Storm surge forecasting: quantifying errors arising from the double-counting of radiational tides

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 as 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 5 use the Global Tide and Surge Model based on Delft-FM to investigate this in the UK and elsewhere, quantifying the weatherrelated tides that may be double-counted in operational forecasts. We show that the global S2 atmospheric tide is captured by the tide-surge model, and observe changes in other key constituents, including M2. We also quantify the extent to which the "Highest Astronomical Tide", which is derived from tide predictions based on observations, may contain weather-related components. 10


Introduction
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.In the UK, 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 (Flowerdew et al., 2010).This approach is taken because the complexity and large range of the UK 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 extra-tropical US, where results of the SLOSH surge model are added to local tidal predictions (National Weather Service, 2018); similarly in Germany, using the BSHsmod model (BSH, 2018); 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 (Taylor and Brassington, 2017).
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.Proudman, 1955).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 1 Ocean Sci.Discuss., https://doi.org/10.5194/os-2018-63Manuscript under review for journal Ocean Sci. Discussion started: 30 May 2018 c Author(s) 2018.CC BY 4.0 License.tide-only model and the harmonic prediction from the tide gauge are equivalent.In fact, the harmonics at the gauge will also be affected by the weather.There is therefore the potential for double-counting of radiational (weather-related) tidal constituents.Specific radiational tides have been studied using response analysis, for example the solar-diurnal S 1 by Ray and Egbert (2004), and semi-diurnal S 2 by Dobslaw and Thomas (2005).Here we use the Global Tide and Surge Model (GTSM) based on Delft-FM, to compare many constituents and their combined effect globally.We find that the double counting of radiational tides has a potential contribution not just on long time scales (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 demonstrate the atmospheric tide at S 2 may be observed in the GTSM model.We also show that the assumption of non-linearity may introduce errors if phase predictions disagree between model and observations.We also quantify the extent to which the Highest and Lowest Astronomical Tide (HAT and LAT) are influenced by weatherrelated tides, and show that in many places several cm of what is reported as HAT is attributable to periodic weather patterns.

Surge forecasting
The harmonic analysis function (Pugh and Woodworth, 2014, Chapter 4) gives the tide prediction H 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 adjustments to amplitude and phase, applied in order to allow for the 18.61 year nodal cycle and 8.85 year longitude of lunar perigee cycle.V n are the phases of the equilibrium tide, which we take as for Greenwich.We use UTC for all times to enable consistency between local gauges and global maps.
The current procedure for forecasting total water level in the UK is as follows: 1. Run a barotropic shelf model (CS3X, currently transitioning to NEMO Surge (O'Neill and Saulter, 2017)) in surge-andtide mode, forced by an ensemble of wind and pressure from the current weather forecast to give timeseries M s (x, t) at each location x.Also run the shelf model in tide-only mode, to get M t (x, t).Get the residual from these models, 2. Find the amplitude A n and phases g n of harmonics at each individual gauge location from past tide records.Derive a tide prediction H g (x g , t) from the gauge harmonics.
3. Forecast the total water level W f as model residual plus gauge harmonic prediction, W f (x g , t) = M r (x g , t)+H g (x g , t).
4. 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 W g based on the mismatch of the peak tide over the last few days.

Selection of tidal constituents
The choice of tidal constituents used for the harmonic analysis varies according to the length of data available.We use 62 harmonics where there is one year's data, 115 for more than one year, as listed in table C1. global model (discussed below), from only 1 month's data, we use 26 independent primary constituents, and a further 8 related constituents.
2.2 Quantifying the effect on forecast of double-counting radiational tides A significant source of error for this method is that a gauge is measuring the total water level, and hence the harmonic prediction H g includes all wave, steric and surge effects.This is not therefore a prediction of the astronomical tide alone.Steric and wave effects are omitted by the model, but M s does include periodic radiational effects, which may be double-counted.We can test a minimum effect of this double-counting purely within the model by using H 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 + H s .
To estimate the error in this model forecast we can once again use the model, assuming 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 tide-only model, figure 1(top).There are several striking features here, 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 cm.We will examine the causes of these below.
If it were possible to avoid the double-counting, and provide astronomical tidal harmonics H t for the observations, this would instead be equivalent to M r + H t , and the error would become M s − (M r + H t )=M t − H t , as shown in figure 1(bottom).Since we're using the model as proxy for observations, if the harmonic prediction is an exact reproduction of the tide-only model then this would be exact.It is less than 5 cm at most UK sites and the monthly cycle has gone, but in the Bristol Channel there is still an error of up to around 0.5 m, indicating that the selection of 62 harmonic constituents are not capturing all of the model tide.This is consistent with Flowerdew et al. (2010) who found an "average [across UK ports] rms error of 7 cm with a maximum value of 29 cm at Newport, in the Bristol Channel", using 50 constituents on the C3SX model.Figure 2 shows the total change in amplitude at Avonmouth varies between the tide-only and surge models by up to 5-8 cm (on a fortnightly cycle between these limits) from this effect alone.This cycle could account for that seen by Byrne et al. (2017), and that in figure 2 of Flowerdew et al. (2010).

