Height systems

The irrigation district wants to extend its main canal from Coulee Ridge toward the Little Belt Mountains, and the design depends on heights along a 25 km line that climbs about 550 m. Leah Tran added up the leveled height differences on the line and was surprised when Ruth Albrecht told her that the sum was not the orthometric height difference between the two ends: it was almost a decimeter short. The canal itself does not care about any particular height system. Water flows from higher to lower potential, and a canal is level when its water surface lies on one level surface. But the numbers written on the drawings must come from some definition of height, and there are several. This lesson explains why leveling alone is not enough, what geopotential numbers are, and how dynamic, orthometric and normal heights are defined from them.

Why leveling depends on the route

Spirit leveling measures the vertical separation between the local level surfaces through two staff positions, set after set. Lesson 5 showed that level surfaces are not parallel: they converge where gravity is stronger, toward the poles and over dense rock. Suppose two benchmarks lie on the same level surface, so that water would not flow between them. Leveling north from one along a route at low elevation and back along a route over a ridge would cross the level surfaces at places where they are differently spaced. The sum of the measured differences around the loop would not be zero, even with perfect instruments. This is the theoretical loop misclosure, and over long lines in mountainous country it reaches several centimeters.

The cure is to weight each measured difference by gravity. The work done against gravity in lifting a unit mass through a leveled difference \(dn\) is \(g\,dn\). Summed around any closed loop, the work is zero, because gravity is a conservative field:

This is why Ruth wanted gravity at the benchmarks. With gravity, leveled differences become potential differences, which are route-independent and form the foundation of every rigorous height system.

Geopotential numbers

The geopotential number of a point P is the difference between the potential of the geoid and the potential at P:

It is positive above the geoid and increases with height. Its unit is the square meter per second squared; the older geopotential unit, 1 gpu = 10 m²/s², makes the numerical value of \(C\) in gpu close to the height in meters, because gravity is close to 10 m/s². Geopotential numbers are the natural heights of physics: all points on a level surface share the same \(C\), and water flows from larger to smaller \(C\). Their drawback is that they are not lengths. Engineers and the public want meters, so each height system divides \(C\) by a gravity value of some kind. The choice of that gravity value distinguishes the systems.

Dynamic heights

The simplest choice is a constant: normal gravity at latitude 45°, \(\gamma_{45} = 9.806\,199\) m/s² on GRS80. The dynamic height is

Points on the same level surface have exactly the same dynamic height, which makes dynamic heights ideal where the main question is whether water will flow. The International Great Lakes Datum, used by Canada and the United States to manage the lakes and the St. Lawrence system, is expressed in dynamic heights for that reason. The drawback is that a dynamic height is not the geometric distance above anything. Wherever mean gravity along the plumb line differs from \(\gamma_{45}\), dynamic heights differ from orthometric heights: a difference of 20 mGal, two thousandths of a percent, changes a 1000 m height by 20 mm, and the differences grow with latitude and height. Leveled differences therefore need dynamic corrections, which can be large on long lines far from 45°.

Orthometric heights

The orthometric height is the geometric length of the plumb line from the geoid to the point. It follows from the geopotential number by dividing by the mean value of gravity along the plumb line inside the Earth:

Gravity inside the topography cannot be measured, so \(\bar g\) must be estimated from surface gravity and a model of the rock density. The Helmert approximation assumes a uniform Bouguer slab of density 2670 kg/m³ and a normal free-air gradient, which gives a mean gravity halfway down the plumb line of

that is, surface gravity plus 0.0424 mGal for every meter of height. Because \(H\) appears on both sides, the computation iterates, but it converges in two or three steps. Helmert orthometric heights are the official heights of both CGVD2013 and NAVD 88. More refined orthometric heights model the actual density and terrain, and they differ from Helmert heights by a few centimeters in high mountains.

Orthometric heights have a clear geometric meaning, and they combine directly with GNSS through \(h = H + N\). Their drawback is that points on the same level surface do not, in general, have the same orthometric height, because the level surfaces are not parallel. Two points on one lake can differ in orthometric height by centimeters.

Normal heights

Mikhail Molodensky avoided the unknown gravity inside the Earth by dividing instead by normal gravity, which can be computed exactly. The normal height is

where \(\bar\gamma\) is the mean normal gravity along the normal plumb line between the ellipsoid and the height \(H^{*}\) above it. Normal heights refer not to the geoid but to a closely related surface, the quasigeoid, and the separation between the quasigeoid and the ellipsoid is the height anomaly \(\zeta\), so that \(h = H^{*} + \zeta\). Over the oceans the quasigeoid and geoid coincide; on land they differ by roughly \(\Delta g_B H/\gamma\), where \(\Delta g_B\) is the Bouguer anomaly. On the prairies the difference is about a decimeter at 1500 m; in high mountains it can exceed a meter. Most European countries use normal heights.

Heights along the canal line

The program below carries a geopotential number along the canal line from BM A, using the leveled differences and the mean of the gravity values at the two ends of each section. At each benchmark it computes the Helmert orthometric height, the normal height and the dynamic height, and compares them with the raw sum of the leveled differences.

