the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Radiational tides: their doublecounting in storm surge forecasts and contribution to the Highest Astronomical Tide
Maialen Irazoqui Apecechea
Andrew Saulter
Kevin J. Horsburgh
Tide predictions based on tidegauge observations are not just the astronomical tides; they also contain radiational tides – periodic sealevel changes due to atmospheric conditions and solar forcing. This poses a problem of doublecounting 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 DelftFM to investigate this in the UK and elsewhere, quantifying the weatherrelated tides that may be doublecounted in operational forecasts. We show that the global S_{2} atmospheric tide is captured by the tideandsurge model and observe changes in other major constituents, including M_{2}. The Lowest and Highest Astronomical Tide levels, used in navigation datums and design heights, are derived from tide predictions based on observations. We use our findings on radiational tides to quantify the extent to which these levels may contain weatherrelated components.
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The operational forecast in several countries of storm surge stillwater levels is based on a combination of a harmonic tidal prediction and a modelderived forecast of the meteorologically induced storm surge component. The forecast is based on the “nontidal residual”, the difference of two model runs with and without weather effects. This is linearly added to the “astronomical prediction” derived from local tidegauge harmonics (Flowerdew et al., 2010). 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 DCSMv6 made it unnecessary (Zijl et al., 2013). It is still in use operationally in the extratropical US, where results of the SLOSH surge model are added to local tidal predictions (National Weather Service, 2018). It is used similarly in Germany with the BSHsmod model (BSH, 2018) and is also used in the new aggregate sealevel forecasting under evaluation in Australia, which also incorporates sealevel anomalies from a global baroclinic model (Taylor and Brassington, 2017).
There are several possible sources of error in this procedure. The purpose of the combined tideandsurge model is to capture the welldocumented nonlinear interactions of the tide and surge. (Proudman, 1955). Yet the forecasting procedure assumes that the nontidal residual may be added linearly to a gaugebased tide prediction. There is also an assumption that the tideonly 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 doublecounting of radiational (weatherrelated) tidal constituents.
In Sect. 2, we show that the doublecounting of radiational tides has a potential contribution to forecasting error not just on long timescales (through S_{a}, S_{sa}) but also on a fortnightly cycle due to variations in S_{2} and in the phase of M_{2}. We also show that the assumption of nonlinearity may introduce errors if phase predictions disagree between model and observations.
Specific radiational tides have been studied using response analysis, for example the solardiurnal S_{1} by Ray and Egbert (2004) and semidiurnal S_{2} by Dobslaw and Thomas (2005). In Sect. 3 we look at more constituents and demonstrate that the atmospheric tide at S_{2} 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 weatherrelated 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 doublecounting 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 O'Neill and Saulter, 2017) in tideandsurge mode forced by an ensemble of wind and pressure from the current weather forecast to give time series M_{s}(x,t) at each location x. Also run the shelf model in tideonly mode to get M_{t}(x,t). Get the residual from these models, ${M}_{r}={M}_{\mathrm{s}}{M}_{\mathrm{t}}$.

At individual tidegauge locations, derive a tide harmonic prediction $\stackrel{\mathrm{\u0303}}{G}({\mathit{x}}_{\mathrm{g}},t)$ based on past records. This is assumed to be more accurate locally than the model tide.

Forecast the total water level F at each location as model residual plus gauge harmonic prediction, $F({\mathit{x}}_{\mathrm{g}},t)={M}_{r}({\mathit{x}}_{\mathrm{g}},t)+\stackrel{\mathrm{\u0303}}{G}({\mathit{x}}_{\mathrm{g}},t)$.

Finally, it has been proposed (Hibbert et al., 2015) that the forecast could apply various “empirical corrections” to nudge the forecast towards the observed level G based on the mismatch of the peak tide over the last few days. However, no formal correction schemes have been implemented.
