Physical Heights of Inland Lakes

Abstract

Inland satellite altimetry has gained traction over the past decade and is now routinely used to monitor the water levels of rivers, lakes and reservoirs. The accuracy of such inland water height measurements, at least from radar altimetry is still relatively poor from a geodetic viewpoint, namely in the range of several decimeter. Accuracies from spaceborne laser altimetry, in particular from the ICESat-2 mission, are at cm-level, however, and further progress in the radar altimetry domain is expected from swath-based altimetry by the SWOT mission, (to be) launched December 2022. With accuracies down to cm-level one needs to reconsider the height system definition of inland lake surfaces as obtained from satellite altimetry. Conventionally one subtracts a global geoid model from the altimetry-derived ellipsoidal height to obtain an orthometric height. Without wind stress, seiches and other time-variable height disturbances the lake water surfaces will conform to equipotential surfaces in the Earth’s gravity field. Thus lake surfaces are surfaces of constant dynamic height, from which follows that a lake surface cannot be a surface of constant orthometric or normal height. Because equipotential surfaces are inherently non-parallel, two points at a lake surface can and will have different orthometric height. Although being well-understood in physical geodesy, we will here model this effect and quantify it for various case studies. We demonstrate that the effects can be as large as a few dm for large lakes at high altitudes, which is an order of magnitude that is relevant in terms of satellite altimetry error levels.

1 Introduction

Satellite altimetry fundamentally provides the range between satellite and water surface. After proper corrections and with knowledge of the spacecraft’s orbital position the basic product of satellite altimetry is the water surface height h above (and normal to) the reference ellipsoid.

By subtracting a geoid N one obtains orthometric height H by virtue of $h=H+N$. Typically a global geoid model is used, which obviously comes with its own model errors (commission, omission). As a consequence of $H=h-N$ these model errors end up fully in the determined orthometric height. For many applications such model geoid errors, considered as bias over the water body of interest, hardly play a role, particularly if variations over time are more interesting than absolute levels. In other applications, the altimetry-derived ellipsoidal heights over a lake are specifically used to improve the local geoid. The underlying assumption in such cases is that the orthometric height over a lake surface be constant.

Satellite altimeters have repeat cycles of 10 nodal days (Topex/Poseidon and Jason family), 35 days (Ers and Envisat family) or 27 days (Sentinel-3 family). These are sampling rates that cannot capture the faster hydrological dynamics, so single-track overpasses are temporally insufficient to create lake height time series. Therefore, over large lakes and reservoirs it is only natural to combine the height information from different groundtracks, either from the same satellite or from different ones, in order to densify the temporal sampling or to extend the length of the time series. After geoid correction, and potentially after inter-satellite bias correction, the height information from the different tracks are combined into a single height time series of the lake, e.g. Bergé-Nguyen et al. (2021), although the tracks refer to different locations on the lake. The combination of track information therefore assumes that the lake surface is a surface of constant orthometric height.

It is known from physical geodesy that the system of orthometric heights does provide unique and physically defined heights, but that surfaces of constant orthometric heights are not equipotential surfaces. That means that even with error-free satellite altimetry and with perfect geoid knowledge the orthometric heights from $H=h-N$ will not be constant over a lake. It also means that satellite-altimetric lake surfaces cannot be used to improve geoids without further precautions.

Although these effects are conceptually known from physical geodesy, it seems that the satellite altimetric literature largely ignores them, presumably because they are small. In this contribution we will formulate the orthometric height variation analytically and quantify its effects numerically.

2 Orthometric Height

The physical height of a surface point P is fundamentally defined through the geopotential number $C_P$, which is the gravity potential difference between the geoid and the surface (Heiskanen and Moritz 1967, §4-4):

$$\displaystyle \begin{aligned} C_P = W_0 - W_P \ . \end{aligned}$$

The potential difference is obtained by the work integral (per unit of mass) in the gravity field $\boldsymbol {g}$ from geoid to point P:

$$\displaystyle \begin{aligned} C_P = \int\limits_P^0 \boldsymbol{g}\cdot\mathrm{d}\boldsymbol{r} \ , \end{aligned}$$

