tide-gauge
network, short-range ensemble forecast RMS spread is of the order of 5–10 cm

The atmospheric tide at

The estimates of the Highest and Lowest Astronomical Tide are influenced by
radiational tides. HAT and LAT are most readily calculated by inspecting long
time series of predicted tides, and if observation-based, these predictions
will include weather-related components. In most places globally this results
in HAT being calculated as higher than the strictly astronomical component
and LAT being lower; however, the opposite is true in the UK. The effects are
of the order of

For many practical purposes it is correct to include predictable seasonal and daily weather-related cycles in the HAT and LAT. However, the separate effects should be understood, as the radiational constituents may be subject to changing weather patterns due to climate change. It is also important not to double-count weather effects if HAT or LAT is used in combination with surge for estimating return-period water levels.

These considerations about HAT would also apply (proportionally less) to other key metrics such as mean high water.

The tidal constituents along the coast, used in the plotting of Figs. 4, 5a and 6a, are provided as a Supplement. For the gridded model results, please contact the authors.

Sites used for analysis showing the order
of coastal points (red to blue points shown above correspond to top to bottom
in Figs.

The coastal points in the model output are spaced roughly every 80 km and
also wherever a tide gauge is situated, according to the GESLA data set

The algorithm for coastal order is as follows.

Define a single global coastline polygon.

This is done using the GSHHG (Global Self-consistent, Hierarchical,
High-resolution Geography) data set

This technique has the benefit of tending to group island chains together in a consistent order. It cannot produce crossing edges. Because polygons are added in distance order, islands near continents are added to their neighbouring coast, and remote mid-ocean islands tend to be clustered and attached to the nearest continent. The coasts of the Pacific, Atlantic and Indian, and Arctic Ocean are all treated clockwise. Antarctica is attached across the Drake Passage and ordered westward. Nearby locations across narrow islands (particularly Sumatra), isthmuses (Panama), and straits (Gibraltar) may be widely separated in the order. But neighbouring points in the order can be expected to have fairly smoothly varying oceanography, with the “bridges” often, although not necessarily, approximating shoals.

As a final step we adjust the starting point of

Rank the coastal points according to the nearest point on the global polygon.

Having defined this coastal order, we can apply it to any coastal data
set, for example tide gauges. We number the vertices

A further advantage here is that having defined the coastal polygon, the same order can be applied to different data sets and models, leading to closely comparable along-coast plots.

Table

Tidal harmonic constituents referred to in this
paper and the maximum change constituents fitted to GTSM tide only
(

Continued.

JW carried out the model runs and post-processing using MIA's recent developments to the GTSM code and global grid. AS advised on Met Office procedures. JW prepared the paper with contributions from all co-authors.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Developments in the science and history of tides (OS/ACP/HGSS/NPG/SE inter-journal SI)”. It is not associated with a conference.

We are grateful for funding from the EU under the Atlantos project, Horizon
2020 grant no. 633211, from the Met Office, and from NERC National
Capability. Some of the results in this paper first appeared as an internal
National Oceanography Centre report