2.1 Tideandsurge model
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 DelftFM (Flexible Mesh) (Irazoqui Apecechea et al., 2018; Verlaan et al., 2015). The version used in this paper has a resolution from around 50 km in the open ocean to around 5 km at the coast. We ran the model in two modes: tide only (M_{t}) and tide and surge (M_{s}). The atmospheric forcing used was the ECMWF (European Centre for MediumRange Weather Forecasts) ERAInterim 6hourly reanalysis (Dee et al., 2011) downloaded at 0.25^{∘} resolution but from a spherical harmonic equivalent to $\sim \mathrm{0.75}{}^{\circ}$. Validation of the major tidal coefficients has been favourable, and although the model underpredicts the effect of tropical cyclones due to coarse temporal and spatial resolution in the weather reanalysis, most surge events are captured. We make the assumption that tropical cyclones at any given location are sufficiently rare that the tidal coefficients fitted over a year should not be very different if those surges are underestimated. Due to limitations of data storage and postprocessing, the output from the model was only saved at high frequency at all grid points for 1 month (January 2012) and a subset of coastal points for the year 2013. All runs were preceded by 11 days of spinup.
2.2 Harmonic analysis and selection of tidal constituents
Harmonic analysis (Pugh and Woodworth, 2014) gives a tidal prediction $\stackrel{\mathrm{\u0303}}{G}$ as
where Z_{0} is the mean of the gauge data, and the amplitudes A_{n} and phases g_{n} are associated with the tidal constituents with astronomically determined frequencies σ_{n}. f_{n}(t) and u_{n}(t) are nodal modulations to amplitude and phase applied in order to allow for the 18.61year nodal cycle and 8.85year longitude of the lunar perigee cycle. V_{n} represents the phases of the equilibrium tide, which we take as for Greenwich, using UTC for all times. Throughout this paper an overhead tilde indicates “time series derived from harmonics”, as the shape is reminiscent of a sine wave.
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 B1. To derive harmonics from the global model from only 1 month of data, we use 26 independent primary constituents and a further 8 related constituents. We will use ${\stackrel{\mathrm{\u0303}}{M}}_{\mathrm{s}}$ and ${\stackrel{\mathrm{\u0303}}{M}}_{\mathrm{t}}$ to indicate harmonic prediction time series from the tideandsurge model and tide model respectively.
2.3 Quantifying the effect on forecast of doublecounting radiational tides
A significant source of error for this method is that a tide gauge is measuring the total water level, and hence the harmonic prediction $\stackrel{\mathrm{\u0303}}{G}$ includes all wave, steric, river levels, and surge effects. This is therefore not a prediction of the astronomical tide alone. Steric, wave, and river effects are omitted by the barotropic model, but M_{s} does include periodic radiational effects, which may be doublecounted. We can test a minimum effect of this doublecounting purely within the model by using ${\stackrel{\mathrm{\u0303}}{M}}_{\mathrm{s}}$, the harmonic prediction of the model including surge, as a proxy for the harmonics of the observations at gauges. Then the forecast procedure can be estimated as ${M}_{r}+{\stackrel{\mathrm{\u0303}}{M}}_{\mathrm{s}}$.
To estimate Δ, the error in this model forecast, we can once again use the model, assuming M_{s}≈G. Hence $\mathrm{\Delta}={M}_{\mathrm{s}}({M}_{r}+{\stackrel{\mathrm{\u0303}}{M}}_{\mathrm{s}})={M}_{\mathrm{t}}{\stackrel{\mathrm{\u0303}}{M}}_{\mathrm{s}}$. That is, the minimum error from the current forecast procedure is equal to the error in the harmonic prediction from the model including surge at estimating the tideonly model; Fig. 1a. There are several striking features here, including annual cycles peaking around March in the Arctic, January in South East Asia, and June in Europe. Fortnightly cycles occur almost everywhere, with amplitudes of several centimetres. We will examine the causes of these below.
