Tides and water levels

An echo sounder measures depth below the transducer, but the water surface it rides on rises and falls with the tide. At the Tacoma berth the difference between mean lower low water and mean higher high water is close to four meters, and large tides span more. A sounding taken at high water and one taken at low water over the same spot differ by the whole tidal range, so before any depth can be charted or compared with a design, it must be reduced to a fixed vertical reference: chart datum, which on U.S. coastal charts is mean lower low water (MLLW). Theo, who has never worked with tides, spends his first week at Northshore learning how water levels are measured, predicted and turned into datums.

Why the tide rises and falls

Tides are caused by the gravitational pull of the Moon and the Sun on the rotating Earth. The Moon's pull is stronger on the side of the Earth nearest to it and weaker on the far side; the difference stretches the oceans into two bulges, one toward the Moon and one away. As the Earth turns beneath them, most coasts pass through two highs and two lows in about 24 hours 50 minutes, the length of the lunar day. The Sun produces a similar but smaller effect. When Sun and Moon line up, at new and full moon, their effects add and tides have their largest range (spring tides); at the quarter moons they partly cancel (neap tides). Because the Moon's orbit is tilted to the equator, the two daily tides are often unequal, giving the mixed tides of the Pacific coast of North America, with a higher high water, a lower high water, a higher low water and a lower low water each day.

The real ocean does not respond like a simple bulge. Tides travel as long waves around ocean basins, shaped by coastlines, depths and the Earth's rotation, and are amplified or damped as they enter bays and inlets. The result is that the tide at each place has its own character, which can only be learned by measuring it. That is the job of tide gauges, and of the long records that national agencies keep at their reference stations.

Measuring water levels

A tide gauge measures the height of the water surface relative to a fixed reference on land. Older gauges used a float in a stilling well, a vertical pipe that damps waves; modern gauges use acoustic or microwave sensors looking down at the water, or pressure sensors on the seabed, with the record sampled every few minutes and sent by satellite or cellular link. A gauge is only useful if its zero is tied to benchmarks ashore by precise leveling, and those ties are checked regularly; otherwise a settling wharf or a bumped sensor would quietly corrupt every reduced depth.

NOAA's National Water Level Observation Network includes about two hundred long-term stations on the coasts and the Great Lakes. Its tidal datums are computed over a 19-year National Tidal Datum Epoch, currently 1983 to 2001, chosen to average out the 18.6-year lunar nodal cycle; a new epoch is being prepared. For a survey away from a long-term station, the surveyor installs a subordinate gauge, levels it to nearby benchmarks and runs it for at least a month, so that its datum can be computed by comparison with the control station. NOAA's specifications for tide and water level requirements set out how. Northshore installs a radar gauge on the terminal's existing wharf, ties it to two NOAA benchmarks and compares it with the Tacoma station.

Transferring a datum to a temporary gauge uses simultaneous comparison. Over the same month, the mean level and the range at the temporary gauge are compared with those at the reference station, whose long-term datums are known. If the temporary gauge's monthly mean sits 0.12 m above the reference's monthly mean and its range is 4 percent larger, the long-term datum at the temporary site is estimated from the reference's datum with those differences applied. A month gives a datum good to a few centimeters in a well-behaved harbor; longer comparisons do better, and a short comparison during a storm or a river freshet can do much worse. The comparison also exposes problems with the gauge itself, such as a sensor that drifts or a stilling well that silts up.

Tidal datums

A tidal datum is a reference level defined by a particular phase of the tide averaged over a long period: mean sea level, mean high water, mean lower low water and so on. Chart datum is chosen low, so that the water seldom falls below it and charted depths are, most of the time, the least depth a mariner will find. Heights on land and clearances under bridges are referred to a high-water datum, so that the charted clearance is the least the mariner can expect.

U.S. charts use MLLW as chart datum on the coasts: the average of the lower of the two daily low waters over the tidal epoch. Because it is an average, the water falls below MLLW on many days, and tide tables then show negative tides. Shorelines are mapped at mean high water (MHW), and bridge and cable clearances on recent charts are referred to mean higher high water (MHHW). On the Great Lakes, where tides are tiny, charts use a low water datum referred to the International Great Lakes Datum of 1985, which is to be replaced by IGLD (2020), scheduled for implementation in 2027.

Tidal datums vary from place to place along a coast, because the tidal range varies. A datum computed at one gauge applies only near that gauge. Between gauges, surveyors use tide zoning or, increasingly, continuous separation models that give the height of chart datum above the ellipsoid everywhere in a region (Lesson 6). Those models make it possible to survey without a gauge in the loop, using GNSS heights directly, but they must be checked against gauges.

