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 . This approach is taken in the UK because the complexity and large range of the tides is such that it has historically been difficult to model them to sufficient accuracy. The same method was applied in the Netherlands until 2015 when improvements to the local surge model DCSM-v6 made it unnecessary . It is still in use operationally in the extratropical US, where results of the SLOSH surge model are added to local tidal predictions . It is used similarly in Germany with the BSHsmod model and is also used in the new aggregate sea-level forecasting under evaluation in Australia, which also incorporates sea-level anomalies from a global baroclinic model .

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. e.g.. Yet the forecasting procedure assumes that the non-tidal residual may be added linearly to a gauge-based tide prediction. There is also an assumption that the tide-only model and the harmonic prediction from the gauge are equivalent. In fact, the harmonics at the gauge will also be affected by the weather, so there is the potential for double-counting of radiational (weather-related) tidal constituents.

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 Sa, Ssa) but also on a fortnightly cycle due to variations in S2 and in the phase of M2. We also show that the assumption of non-linearity may introduce errors if phase predictions disagree between model and observations.

Specific radiational tides have been studied using response analysis, for example the solar-diurnal S1 by and semi-diurnal S2 by . In Sect. 3 we look at more constituents and demonstrate that the atmospheric tide at S2 may be observed in the GTSM.

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 ) in tide-and-surge mode forced by an ensemble of wind and pressure from the current weather forecast to give time series Ms(x,t) at each location x. Also run the shelf model in tide-only mode to get Mt(x,t). Get the residual from these models, Mr=Ms-Mt.

Figure  shows the vector difference in individual constituents between tide-and-surge and tide-only models run for 2013 along the coast globally. With some exceptions in the Arctic and Antarctic, the effect on Sa is around 5–20 cm, with around half that effect on Ssa, although in the Indian Ocean there is a change to Sa only. Since the model was only run for 1 year, Sa may not be representative of all years, but Fig.  indicates typical changes. In the Baltic, the seasonal change is wind forced, but elsewhere it is consistent with the annual and semi-annual cycles in sea-level atmospheric pressure .

MSf is affected by the surge component, as a side effect of the interaction between M2 and S2. This is because MSf is the fortnightly constituent which arises from the combination of M2 and S2, with a speed equal to the difference of their speeds. MS4 is the counterpart to this, with a speed equal to the sum of the speeds of M2 and S2 . Less explicable is the effect on Mm and Mf, but it may be due to insufficient separation with MSf over a relatively short record. Another possibility is that non-tidal power in the tide-and-surge model is leaking into Mm and Mf estimates. Eliminating this would require a many-year model run.

The diurnal constituents K1 and O1 are affected by less than 5 cm and are only changed regionally in the Antarctic. S1, however, is everywhere less than 0.1 cm in the tide-only model, but with the surge model peaks at 0.5 cm in northern Australia, the broadest regional effect being 0.2–0.3 cm in South East Asia, consistent with the findings of .

It may come as a surprise that constituents such as M2, which has a purely lunar frequency, could possibly be affected by the weather. There is a very small atmospheric tide at M2, peaking at the Equator at about 0.1 mbar . But more significant is the non-linear interaction of surge and tide. The surge may consistently advance the phase of the tide during low-pressure events and certain wind configurations. A high-pressure system could delay the phase of the tide, but there is asymmetry between these events, so there is a net bias on the phase when the weather is included.

The effect on higher-order constituents is everywhere less than 5 cm. The maximum difference in the UK and globally for each constituent is given in Appendix . In the UK, the constituents affected the most by including the surge are S2, Ssa, M2, Sa, Mm, MS4, MSf, and Mf, with a maximum change of >2 cm, and a further 19 constituents change 1–2 cm. Globally, Sa and Ssa are far more significant, but S2, Mf, M2, Mm, MSf, S1, K1, K2, O1, MA2, and MS4 all change more than 4 cm (somewhere on the global coast). A vector difference of 13 cm in S2 is seen in north-west Australia.

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.

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 . Once the phases and amplitudes An and gn are known, G̃(t) is fully determined by Eq. (), and the future HAT and LAT are given by max(G̃(t)) and min(G̃(t)). But because of the overlap in phase of the forcing between the constituents and the fn and un nodal modulations, it is not trivial to write HAT or LAT algebraically. They are therefore determined by inspection of the predicted tides, preferably over a 18.6-year nodal cycle. Figure a shows the range, HAT minus LAT, when we do this by synthesising a predicted tide at 15 min intervals over 18.6 years globally. Radiational effects are omitted from this figure, which is based on a tide-only run. Since the GTSM data were limited to 1 month, it uses only 34 constituents, therefore omitting S1 and the long-period contributions to HAT and LAT.

An approximate calculation of range as 2(M2+S2+O1+K1) is occasionally used e.g., but the error due to this can be over 1 m (Fig. b). N2 is a significant contributor, at about 20 % of M2 in many sites worldwide. A few tens of centimetres are accounted for by the omission of the nodal modulations, and there are also the shallow-water constituents at the coast.

Figure c shows the effect on HAT and LAT of including surge in the GTSM. Coastal locations are shown and 62 constituents used. In many places around the world the HAT is higher when the tide-and-surge model is used. So the observation-based HAT has been raised by some radiational component. But in most of the UK, the HAT goes down when the tide-and-surge model is used to generate the tidal predictions. This is because the peak of the weather-related components does not coincide with the maximum astronomical effects alone. This implies that since the tide-gauge predictions include surge, the observation-based HAT in the UK is actually about 10 cm lower than true astronomical-only tidal height.

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. c) both LAT and HAT go down when the surge model is used to generate the tidal predictions, so the quoted LAT and HAT are actually about 10 cm lower than astronomical only.

The most extreme changes shown in Figure c are in the Arctic and Antarctic and should be interpreted with some caution as these areas are the least well represented in the model.

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 M2, may be affected by the surge, perhaps owing to asymmetry in phase changes of the tide under high- and low-pressure weather systems.

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 S2 harmonic. Further errors in total water level and skew surge arise directly from the linear addition of the harmonic prediction to the non-linear residual, particularly where there is a phase difference between model and gauge tidal harmonics.

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 . It is noted that, in the UK, the majority of coastal flood events occur around peak spring tides , for which the sensitivity to any errors in the M2–S2 phase relationship is arguably at its highest.

The atmospheric tide at S2 is present in the ERA-Interim forcing, and the ocean response to it, with an amplitude of about 1–5 cm, can be seen in the difference between the model results with and without surge. There is hence an argument for including an atmospheric tide forcing in a “tide-only” model, and this is being explored by . In this case, care would need to be taken to omit the direct atmospheric tide forcing in the tide-and-surge version to avoid a different form of double-counting.

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 ∼10 cm.

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.