If it were possible to avoid the doublecounting and provide astronomical tidal harmonics for the observations, the prediction would instead be equivalent to ${M}_{r}+{\stackrel{\mathrm{\u0303}}{M}}_{\mathrm{t}}$ and the error would become $\mathrm{\Delta}={M}_{\mathrm{s}}({M}_{r}+{\stackrel{\mathrm{\u0303}}{M}}_{\mathrm{t}})$=${M}_{\mathrm{t}}{\stackrel{\mathrm{\u0303}}{M}}_{\mathrm{t}}$, as shown in Fig. 1b. Since we are using the model as a proxy for observations, if the harmonic prediction were an exact reproduction of the tideonly model then Δ=0. In practice Δ<5 cm at most UK sites and the monthly cycle has gone, but in the Bristol Channel there is still an error of around 50 cm, indicating that the 62 harmonic constituents are not capturing all of the model tide and further shallowwater constituents may be required. This is consistent with the conclusions of Flowerdew et al. (2010), who found an “average (across UK ports) RMS error (in harmonic prediction of a tideonly run) of 7 cm with a maximum value of 29 cm at Newport, in the Bristol Channel”, using 50 constituents on the CS3X model.
2.4 Fortnightly cycle arising from small changes to S_{2} phase
M_{2} has a period of 12.42 h and S_{2} exactly 12 h. They move in and out of phase with each other twice in a lunar month, resulting in the spring–neap cycle. A small change in phase to the S_{2} harmonic would result in a change of which days it is in phase with M_{2} and hence a substantial change in total tidal amplitude at a given date. For example, near Avonmouth in the Bristol Channel, S_{2} derived from M_{s} has an amplitude 3.5 cm greater than S_{2} derived from M_{t}; however, there is a phase change of around 3.5^{∘}, so the tide arrives 7 min later. Figure 2 shows how this and smaller changes in M_{2} account for differences between ${\stackrel{\mathrm{\u0303}}{M}}_{\mathrm{t}}$ and ${\stackrel{\mathrm{\u0303}}{M}}_{\mathrm{s}}$ of up to 5–8 cm on a fortnightly cycle between these limits. This can account for about half the error in forecasted high water at Avonmouth, which varies between 5 and 20 cm on a fortnightly cycle Byrne et al. (2017). Similar variation in error of the forecast was seen by Flowerdew et al. (2010).
2.5 Quantifying surgeforecasting error due to disregarding nonlinearity
The forecasting approach of the linear addition of a nonlinear model residual to a harmonic prediction, $F={M}_{r}+\stackrel{\mathrm{\u0303}}{G}$, can also cause errors. Disagreements in phase between the model tide M_{t} and harmonic prediction from the gauge $\stackrel{\mathrm{\u0303}}{G}$ affect the forecast of an individual surge event.
Consider a simplified example in which the tide can be modelled by a single constituent, M_{t}=Acos(σt). Suppose there is a storm surge in which there is a uniform additional water level A_{s} and an advancement of the tide of t=δ, so the tideandsurge model is ${M}_{\mathrm{s}}={A}_{\mathrm{s}}+A\mathrm{cos}\left(\mathit{\sigma}\right(t+\mathit{\delta}\left)\right).$ As before, the model residual is given by ${M}_{r}={M}_{\mathrm{s}}{M}_{\mathrm{t}}$.
Suppose the harmonic prediction at the gauge agrees in amplitude to the tideonly model, but has slightly different phase: $\stackrel{\mathrm{\u0303}}{G}=A\mathrm{cos}\left(\mathit{\sigma}\right(t+\mathit{\u03f5}\left)\right).$
The skew surge is defined as the difference between the maximum water level, here max(M_{s}), and $max\left(\stackrel{\mathrm{\u0303}}{G}\right)$. The error in the skew surge forecast is $E=max({M}_{r}+\stackrel{\mathrm{\u0303}}{G})max\left({M}_{\mathrm{s}}\right)$. Substituting in and assuming phase changes are small, we find A_{s} cancels out and can show analytically that
This is illustrated in Fig. 3, with A=3 m, $\mathit{\sigma}=\mathrm{2}\mathit{\pi}/\mathrm{12.42}\phantom{\rule{0.125em}{0ex}}{\mathrm{h}}^{\mathrm{1}}$ (M_{2}), and the surge advancing the tide by δ=30 min. The residual M_{r} is decreasing during high water due to the advanced tide. So if the observed harmonics have high water later than the model (ϵ=5 min), the forecast skew surge is underestimated by 3 cm. If the observed harmonics predict high water earlier than the model ($\mathit{\u03f5}=\mathrm{5}$ min), the forecast skew surge is overestimated by 3 cm.