which is a general path integral, typically evaluated over the surface of the Earth. In practice it is discretized as $\sum _i g_i l_i$, i.e. the sum over the product of leveling increments $l_i$ and surface gravity $g_i$ along the leveling line. The geopotential number can be evaluated conceptually also along the plumbline between the surface point P and its footpoint $P_0$ on the geoid. With $\boldsymbol {g}$ being tangent to the local plumbline, the scalar product of two vectors under the integral reduces to a product of two scalars:

$$\displaystyle \begin{aligned} C_P = \int\limits_P^{P_0} \boldsymbol{g}\cdot\mathrm{d}\boldsymbol{r} = -\int\limits_P^{P_0} g \, \mathrm{d} H = \int\limits_{P_0}^P g \, \mathrm{d} H \ . \end{aligned}$$

By simultaneously multiplying and dividing by the length of this stretch of plumbline, called $H_P$, one arrives at:

$$\displaystyle \begin{aligned} C_P = H_P \left[ \frac{1}{H_P} \int\limits_{P_0}^P g \, \mathrm{d} H \right] = H_P\, \bar{g}_P \ , \end{aligned}$$

in which $\bar {g}_P$ denotes the average gravity along the plumbline between surface point P and the geoid. Finally, the above equation is recast into the definition of orthometric height:

$$\displaystyle \begin{aligned} {} H_P = \frac{C_P}{\bar{g}_P} \ , \end{aligned} $$

(1)

i.e. the height of surface point P above the geoid, measured along the curved plumbline. Although the orthometric height constitutes a clear and physical definition of height above the geoid, two weaknesses are pointed out in the geodetic literature:

• 1. The orthometric height is the length of the plumbline, which is fundamentally curved. Hence, strictly speaking $h \neq H + N$. However, given the typical size of deflections of the vertical in the order of several arc seconds, this effect of mismatch between plumbline and ellipsoid normal is minor relative to the error level in satellite altimetry.

• 2. Gravity is unknown along the plumbline inside the Earth’s crust. It can only be approximated by a model (or hypothesis) of the nearby density distribution. An often-used model is the Poincaré-Prey reduction, which assumes a linear gravity decay between surface and geoid, so that the mean gravity $\bar {g}_P$ equals the gravity value at mid-height $g(\frac {1}{2}H_P)$. To obtain this value a second assumption is made, namely that the topography around P is flat, so that the topographic effect can be computed through a Bouguer-plate. The Poincaré-Prey reduction then consists of (a) removing a Bouguer-plate of thickness $\frac {1}{2}H_P$, (b) going down the plumbline towards the mid-point in free air, and (c) restoring the Bouguer-plate. The resulting height is called Helmert orthometric height and is obviously an approximation to the true length of the curved plumbline. A reduction of the approximation error may be achieved if one uses a better topographic model and sub-surface density information.

These weaknesses are physical geodesy textbook material and will not concern us in this contribution.

3 Orthometric Height Variation at Lake Surface

A further consequence of plumblines being curved does concern us here, though. With equipotential surfaces being perpendicular to the plumbline, a plumbline curvature will necessarily lead to equipotential surfaces at altitude being not parallel to the geoid. In other words, different points on the equipotential surface at altitude have different distances to the geoid, i.e. different orthometric heights. Hence, an equipotential surface is fundamentally not a surface of constant orthometric height. The only exception is the geoid itself with a constant $H=0$.

This geometric description becomes mathematically obvious in definition (1): the geopotential number will be constant at the equipotential surface, but the mean gravity will spatially vary, depending on surrounding topography or on lake bathymetry or on density variation in the crust. If we identify the surface of a lake or reservoir as an equipotential surface, we thus must conclude that the orthometric height of the lake surface is spatially variable. The main assumption made here is that the lake or reservoir is at rest. Any wind stress or dynamic lake topography would violate the assumption.

