Mariner Studio

Tide Prediction Algorithms: How We Calculate Future Tides

The tide table on your phone isn’t a guess—it’s the result of 150 years of mathematical refinement. Every time you check whether you’ll have enough water to cross a bar or clear a bridge, you’re looking at predictions calculated by algorithms that combine celestial mechanics, local geography, and centuries of observation data.

Understanding how tide predictions work transforms them from mysterious numbers into tools you can use with confidence. Let’s explore the mathematics behind those tide tables.

The foundation: harmonic analysis

Tide prediction relies on a mathematical technique called harmonic analysis, which treats tides as the sum of many predictable wave patterns. Think of it like a musical chord—just as a chord is made up of individual notes, the tide at any location is the sum of dozens of individual tidal components.

Each component represents a different astronomical force. The moon’s gravitational pull creates one set of waves. The sun creates another. The moon’s varying distance from Earth creates yet another. The angle between the sun and moon creates more. Each force contributes a predictable wave pattern with its own amplitude (height) and period (timing).

Sir George Darwin (Charles Darwin’s son) and Lord Kelvin developed this approach in the 1860s and 1870s. Their insight was revolutionary: instead of trying to predict the complex physics of water movement, they could measure the actual tide at a location for a year, mathematically decompose it into constituent waves, and then use those wave patterns to predict future tides indefinitely.

The elegance of this method is that it works without needing to understand why a particular location has the tidal characteristics it does. The local geography, water depth, coastline shape, and bottom friction all affect tides in complex ways—but they’re all captured in the measured amplitudes and phases of the constituents.

The 37 tidal constituents (and then some)

NOAA’s standard tide prediction algorithm uses 37 primary tidal constituents, though different tide stations may use more or fewer depending on their specific characteristics. Here are the most important ones:

Principal constituents

M2 (Principal lunar semi-diurnal) – Period: 12.42 hours
The largest component at most locations. This represents the moon’s gravitational pull creating two high tides and two low tides each lunar day. The moon orbits slightly slower than Earth rotates, so the tidal day is 24 hours and 50 minutes—which is why high tide occurs 50 minutes later each day.

S2 (Principal solar semi-diurnal) – Period: 12.00 hours
The sun’s gravitational effect. About half the amplitude of M2 at most locations, but crucial for understanding spring and neap tides. When M2 and S2 align (new and full moons), you get spring tides with larger ranges. When they’re 90 degrees out of phase (quarter moons), you get neap tides with smaller ranges.

N2 (Larger lunar elliptic) – Period: 12.66 hours
The moon’s orbit isn’t circular—it’s elliptical. When the moon is closer (perigee), its gravitational pull is stronger. N2 captures this variation, which creates a monthly cycle in tidal range separate from the spring-neap cycle.

K1 (Lunar-solar diurnal) – Period: 23.93 hours
At some locations, particularly in the tropics and some Pacific islands, this diurnal (once daily) component dominates. It represents the declination of the moon and sun—how far they are north or south of the equator.

Why location matters

The relative strength of these constituents varies dramatically by location. In the Atlantic, M2 typically dominates, creating nice symmetric semi-diurnal tides (two highs and two lows per day). In the Gulf of Mexico, K1 is often stronger than M2, creating diurnal tides (one high and one low per day). Mixed tides, common on the U.S. Pacific coast, occur when both semi-diurnal and diurnal constituents are significant.

This is why you can’t just take a tide table from one location and apply it somewhere else, even if they’re relatively close. The local geography affects which constituents are amplified and which are dampened.

The prediction calculation

Once we know the constituents for a station, predicting the tide at any future time becomes a straightforward calculation:

Tide Height = MLW + Σ (Amplitude × cos[(Speed × t) + Phase])

Where:

  • MLW is Mean Low Water (the baseline)
  • Amplitude is the height contribution of each constituent
  • Speed is how fast that constituent cycles (its frequency)
  • t is the time we’re predicting for
  • Phase is where that constituent started in its cycle (referenced to a specific date)
  • Σ means we sum this calculation for all constituents

For a station with 37 constituents, we calculate this cosine function 37 times and add them together. Modern computers make this trivial, but in the early 1900s, this was done with mechanical tide-predicting machines—elaborate brass instruments with gears representing each constituent.