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 ϵ to have the same sign as δ, as the gauge registers water levels more like the tideandsurge model than the tideonly model and the harmonic predictions would follow suit.
Figure 4 shows the vector difference in individual constituents between tideandsurge and tideonly models run for 2013 along the coast globally. With some exceptions in the Arctic and Antarctic, the effect on S_{a} is around 5–20 cm, with around half that effect on S_{sa}, although in the Indian Ocean there is a change to S_{a} only. Since the model was only run for 1 year, S_{a} may not be representative of all years, but Fig. 4 indicates typical changes. In the Baltic, the seasonal change is wind forced, but elsewhere it is consistent with the annual and semiannual cycles in sealevel atmospheric pressure (Chen et al., 2012).
MS_{f} is affected by the surge component, as a side effect of the interaction between M_{2} and S_{2}. This is because MS_{f} is the fortnightly constituent which arises from the combination of M_{2} and S_{2}, with a speed equal to the difference of their speeds. MS_{4} is the counterpart to this, with a speed equal to the sum of the speeds of M_{2} and S_{2} (Pugh and Woodworth, 2014). Less explicable is the effect on M_{m} and M_{f}, but it may be due to insufficient separation with MS_{f} over a relatively short record. Another possibility is that nontidal power in the tideandsurge model is leaking into M_{m} and M_{f} estimates. Eliminating this would require a manyyear model run.
The diurnal constituents K_{1} and O_{1} are affected by less than 5 cm and are only changed regionally in the Antarctic. S_{1}, however, is everywhere less than 0.1 cm in the tideonly 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 Ray and Egbert (2004).
It may come as a surprise that constituents such as M_{2}, which has a purely lunar frequency, could possibly be affected by the weather. There is a very small atmospheric tide at M_{2}, peaking at the Equator at about 0.1 mbar (Schindelegger and Dobslaw, 2016). But more significant is the nonlinear interaction of surge and tide. The surge may consistently advance the phase of the tide during lowpressure events and certain wind configurations. A highpressure 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 higherorder constituents is everywhere less than 5 cm. The maximum difference in the UK and globally for each constituent is given in Appendix B. In the UK, the constituents affected the most by including the surge are S_{2}, S_{sa}, M_{2}, S_{a}, M_{m}, MS_{4}, MS_{f}, and M_{f}, with a maximum change of >2 cm, and a further 19 constituents change 1–2 cm. Globally, S_{a} and S_{sa} are far more significant, but S_{2}, M_{f}, M_{2}, M_{m}, MS_{f}, S_{1}, K_{1}, K_{2}, O_{1}, M_{A2}, and MS_{4} all change more than 4 cm (somewhere on the global coast). A vector difference of 13 cm in S_{2} is seen in northwest 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.
3.1 S_{2} atmospheric tide
Some of the difference between the harmonics of surge and tideonly models is directly attributable to the atmospheric tides. The global atmospheric pressure field contains S_{2} variations with an amplitude of about 1.25cos^{3}ϕ mbar for latitude ϕ (Pugh and Woodworth, 2014). GTSM air pressure and wind forcing is taken from the ERAInterim data set (Appendix A), and the ocean response to that forcing at S_{2} is contained in the difference between the harmonic predictions of the M_{s} and M_{t} model runs (Fig. 5). It is consistent with response analysis based on the S_{2} tides seen in ECMWF reanalysis data (Dobslaw and Thomas, 2005) and in a twolayer model forced by eight constituents (Arbic, 2005). The 6 h sampling prevents ERAInterim forcing from capturing the S_{2} atmospheric tide correctly (Dobslaw and Thomas, 2005), but the analysis in this paper is selfconsistent with the forcing used.