Consider now the difference in orthometric height between two points, P and Q, on the lake surface:

$$\displaystyle \begin{aligned} \Delta H_{PQ} = H_Q - H_P = \frac{C_Q}{\bar{g}_Q} - \frac{C_P}{\bar{g}_P} \ . \end{aligned}$$

With some manipulation, and setting $C_Q=C_P$,

$$\displaystyle \begin{aligned} \Delta H_{PQ} = \frac{\bar{g}_P}{\bar{g}_P} \frac{C_Q}{\bar{g}_Q} - \frac{C_P}{\bar{g}_P} \frac{\bar{g}_Q}{\bar{g}_Q} = \frac{\bar{g}_P - \bar{g}_Q}{\bar{g}_Q} \frac{C_P}{\bar{g}_P} \ , \end{aligned}$$

we arrive at

$$\displaystyle \begin{aligned} {} \Delta H_{PQ} = \frac{\bar{g}_P - \bar{g}_Q}{\bar{g}_Q} H_P \approx \frac{g_P - g_Q}{g_Q} H_P \ . \end{aligned} $$

(2)

In the latter approximation the mean gravity values along the plumbline are replaced by their surface values. Appendix 1 justifies this step numerically.

The orthometric height variation $H_Q - H_P$ (2) is proportional to the gravity difference $g_P - g_Q$ (in this order). This makes sense, as scalar gravity represents the gradient of the potential along the plumbline. Loosely speaking it represents the density of equipotential surfaces along the plumbline. The higher the gravity, the denser the level surfaces and, hence, the lower the height. The orthometric height variation is also proportional to the height of the lake itself. This also makes sense: the longer the plumbline, the more damage curvature can do.

A numerical rule-of-thumb can be derived from (2) by setting its denominator to $10\mathrm {m}/\mathrm {s}^2$. If the gravity variations are then given in units of mGal and the height in km, the left hand side is provided in mm. This rule-of-thumb resembles (Heiskanen and Moritz 1967, eqn. (4-34))

$$\displaystyle \begin{aligned} \delta H_{\mathrm{mm}} \ \dot{=}\ \delta\bar{g}_{\mathrm{mgal}} H_{\mathrm{km}} \ , \end{aligned}$$

although that equation was meant to evaluate the effect of an error in mean gravity $\bar {g}$ on H.

4 Quantification: Case Studies

In order to quantify how serious the orthometric height variation can get, we will have a look at a number of case studies of lakes either with large gravity variation at its surface or at large altitude. The number of case studies is limited, however, because gravimetry over lakes is rare. And if data exist, their availability may be restricted.

It would not make sense to synthesize lake surface gravity from global geopotential models if real lake gravimetry did not enter such models. In such cases the global models would act as interpolators, smoothening the gravity field over the lake while underestimating the gravity variation.

The “lake” with the potentially largest gravity variation would be the global ocean. Despite a considerable gravity range of about 500 mGal, the oceans, being the embodiment of the geoid, have orthometric heights very close to zero. Therefore the orthometric height variation is near-zero, too.

4.1 Lake Vänern, Sweden

Lake Vänern has a surface area of $5650\mathrm {km}^2$. At a surface elevation of just 44 m one cannot expect a large orthometric height variation, but the high quality of the gravity data material makes for an interesting case study nonetheless. Landmäteriet, the national mapping, cadastral and land registration authority of Sweden, performed gravimetry over the ice in 2011 by hovercraft while the lake was frozen.

With a gravity range of about 50 mGal and a surface altitude of just 44 m the range of orthometric height variation is only 2 mm (Fig. 1). In Eq. (2) the points P and Q can be chosen freely. Here Q represents any of the surface points, whereas P is selected such that the height variation is centered. Although the orthometric height variation is negligible in the context of satellite altimetry, the case study demonstrates that the flattening alone explains the North-South gravity variation to a large extent. Consequently, the orthometric height variations can be approximated partially by the normal gravity already. For Lake Vänern this would amount to about 60% of the full effect (not shown here).

Fig. 1

Gravity (top) and corresponding orthometric height variation (bottom) over Lake Vänern

4.2 Lake Michigan, USA

The GRAV-D project of NOAA/NGS, the US National Geodetic Survey, provides a wealth of airborne gravity data. Over the wider area of the Great Lakes the data have been downward continued by Li et al. (2016). Michigan Lake was selected here as case study because of its North-South extension of nearly 500 km (Fig. 2).

Fig. 2

Gravity (top) and corresponding orthometric height variation (bottom) over Lake Michigan

Due to its sheer size one can expect Lake Michigan to have a considerable gravity variation. Indeed, it shows a range of about 350 mGal. Even after subtracting the normal field at the surface, the gravity disturbances still show a range of 80 mGal (not shown here). Despite a moderate surface altitude of about 140 m the large gravity range guarantees an orthometric height variation of the lake surface of about 5 cm, a level that starts to become interesting in radar altimetry, and that certainly is relevant to laser altimetry.