Quantifying surge-forecasting error due to disregarding non-linearity
The approach of linear addition of the harmonic prediction to the non-linear residual, W f = M r + H g , itself carries a risk of error, even if the harmonic prediction did not include any radiational forcing.Consider the following simplified example.Suppose the model tide has an M 2 amplitude of 3 m (ignore all other constituents), and there is a surge which has the effect of adding a constant amplitude 0.2 m and advancing the tide by a constant 30 min.Suppose also that the harmonics of the observed tide have the same amplitude as the tide-only model, but are out of 3 The difference of specific harmonics Figure 4 shows the vector difference in individual constituents between surge and tide-only models run for 2013, along the coast 5 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 there is an annual effect only in South-East Asia.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 effect on M m and M f , but it may be due to 10 insufficient separation with MS f over a relatively short record.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 1 mm in the tide-only model, but with the surge model peaks at 5 mm in northern Australis, the broadest regional effect being has 2-3 mm, in South-East Asia, consistent with the findings of (Ray and Egbert, 2004).C1.See appendix B for explanation of coastal axis.
It may come as a surprise that constituents such as M 2 , which has a purely lunar frequency, could possibly be effected by the weather.But this is where the non-linear interaction of surge and tide comes into play.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 The Highest Astronomical Tide (HAT) is used internationally for flood-forecasting references levels and in navigation for clearance under bridges.HAT can be used in structural design with skew surge as an independent variable for determining return period water levels.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, H(t) is fully determined for all time by equation ( 1), and the future HAT and LAT are given by max(H(t)) and min(H(t)).But because of the overlap in phase of the forcing between the constituents, and the f n and u n nodal adjustments, 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 6a shows the range, HAT minus LAT, when we do this by synthesising a predicted tide at 15 minute intervals over 18.6 years, globally.
The quick calculation of Range = 2(M 2 +S 2 +O 1 +K 1 ) is occasionally used (e.g.Yotsukuri et al., 2017), but the error due to this can be over 1 metre (figure 6b), and although in the open ocean most of this is due to omission of the nodal tide, near the coast up to 0.5 m of the maximum range can be due to the shallow water constituents.Radiational effects are omitted from figure 6a, which is based on a tide-only run.Also, this analysis was limited to 1 month, so uses only 34 constituents, therefore omitting S 1 and the long-period contributions to HAT and LAT.
Figure 6c shows the effect on HAT and LAT using the constituents derived with surge in the GTSM.In most of the UK, the HAT goes down when the surge model is used to generate the tidal predictions, rather than going up.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.But in many places round the world the HAT is higher when the surge model is used.So the observation-based HAT actually has been raised by some radiational component.
LAT tends to move the opposite way, so in most places the maximum tidal range is increased by using the tide+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 figure 6c) both LAT and HAT go down when the surge model is used to generate the tidal predictions, so the quoted LAT and HAT is actually about 10cm 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 understood 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's about a 7cm increase in HAT and 3cm 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.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 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, the forecaster 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-wide set of Class A ports short range ensemble forecast RMS spread is of order 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), where 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 ERA Interim forcing, and the ocean response to it, with amplitude about 1-5 cm, can be seen in the difference between the model results with and without surge.There is therefore an argument for including an atmospheric tide forcing in a "tide-only" model, and this is being explored by (Irazoqui Apecechea et al., 2018 (In review)).In this case, care would need to be taken to omit the direct atmospheric tide forcing in the tide+surge version, to avoid a different form of double-counting.
The estimates of 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 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 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 are used in combinations 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.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.
2) 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 [1, ..., K].For each of the tide 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 coarse resolution 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

Appendix C: Tidal constituents
Table C1 lists the constituents used in this paper.For short records, related constituents are used, and we fit 34 constituents with only 26 independent terms.We follow the usual convention of a for annual, f fortnightly, 2 for approximately semi-diurnal etc., which are given as subscripts in the main text.

Figure 1 .
Figure 1.Time series of error (m) in harmonic prediction with 62 constituents (top panel) including surge and (bottom panel) tide-only at estimating the tide-only model, GTSM 2013 only.See appendix B for explanation of vertical coastal axis.

phase by 5 Figure 3 .
Figure 3.A surge is imposed which adds a constant amplitude of 20 cm and advances the tide by a constant 30 min.If the harmonics of the observations differ in phase by 5 min from the model a forecast error of ±3 cm will result as shown.Lower panels are magnified to show the high water.

Figure 4 .
Figure 4. Vector difference (m, offset) between coefficients fitted to model including surge or tide only, GTSM 2013 only, 62 constituents fitted.This is is the breakdown into constituents of the the difference between panels of figure 1.The maximum effect for these harmonics and others are given in tableC1.See appendix B for explanation of coastal axis.

5Figure 5 .Figure 6 .
Figure 5. Amplitude (m) of S2 difference between between coefficients fitted to GTSM model including surge or tide only.First panel: coastal data only, whole of 2013; second panel: 26 primary coefficients fitted to January 2012 only.
Ocean Sci.Discuss., https://doi.org/10.5194/os-2018-63Manuscript under review for journal Ocean Sci. Discussion started: 30 May 2018 c Author(s) 2018.CC BY 4.0 License.There are substantial changes in tidal constituents fitted to tide-only and tide-and-surge 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 low pressure weather systems.Some effects of the weather on tides are double-counted in the forecast procedure used in the UK, where 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 Ocean Sci.Discuss., https://doi.org/10.5194/os-2018-63Manuscript under review for journal Ocean Sci. Discussion started: 30 May 2018 c Author(s) 2018.CC BY 4.0 License.

Figure B1 .
Figure B1.Sites used for analysis and coastal ordering (Red to Blue is top-to-bottom of other plots)

Table C1 .
Tidal Harmonic Constituents referred to in this paper, and the maximum change between models run tide-only or with surge forcing, at coastal locations, as from figure 4.