The Highest Astronomical Tide (HAT) is used internationally for floodforecasting 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 returnperiod water levels. The Lowest Astronomical Tide (LAT) is also an important parameter recommended for use as the datum on navigation charts (IHO, 2017). Once the phases and amplitudes A_{n} and g_{n} are known, $\stackrel{\mathrm{\u0303}}{G}\left(t\right)$ is fully determined by Eq. (1), and the future HAT and LAT are given by $\mathrm{max}\left(\stackrel{\mathrm{\u0303}}{G}\right(t\left)\right)$ and $\mathrm{min}\left(\stackrel{\mathrm{\u0303}}{G}\right(t\left)\right)$. But because of the overlap in phase of the forcing between the constituents and the f_{n} and u_{n} 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 18.6year nodal cycle. Figure 6a 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 tideonly run. Since the GTSM data were limited to 1 month, it uses only 34 constituents, therefore omitting S_{1} and the longperiod contributions to HAT and LAT.
An approximate calculation of range as $\mathrm{2}({\mathbf{M}}_{\mathrm{2}}+{\mathbf{S}}_{\mathrm{2}}+{\mathbf{O}}_{\mathrm{1}}+{\mathbf{K}}_{\mathrm{1}})$ is occasionally used (Yotsukuri et al., 2017), but the error due to this can be over 1 m (Fig. 6b). N_{2} is a significant contributor, at about 20 % of M_{2} in many sites worldwide. A few tens of centimetres are accounted for by the omission of the nodal modulations, and there are also the shallowwater constituents at the coast.
Figure 6c 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 tideandsurge model is used. So the observationbased HAT has been raised by some radiational component. But in most of the UK, the HAT goes down when the tideandsurge model is used to generate the tidal predictions. This is because the peak of the weatherrelated components does not coincide with the maximum astronomical effects alone. This implies that since the tidegauge predictions include surge, the observationbased HAT in the UK is actually about 10 cm lower than true astronomicalonly tidal height.
LAT tends to move the opposite way, so in most places the maximum tidal range is increased by using the tideandsurge model. That is, the true astronomicalonly tidal range is slightly less than that quoted from harmonics based on predictions. In Scotland (just above Liverpool in Fig. 6c) 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 6c 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 tideonly 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 tideonly and tideandsurge model results. Even constituents with purely lunar frequencies, including M_{2}, may be affected by the surge, perhaps owing to asymmetry in phase changes of the tide under high and lowpressure weather systems.
Some effects of the weather on tides are doublecounted in the forecast procedure used in the UK, in which model residuals are added to gaugebased 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 tideonly 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 S_{2} harmonic. Further errors in total water level and skew surge arise directly from the linear addition of the harmonic prediction to the nonlinear 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 nontidal residual. Where, for example, such a comparison may lead to the observed residual falling outside the bounds of an ensemble of forecast nontidal 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 postprocessing system. For comparison, across the UK tidegauge network, shortrange ensemble forecast RMS spread is of the order of 5–10 cm (Flowerdew et al., 2013). It is noted that, in the UK, the majority of coastal flood events occur around peak spring tides (Haigh et al., 2015), for which the sensitivity to any errors in the M_{2}–S_{2} phase relationship is arguably at its highest.
The atmospheric tide at S_{2} is present in the ERAInterim 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 “tideonly” model, and this is being explored by Irazoqui Apecechea et al. (2018). In this case, care would need to be taken to omit the direct atmospheric tide forcing in the tideandsurge version to avoid a different form of doublecounting.
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 observationbased, these predictions will include weatherrelated 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 weatherrelated 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 doublecount weather effects if HAT or LAT is used in combination with surge for estimating returnperiod 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.