# Geopotential numbers and four kinds of height along a line of spirit levels
import math

a, f = 6378137.0, 1 / 298.257222101
ge, k, e2 = 9.7803267715, 0.001931851353, f * (2 - f)
def gamma0(lat):
    s2 = math.sin(math.radians(lat)) ** 2
    return ge * (1 + k * s2) / math.sqrt(1 - e2 * s2)
gamma45 = gamma0(45.0)

# Benchmarks southward from BM A towards the Little Belt Mountains: latitude, measured gravity (m/s^2),
# and the measured height difference from the previous benchmark (m)
line = [("BM A", 47.5182, 9.805761, None),
        ("K1", 47.4604, 9.805522, 95.312),
        ("K2", 47.3987, 9.805051, 211.744),
        ("K3", 47.3308, 9.804511, 240.128)]
H_A = 902.418                                       # published orthometric height of BM A

# Helmert: mean gravity along the plumb line = surface g + 0.0424 mGal per m of height
def gbar_helmert(g, H): return g + 0.0424e-5 * H                # half the 0.0848 mGal/m Poincare-Prey gradient
C = H_A * gbar_helmert(line[0][2], H_A)                         # geopotential number of BM A
raw = H_A
rows = []
for i, (name, lat, g, dn) in enumerate(line):
    if dn is not None:
        g_prev = line[i - 1][2]
        C += dn * (g + g_prev) / 2                   # dC = mean gravity along the section times dn
        raw += dn
    # Helmert orthometric height: solve H = C / (g + 0.0424 mGal/m * H) by iteration
    H = C / g
    for _ in range(5):
        H = C / gbar_helmert(g, H)
    # Normal height: C divided by mean normal gravity along the normal plumb line
    Hn = C / gamma0(lat)
    for _ in range(5):
        Hn = C / (gamma0(lat) * (1 - (1 + f + 0.003449786 - 2 * f * math.sin(math.radians(lat)) ** 2) * Hn / a + Hn * Hn / (a * a)))
    Hdyn = C / gamma45
    rows.append((name, C, raw, H, Hn, Hdyn))

print("name   C (m^2/s^2)   sum of dn    Helmert H    normal H*    dynamic")
for name, C, raw, H, Hn, Hdyn in rows:
    print(name.ljust(5), format(C, "12.3f"), format(raw, "11.4f"), format(H, "12.4f"), format(Hn, "12.4f"), format(Hdyn, "11.4f"))
name, C, raw, H, Hn, Hdyn = rows[-1]
print()
print("at", name, ": orthometric correction (H - sum dn) =", format(1000 * (H - raw), "+.1f"), "mm")
print("normal minus orthometric =", format(1000 * (Hn - H), "+.1f"), "mm;  dynamic minus orthometric =", format(1000 * (Hdyn - H), "+.1f"), "mm")
print("geopotential numbers in geopotential units (1 gpu = 10 m^2/s^2):", round(C / 10, 4))

name   C (m^2/s^2)   sum of dn    Helmert H    normal H*    dynamic
BM A      8849.241    902.4180     902.4180     902.3315    902.4129
K1        9783.836    997.7300     997.7453     997.6497    997.7195
K2       11860.046   1209.4740    1209.5221    1209.4063   1209.4438
K3       14214.449   1449.6020    1449.6957    1449.5560   1449.5370

at K3 : orthometric correction (H - sum dn) = +93.7 mm
normal minus orthometric = -139.8 mm;  dynamic minus orthometric = -158.7 mm
geopotential numbers in geopotential units (1 gpu = 10 m^2/s^2): 1421.4449

At K3, 547 m above BM A, the raw sum of the leveled differences gives 1449.602 m, but the Helmert orthometric height is 1449.696 m. The 94 mm difference is the orthometric correction for this line. It arises because the line climbs into an area where gravity is about 125 mGal weaker, so the level surfaces are farther apart there than at BM A. The normal height at K3 is 140 mm below the orthometric height, reflecting the negative Bouguer anomaly of the region, and the dynamic height is 159 mm below it, because at 1450 m the mean gravity along the plumb line is smaller than the 45° value used for dynamic heights. For the canal, the differences matter: a 94 mm error in the head at the upper end would change the design gradient noticeably.

The orthometric correction

In practice, national leveling networks are adjusted in geopotential numbers, and the heights are published in one system. A surveyor who levels a short line between published benchmarks applies an orthometric correction to the observed differences so that they agree with published orthometric heights. For a section from A to B it can be written

with \(\gamma_0\) an arbitrary constant, usually \(\gamma_{45}\). On flat ground the correction is negligible; on lines that climb hundreds of meters it reaches centimeters, as the program shows. For small sites with modest relief, such as the Coulee Ridge subdivision, it is a fraction of a millimeter and can be ignored.

Ellipsoidal heights complete the picture. They are purely geometric, come directly from GNSS, and have nothing to do with the flow of water. Two points with the same ellipsoidal height can differ in orthometric height by a meter if the geoid slopes between them. They are converted to physical heights through a geoid or quasigeoid model.