<p>You are looking for an easy solution to the local maxima of
<span class="math-container">$$\sum_i a_i \sin(\theta_i+\omega_i t)$$</span>
This means solving for solutions to
<span class="math-container">$$\sum_i a_i \omega_i \cos(\theta_i+\omega_i t) = 0$$</span>
This is a transcendental equation with multiple terms with different frequencies. While there are many ways to find such maxima that vary in computational complexity and computational cost, there typically is no nice and easy way to find such maxima.</p>

CC BY-SA 4.0
2020-07-30T02:30:50.930Z
When you talk about "ways to find [local] maxima", are you talking about numerical solutions like Newton's method etc?
naught101
CC BY-SA 4.0
2020-07-30T01:05:41.203Z
2020-07-30T01:05:41.203Z
David Hammen
<p>It depends on what you are trying to achieve.</p>
<p>If you need exact figures for the maximum tidal height each day/year at a given location, there are no simple analytical solutions due to the complex nature of tides.</p>
<p>If you are interested in ballpark figures (which are good enough for most recreational, but also some engineering applications), the maximum annual tidal height can be approximated by tidal datums, either the Mean High Water Springs (MHWS) or Mean Higher High Water (MHHW) depending on the tidal regime. If you would want to be extremely conservative, you could use Highest Astronomical Tide (HAT). Also taking the mean of MHWS/MHHW and HAT could be an option. This approximation would be suitable for 'regular' coastal and offshore locations. Complex bathymetry (e.g. funnel-shaped estuary) is trickier.</p>
<p>This could potentially help you save some of the computational effort, as you would need to calculate the tidal datums once per location (rather than once per year per location).</p>

CC BY-SA 4.0
2020-07-27T14:52:50.087Z
Clarified the question: only interested in astronomical tides (local bathymetric effects would be nice, but not essential). I'm interested in making a dataset that contains distribution parameters for annual maximum high tide for many locations around the world. So I only need that one number per year. HAT is not useful, because it only occurs every few of decades. I guess I could maybe infer the distribution from MHWS/MHHW and HAT though? But then, I'd also need to know those values, and I'm not sure if they're calculable analytically either.
naught101
CC BY-SA 4.0
2020-07-27T18:16:00.530Z
You can calculate [tidal constituents/harmonics](https://en.wikipedia.org/wiki/Theory_of_tides#Tidal_constituents) from water level time series. Check ([t_tide](http://www.omg.unb.ca/GGE/5013_LABS/T_Tide_CompAndGeo.pdf)). Also the software you linked in your initial post (OTPS) seems to provide tidal constituents from gridded model output. To calculate MHWS etc, see formulas on page 6 of [this R package documentation](https://cran.r-project.org/web/packages/TideHarmonics/vignettes/austides.pdf) (sorry for the weird reference). This is a brute force method though :D.
Ingvar Lukas
CC BY-SA 4.0
2020-07-28T00:50:27.363Z
Yeah, OK. But even then, I'd still have to figure out how to infer a maximum annual tide height distribution from those, and that seems pretty non-obvious. I guess I'd have to do it empirically/numerically anyway. Seems like my best bet is really going to be to DIY..
naught101
CC BY-SA 4.0
2020-07-27T13:38:07.927Z
2020-07-27T13:38:07.927Z
Ingvar Lukas
<p>It's possible to predict tide at a given location and at a given time reasonably well (e.g. <a href="https://earthscience.stackexchange.com/questions/15313/predicting-low-tide">predicting low tide</a>). And there are tools available that make it possible to do this calculation for any location on earth at any time, such as <a href="https://www.tpxo.net/otps" rel="nofollow noreferrer">OTPS</a>.</p>
<p>Given that, it's possible calculate the annual maximum tide by calculating (sub)hourly data, and then taking the maximum. However, this is a pretty expensive way to do things - you're calculating 8.7k heights just to calculate a single maximum, and the calculations themselves are not particularly cheap. This quickly becomes a problem if you want to calculate a lot of locations.</p>
<p>Annual maximum tide is not the same year to year, and from memory, the tides have a long-term cyclic nature of around 18 years (that is the lowest common multiple of the periods of all of the cyclic components).</p>
<p><strong>Is there a way of either analytically calculate the maximum tide height on a given day, or for a given year; or to calculate/estimate the <em>time</em> of that maximum (from which it would be possible to calculate the height)?</strong></p>
<p>I'm only interested in the long-term predictable component of tides - astronomical tides (and perhaps other things, like local orographic effects?). Not interested in storm surge or wave effects, or other things that are hard to predict in the long term.</p>

CC BY-SA 4.0
2020-07-28T10:28:23.307Z
You are looking for an easy solution to the local maxima of $$\sum_i a_i \sin(\theta_i + \omega_i t)$$ While there are many ways to find  such maxima that vary in computational complexity and computational cost, there is no nice and easy way to find such maxima.
David Hammen
CC BY-SA 4.0
2020-07-29T23:54:11.000Z
@DavidHammen: While the other answers are interesting and potentially useful, I guess your comment is the actually correct answer to the question. Post it as an answer, and I'll accept it.
naught101
<tides><predictability><spectral-analysis>
Is it possible to predict the maximum daily/annual tide height without calculating hourly heights?
naught101
# Is it possible to predict the maximum daily/annual tide height without calculating hourly heights?