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 (Woodworth et al., 2017). Due to automatic procedures to select output sites, a few may be incorrectly sited at model dry sites – these are clearly seen in plots as lacking sufficient highfrequency variability. The alongcoast plots are ordered approximately from west to east around the world coastline, starting and ending at Alaska. The order is indicated in Fig. A1.
The algorithm for coastal order is as follows.

Define a single global coastline polygon.
This is done using the GSHHG (Global Selfconsistent, Hierarchical, Highresolution Geography) data set (Wessel and Smith, 1996) version gshhg2.3.6 (available at: https://www.ngdc.noaa.gov/, last access: 19 August 2016). We use the coarse resolution, with only Level 1 (coastline) and Level 6 (Antarctic Ice Shelf), although consistent results for this technique can be obtained including enclosed lakes. To merge the separate land masses and islands into a weakly simple polygon topologically equivalent to a disc, we start with a single land mass and add others in turn using pairs of identical edges as “bridges”. We start with the main land mass of Eurasia L_{1} and find the closest vertex l to a vertex p from any of the remaining polygons [P_{2},…P_{N}]. Suppose p belongs to polygon P_{j}. Then we add P_{j} to L_{1} using two new edges $\overrightarrow{lp}$ and $\overrightarrow{pl}$ to give a new merged polygon L_{2}. The vertices of L_{2} are then $\left[{L}_{\mathrm{1}}\right(\mathrm{1}:l),{P}_{j}(p:\mathrm{end},\mathrm{1}:p\mathrm{1}),{L}_{\mathrm{1}}(l:\mathrm{end}\left)\right]$. Now repeat, searching for the nearest point in L_{2} to any vertex in the remaining polygons $[{P}_{\mathrm{2}},\mathrm{\dots},{P}_{j\mathrm{1}},{P}_{j+\mathrm{1}},\mathrm{\dots}{P}_{N}]$. It is necessary for all initial polygons to be defined in the same sense (anticlockwise). If inland seas (Level 2) are included, they should be defined clockwise. The GSHHG data are consistent with this definition. The distance for nearest points is defined as arc length on a sphere.
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 midocean 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 L_{2} to be in Alaska for convenience of mapping.

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 $[\mathrm{1},\mathrm{\dots},K]$. For each of the gauge locations T we find the nearest vertex k and then rank the gauges according to T_{k}. In the event of gauges being much closer than the resolution of the vertices, a quick method for refinement is to linearly interpolate with extra vertices along polygon edges. Some problems may also occur with islands not in the coarseresolution data, which will tend to jump to the nearest coast.
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 alongcoast plots.
Table B1 lists the constituents used in this paper. For the 1month model run, related constituents are used, and we fit 34 constituents with only 26 independent terms. 62 constituents are used for the 1year run. The list of 115 usually applied to 18.6year data is used only as a check on the stability of the result in Sect. 3 and is provided in the Supplement.
JW carried out the model runs and postprocessing 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 coauthors.
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 interjournal 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 (Williams et al., 2018). We thank Martin
Verlaan of Deltares and Clare O'Neill for model development work and Phil
Woodworth, Richard Ray, and two anonymous reviewers for helpful suggestions
during the final preparation of the paper.
Edited by: Richard Ray
Reviewed by: two anonymous referees
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 Abstract
 Introduction
 Surge forecasting
 The difference of specific harmonics
 Highest Astronomical Tide and Lowest Astronomical Tide
 Conclusions
 Data availability
 Appendix A: Ordering of model sites around the coast
 Appendix B: Tidal constituents
 Author contributions
 Competing interests
 Special issue statement
 Acknowledgements
 References
 Supplement
 Abstract
 Introduction
 Surge forecasting
 The difference of specific harmonics
 Highest Astronomical Tide and Lowest Astronomical Tide
 Conclusions
 Data availability
 Appendix A: Ordering of model sites around the coast
 Appendix B: Tidal constituents
 Author contributions
 Competing interests
 Special issue statement
 Acknowledgements
 References
 Supplement