4.3 Issyk Kul, Kyrgyzstan

With a surface elevation of about 1600 m, a length of 178 km and surrounded by mountain ranges, Issyk Kul might have been an interesting case study. As an endorheic lake Issyk Kul would not suffer from drainage-related surface slope and would suitably fulfil the condition of being an equipotential surface. However, no observed gravity data is available. The example is used here to demonstrate the use of gravity data synthesized from a global geopotential model. Since no lake gravimetry was ingested into the global geopotential model egm2008 used here, the gravity variation over the lake will be too smooth. Moreover it shows interpolation artefacts. As a result the orthometric height variation will be underestimated and will also contain artefacts (Fig. 3).

Fig. 3

Gravity (top) and corresponding orthometric height variation (bottom) over Issyk Kul

4.4 Salar de Uyuni, Bolivia

The Salar de Uyuni is not a lake, it is a salt flat high up in the Andes at an altitude of 3700 m. Occasional rainfall creates a thin layer of water that solves the top salt layer and restores topographical deformations back to an equipotential surface before it evaporates, cf. (Borsa et al. 2008a). Gravimetry was part of geophysical prospecting in the 1970s (Cady and Wise 1992). Borsa et al. (2008b) assessed the equipotential surface properties of this salt flat.

The gravity variation range of about 80 mGal is less impressive than in the case study of Lake Michigan. However, the high altitude creates a large orthometric height variation in the range of 3 dm (Fig. 4).

Fig. 4

Gravity (top) and corresponding orthometric height variation (bottom) over Salar de Uyuni

5 Conclusions and Outlook

Since a water surface at rest conforms to the local gravity field potential, a lake surface will be an equipotential surface. As a consequence it cannot be a surface of constant orthometric height. We here formulated the orthometric height variation across the lake surface. A rule-of thumb says that its effect (in units of mm) is calculated as the amount of gravity variation at the surface (in mGal) times the overall orthometric height of the lake surface (in km). For several case studies we have shown that orthometric height variation can amount to 2–3 dm for lakes at high altitudes. However, for most lakes worldwide the orthometric height variation effect will be around the mm- to cm-level, i.e. hardly relevant in the context of satellite radar altimetry.

Because North-South gravity variation is to a large extent due to the flattening of the Earth, the orthometric height variation can be modeled to that extent accordingly. In the case studies shown here the flattening-induced orthometric height variation can explain roughly 50–60% of the total effect, the remainder due to density-induced gravity anomaly.

The database of globally available lake gravity is poor. Using gravity information derived from global geopotential models will not be an alternative as these models lack spatial resolution and will underestimate gravity variation over the lake, if lake gravimetry has not been ingested into the model. A further alternative, namely deriving gravity information from satellite altimetry itself, a well-established technique over the open ocean and big lakes, was not part of this study, and must be the object of further exploration.

The orthometric height variation at the lake surface is a geometric expression of the fact that equipotential surfaces are not parallel to each other. Therefore, as a word of warning to geoid modelers: it would be a mistake to use a geometrically derived lake surface, e.g. from laser altimetry, as a proxy for the geoid in areas where gravity information is sparse. The surface of the lake at rest is simply not parallel to the geoid.

Appendices

Appendix 1: Approximation

In Eq. (2) the mean gravity $\bar {g}_P$ was replaced by surface gravity $g_P$. It will be shown here that this approximation is allowed when comparing with Poincaré-Prey reduced gravity, which is the next-best approximation of mean gravity. In Sect. 2 this reduction was explained by the 3 steps: (a) removing a Bouguer plate of thickness $\frac {1}{2}H_P$, (b) going down to the mid-point along the plumbline in free air, and (c) restoring the Bouguer plate. This leads to the approximation:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \bar{g}_P & = &\displaystyle g(\textstyle\frac{1}{2}H_P) \\ & = &\displaystyle g_P - \textstyle\textsc{bo}\cdot\frac{1}{2}H_P + \textsc{fa}\cdot\frac{1}{2}H_P - \textsc{bo}\cdot\frac{1}{2}H_P \\ & = &\displaystyle g_P + (\textstyle \frac{1}{2}\textsc{fa} - \textsc{bo}) H_P \\ & = &\displaystyle g_P + \textsc{pp}\cdot H_P \ . \end{array} \end{aligned} $$