Example calculation (simplified)

Let’s predict the tide for a hypothetical station at noon today using just the three largest constituents:

Station constituents:

  • M2: Amplitude = 1.2m, Speed = 28.984°/hour, Phase = 45°
  • S2: Amplitude = 0.3m, Speed = 30.000°/hour, Phase = 20°
  • N2: Amplitude = 0.2m, Speed = 28.440°/hour, Phase = 60°

At 12 hours after the reference time:

M2 contribution = 1.2 × cos[(28.984 × 12) + 45] = 1.2 × cos(392.8°) = 1.02m
S2 contribution = 0.3 × cos[(30.000 × 12) + 20] = 0.3 × cos(380°) = 0.29m
N2 contribution = 0.2 × cos[(28.440 × 12) + 60] = 0.2 × cos(401.3°) = 0.17m

Predicted tide height = MLW + 1.02 + 0.29 + 0.17 = MLW + 1.48m

The actual calculation uses 37+ constituents and is more precise, but the principle is identical.

How Mariner Studio uses NOAA’s predictions

Mariner Studio pulls tide predictions directly from NOAA’s API, which provides pre-calculated predictions for thousands of stations worldwide. These predictions are generated using the harmonic method described above, with station-specific constituents derived from years of measurements.

When you view a tide graph in the app, you’re seeing these harmonic predictions visualized over time. The smooth curve you see is literally the sum of 37 sine waves. The predictions are typically calculated for every 6 minutes throughout the day, giving you precise timing for any tidal height.

NOAA publishes the constituent data for each station, which means the predictions can be regenerated indefinitely into the future. The sun and moon’s positions are predictable for thousands of years, so the only limitation on tide predictions is knowing the local constituents—which are assumed to be stable over time.

Accuracy and limitations

Harmonic tide predictions are remarkably accurate under normal conditions, typically within 10-15 cm (4-6 inches) of actual observed tides. But there are situations where predictions diverge from reality:

Weather effects

The harmonic method predicts astronomical tides—the water level caused by gravitational forces alone. It doesn’t account for:

  • Storm surge – Low pressure and wind can raise or lower water levels by several feet
  • River discharge – Heavy rainfall upstream affects estuarine water levels
  • Sustained winds – Days of strong onshore or offshore wind change sea level
  • Barometric pressure – Every 1 millibar change in pressure affects sea level by roughly 1 cm

This is why professional mariners check both predicted tides and real-time water level observations. Mariner Studio displays NOAA tide predictions, but always verify actual conditions when safety matters. A 3-foot predicted high tide combined with a storm surge becomes a 6-foot high tide.

Long-term changes

The harmonic constituents assume the local geography is stable. But coastlines change:

  • Dredging changes water depth and flow patterns
  • Land subsidence or uplift changes relative sea level
  • Sea level rise gradually increases the baseline
  • Inlet changes affect how tidal energy enters a bay

NOAA periodically re-analyzes stations (typically every 19 years—the Metonic cycle that captures all major tidal variations). Stations in rapidly changing environments may show drift between observations and predictions over time.

Shallow water effects

In very shallow or complex waters, non-linear effects become significant. Overtides and compound tides—constituents that are multiples or combinations of the basic astronomical constituents—become important. These are especially significant in:

  • Shallow estuaries
  • Complex inlet systems
  • Areas with strong tidal currents
  • Locations with significant bottom friction

NOAA’s predictions include many of these compound constituents, but very shallow or unique locations may show larger prediction errors.

The 19-year tidal epoch

You might wonder why NOAA uses 19 years of data to establish tidal constituents. This period, called the Metonic cycle, captures all significant variations in the moon’s orbit and the sun-moon relationship.

Over 19 years:

  • The lunar nodes complete their 18.6-year cycle
  • The moon’s apsidal precession (the rotation of its elliptical orbit) completes
  • The spring-neap cycle repeats in the same pattern relative to the calendar
  • Most long-period tidal constituents complete integer numbers of cycles

After 19 years, the astronomical configuration essentially repeats, making it the ideal period for determining the local tidal characteristics. Stations with shorter observation periods can produce usable predictions, but 19 years captures the complete picture.