Harmonic analysis and prediction

The tide at a place can be described as the sum of many sinusoidal constituents, each tied to an astronomical cycle. The principal lunar semidiurnal constituent, M2, has a period of 12.42 hours; the principal solar semidiurnal S2 has exactly 12 hours; N2 accounts for the Moon's varying distance; K1 and O1 are the main diurnal constituents. Harmonic analysis of a long record gives the amplitude and phase of each constituent at that place, and the prediction is then

where \(Z_0\) is the mean level above chart datum, \(A_i\) and \(g_i\) are the amplitude and phase lag of constituent \(i\), \(\omega_i\) its angular speed, and the nodal factors \(f_i\) and \(u_i\) and the astronomical argument \(V_0\) adjust for the slowly varying astronomy. National agencies analyze dozens of constituents; five carry most of the energy at a typical Pacific harbor. Predictions are excellent for astronomical tides, but the actual water level also responds to weather: onshore winds and low air pressure raise it, offshore winds and high pressure lower it, and river flow adds its own variation. The difference between observed and predicted levels can reach tens of centimeters in a storm, which is why surveys reduce soundings with observed water levels, not predictions.

Reducing soundings

The traditional reduction subtracts the water level above chart datum at the moment of each sounding from the depth measured below the water surface (after correcting for the transducer's draft and the vessel's squat and heave):

where \(\eta(t)\) is the observed water level above chart datum. When the water is below chart datum, \(\eta\) is negative and the reduced depth is greater than the measured one. If the survey area is some distance from the gauge, the tide arrives there earlier or later and with a different range; tide zoning divides the area into zones, each with a time correction and a range ratio relative to the gauge. Errors in zoning are often the largest single source of vertical uncertainty in a traditional survey, which is one reason ellipsoid-referenced surveying has spread so quickly.

import math
# Illustrative harmonic constants for the harbour gauge: (name, speed deg/h, amplitude m, phase deg)
CON = [('M2', 28.9841042, 1.10, 110.0), ('S2', 30.0000000, 0.27, 140.0),
       ('N2', 28.4397295, 0.21, 85.0), ('K1', 15.0410686, 0.83, 255.0),
       ('O1', 13.9430356, 0.47, 240.0)]
Z0 = 1.50   # mean water level above MLLW, m
def tide(t_h):
    """Predicted water level above MLLW (m) at t hours from the start of the record."""
    return Z0 + sum(a * math.cos(math.radians(w * t_h - g)) for _, w, a, g in CON)

# One year of hourly predictions: find the lowest level, mean lower low water and the range
levels = [tide(t) for t in range(0, 24 * 365)]
low = min(levels); high = max(levels)
daily_low = [min(levels[d * 24:(d + 1) * 24 + 1]) for d in range(364)]
print(f"lowest predicted level in the year {low:+.2f} m, highest {high:+.2f} m")
print(f"mean of daily lowest waters {round(sum(daily_low)/len(daily_low), 2) + 0.0:+.2f} m above MLLW")

# Tide reduction: soundings measured from the water surface, logged with time
obs = [(9.25, 17.62), (9.50, 17.41), (9.75, 17.19), (10.00, 16.98)]  # (hour, depth below surface m)
print(" hour   measured   water level   reduced to MLLW")
for t, d in obs:
    wl = tide(t)
    print(f"{t:5.2f}   {d:7.2f}   {wl:9.2f}     {d - wl:8.2f}")

# Zoning: the survey area lies between gauges; apply a time lag and a range ratio
lag_h, ratio = 0.25, 1.06
t = 9.50
wl_zone = Z0 + ratio * (tide(t - lag_h) - Z0)
print(f"zone-corrected water level at {t} h: {wl_zone:.2f} m (gauge {tide(t):.2f} m)")

lowest predicted level in the year -0.58 m, highest +4.32 m
mean of daily lowest waters +0.00 m above MLLW
 hour   measured   water level   reduced to MLLW
 9.25     17.62       -0.46        18.08
 9.50     17.41       -0.45        17.86
 9.75     17.19       -0.41        17.60
10.00     16.98       -0.35        17.33
zone-corrected water level at 9.5 h: -0.58 m (gauge -0.45 m)

With these illustrative constants, the mean of the daily lowest waters comes out at 0.00 m, as MLLW requires, while the lowest predicted level is 0.58 m below it: on spring tides the water falls well below chart datum. The four soundings were taken near low water, when the level was 0.35 to 0.46 m below MLLW, so the reduced depths are greater than the measured ones. The zone correction shows how much a quarter-hour lag and a 6 percent range difference matter: 0.13 m, about half the vertical allowance for a high-order survey.

Water levels inland

Inland waters have no tide worth the name, but their levels still change. The project's reservoir rises and falls by meters over a season as water is stored and released, and the Sun River rises with snowmelt and summer storms. For the reservoir sonar survey, Grant reads the project's reservoir gauge at the start, middle and end of each day and Leah reduces soundings to elevations on the project's vertical datum (NAVD 88) rather than to depths below the water surface. On the river, the water surface slopes downstream, so a single gauge reading does not apply along the whole reach; Grant measures the water surface with RTK at the start and end of each cross-section, or uses GNSS heights on the boat directly.