(Asked by: username_0 on 2020-07-27)

It's possible to predict tide at a given location and at a given time reasonably well (e.g. [predicting low tide](https://earthscience.stackexchange.com/questions/15313/predicting-low-tide)). And there are tools available that make it possible to do this calculation for any location on earth at any time, such as [OTPS](https://www.tpxo.net/otps).

Given that, it's possible calculate the annual maximum tide by calculating (sub)hourly data, and then taking the maximum. However, this is a pretty expensive way to do things - you're calculating 8.7k heights just to calculate a single maximum, and the calculations themselves are not particularly cheap. This quickly becomes a problem if you want to calculate a lot of locations.

Annual maximum tide is not the same year to year, and from memory, the tides have a long-term cyclic nature of around 18 years (that is the lowest common multiple of the periods of all of the cyclic components).

**Is there a way of either analytically calculate the maximum tide height on a given day, or for a given year; or to calculate/estimate the *time* of that maximum (from which it would be possible to calculate the height)?**

I'm only interested in the long-term predictable component of tides - astronomical tides (and perhaps other things, like local orographic effects?). Not interested in storm surge or wave effects, or other things that are hard to predict in the long term.

**Tags:** tides, predictability, spectral-analysis

> **Comments:**
> **username_1**: You are looking for an easy solution to the local maxima of $$\sum_i a_i \sin(\theta_i + \omega_i t)$$ While there are many ways to find  such maxima that vary in computational complexity and computational cost, there is no nice and easy way to find such maxima.  
> **username_0**: @username_1: While the other answers are interesting and potentially useful, I guess your comment is the actually correct answer to the question. Post it as an answer, and I'll accept it.  

---

## 2 answers

### Answer by username_1 on 2020-07-30. Score: 2 | Accepted answer

You are looking for an easy solution to the local maxima of
$$\sum\_i a\_i \sin(\theta\_i+\omega\_i t)$$
This means solving for solutions to
$$\sum\_i a\_i \omega\_i \cos(\theta\_i+\omega\_i t) = 0$$
This is a transcendental equation with multiple terms with different frequencies. While there are many ways to find such maxima that vary in computational complexity and computational cost, there typically is no nice and easy way to find such maxima.

> **Comments:**
> **username_0**: When you talk about "ways to find [local] maxima", are you talking about numerical solutions like Newton's method etc?  

---

### Answer by username_2 on 2020-07-27. Score: 3

It depends on what you are trying to achieve.

If you need exact figures for the maximum tidal height each day/year at a given location, there are no simple analytical solutions due to the complex nature of tides.

If you are interested in ballpark figures (which are good enough for most recreational, but also some engineering applications), the maximum annual tidal height can be approximated by tidal datums, either the Mean High Water Springs (MHWS) or Mean Higher High Water (MHHW) depending on the tidal regime. If you would want to be extremely conservative, you could use Highest Astronomical Tide (HAT). Also taking the mean of MHWS/MHHW and HAT could be an option. This approximation would be suitable for 'regular' coastal and offshore locations. Complex bathymetry (e.g. funnel-shaped estuary) is trickier.

This could potentially help you save some of the computational effort, as you would need to calculate the tidal datums once per location (rather than once per year per location).

> **Comments:**
> **username_0**: Clarified the question: only interested in astronomical tides (local bathymetric effects would be nice, but not essential). I'm interested in making a dataset that contains distribution parameters for annual maximum high tide for many locations around the world. So I only need that one number per year. HAT is not useful, because it only occurs every few of decades. I guess I could maybe infer the distribution from MHWS/MHHW and HAT though? But then, I'd also need to know those values, and I'm not sure if they're calculable analytically either.  
> **username_2**: You can calculate [tidal constituents/harmonics](https://en.wikipedia.org/wiki/Theory_of_tides#Tidal_constituents) from water level time series. Check ([t_tide](http://www.omg.unb.ca/GGE/5013_LABS/T_Tide_CompAndGeo.pdf)). Also the software you linked in your initial post (OTPS) seems to provide tidal constituents from gridded model output. To calculate MHWS etc, see formulas on page 6 of [this R package documentation](https://cran.r-project.org/web/packages/TideHarmonics/vignettes/austides.pdf) (sorry for the weird reference). This is a brute force method though :D.  
> **username_0**: Yeah, OK. But even then, I'd still have to figure out how to infer a maximum annual tide height distribution from those, and that seems pretty non-obvious. I guess I'd have to do it empirically/numerically anyway. Seems like my best bet is really going to be to DIY..  

---