Beyond basic predictions: subordinate stations

Not every location has a tide gauge and a full set of constituents. NOAA maintains about 3,000 subordinate stations—locations where tide predictions are derived from a nearby reference station using time and height corrections.

For example, a marina 15 miles up a river from a reference station might have:

  • High tide 1 hour 20 minutes later than the reference station
  • Heights that are 85% of the reference station values

These corrections are determined empirically through short-term observations. While less accurate than a dedicated station, subordinate station predictions are usually good enough for navigation planning.

Mariner Studio indicates whether you’re viewing a reference station (with measured constituents) or a subordinate station (with derived predictions). For critical transits in tide-sensitive waters, reference stations provide more reliable data.

Modern developments

While the harmonic method remains the foundation of tide prediction, modern approaches are adding capability:

Hydrodynamic models

Numerical models that simulate water movement using physics-based equations can predict tides in locations without observations. These models are especially valuable for:

  • Forecasting storm surge combined with astronomical tides
  • Predicting tides in data-sparse regions
  • Understanding how coastal changes affect tides
  • Planning offshore structures

Machine learning

Recent research explores using machine learning to improve predictions by:

  • Learning residual patterns not captured by harmonic analysis
  • Incorporating weather effects automatically
  • Adapting to coastal changes faster
  • Identifying anomalous tides that warrant investigation

However, these approaches supplement rather than replace harmonic analysis. The physics-based harmonic method remains the gold standard because it’s transparent, explainable, and works reliably with historical data.

Practical implications for mariners

Understanding tide prediction algorithms helps you use tide data more effectively:

Trust the timing more than the height – The phase relationships between constituents are very stable, making the timing of high and low tide extremely reliable. The heights can be affected by weather, but the clock-like progression of tides is dependable.

Know your datum – All predictions reference a specific tidal datum (usually MLLW). Understanding tidal datums helps you interpret chart depths and clearances correctly.

Recognize spring-neap patterns – The 14-day spring-neap cycle comes directly from the S2 constituent getting in and out of phase with M2. This isn’t randomness—it’s predictable mathematics.

Plan with the long view – Because predictions are generated from stable astronomical patterns, you can confidently plan passages months in advance. The predictions for next summer are just as reliable as today’s.

Verify with observations – When weather matters, check real-time water levels against predictions. A large difference indicates storm surge, unusual river discharge, or other non-astronomical effects that could affect your transit.

The reliability you can count on

When you tap a tide station in Mariner Studio and see predictions extending days or weeks into the future, you’re looking at calculations based on mathematical certainty. The moon’s position three weeks from Tuesday at 14:47 is as predictable as the sunrise. The local tidal response to that moon position is captured in constituents measured over years.

This is why tide predictions are so reliable—they’re not forecasts of a chaotic system like weather. They’re calculations of a deterministic system. The celestial mechanics are unchanging, and the local geography changes slowly.

The algorithm that generates these predictions is elegant in its simplicity: sum up three dozen cosine waves, each representing a different astronomical force. That a method developed in the 1860s remains our primary tool in 2025 speaks to its fundamental soundness.

Next time you check a tide table, remember you’re reading the output of mathematics refined over 150 years, based on observations taken over 19 years, calculating forces that have operated since the moon first entered orbit. It’s one of the most reliable predictions you’ll encounter in navigation.

Key takeaway

Tide predictions use harmonic analysis to decompose measured tides into 37+ constituent waves, each representing a different astronomical force. These constituents, once established, allow indefinite future predictions by summing cosine functions representing each force. The method is remarkably accurate under normal weather conditions and provides the reliable predictions you see in Mariner Studio.

Related features & learning

About tide prediction in Mariner Studio: The app displays NOAA’s official tide predictions for thousands of stations worldwide, calculated using the harmonic method described in this article. Predictions are updated regularly and cover multiple days ahead, giving you reliable planning data for any passage.