The conventional value for the free-air gradient $\textsc {fa}= 0.3086\text{mGal}/\text{m}$. For the crustal density one conventionally takes $\rho =2670\mathrm {kg}/\mathrm {m}^3$ leading to a Bouguer gradient of $\textsc {bo}= 0.1119\mathrm {mGal}/\mathrm {m}$. Hence the Poincaré-Prey gradient becomes

$$\displaystyle \begin{aligned} \textsc{pp} = 0.0424\mathrm{mGal}/\mathrm{m} = 4.24\cdot 10^{-7}\mathrm{s}^{-2} = 424\mathrm{E} \ . \end{aligned}$$

Let us see now how this small correction gradient affects the orthometric height variation (2).

$$\displaystyle \begin{aligned} \begin{array}{rcl} \Delta H_{PQ} & =&\displaystyle \frac{\bar{g}_P - \bar{g}_Q}{\bar{g}_Q} H_P \\ {} & \approx&\displaystyle \frac{g_P - g_Q - \textsc{pp}\cdot\Delta H_{PQ}} {g_Q + \textsc{pp}\cdot H_Q} H_P \ . \end{array} \end{aligned} $$

(3)

The orthometric height variation $\Delta H_{PQ}$ has been shown to be at mm- to cm-level for most lakes. Only in extreme cases like the Salar de Uyuni does it become a few dm. After multiplication with pp a very small number remains that does little to change the numerator. In the denominator pp is multiplied by the lake height itself. But even if the lake is situated at a few km altitude, the resulting correction is still small relative to the full gravity value.

Of course, the Poincaré-Prey reduction is a first approximation to determining the mean gravity along the plumbline. An improved approximation would require topographic, bathymetric and density information. It is believed that such a refinement will not numerically change the presented results. Hence, the approximation of Eq. (2) seems more than sufficient.

Appendix 2: Normal Height Variation

Along the same lines as above it can be derived how much normal height $H^{\mathrm {n}}$ will vary along a lake surface:

$$\displaystyle \begin{aligned} {} \Delta H_{PQ}^{\mathrm{n}} = \frac{\bar{\gamma}_P - \bar{\gamma}_Q}{\bar{\gamma}_Q} H_P^{\mathrm{n}} \ . \end{aligned} $$

(4)

The mean normal gravity along the plumbline in the normal field is measured from the ellipsoid footpoint $P_0$ upwards:

$$\displaystyle \begin{aligned} \bar\gamma_P = \gamma(\textstyle\frac{1}{2}H_P^{\mathrm{n}}) = \gamma_{P_0} - \textstyle\frac{1}{2}\textsc{fa}\cdot H_P^{\mathrm{n}} \ . \end{aligned}$$

Inserting this mean normal gravity for P and Q into Eq. (4) leads to a similar discussion as above. Despite the fact that the free-air gradient fa is numerically larger than pp, we conclude that such a refinement can be neglected and that the normal height variation at the lake surface can be approximated by

$$\displaystyle \begin{aligned} \begin{array}{rcl} \Delta H_{PQ}^{\mathrm{n}} & \approx &\displaystyle \frac{\gamma_{P_0} - \gamma_{Q_0}}{\gamma_{Q_0}} H_P^{\mathrm{n}} \ , \end{array} \end{aligned} $$

(5)

$$\displaystyle \begin{aligned} \begin{array}{rcl} & \mbox{or}&\displaystyle \\ \Delta H_{PQ}^{\mathrm{n}} & \approx &\displaystyle \frac{\gamma_P - \gamma_Q}{\gamma_Q} H_P^{\mathrm{n}} \ . \end{array} \end{aligned} $$

(6)

The former version makes use of normal gravity at the footpoint on the ellipsoid. The latter version, which uses normal gravity at the lake surface, is exactly the part of the orthometric height variation due to flattening. Thus, it can be predicted by the normal gravity variation with latitude. Alternatively, one can resort to formulas for $\bar \gamma$ as provided in Heiskanen and Moritz (1967, §4-5).