Fundamental Ambiguity in the Definition of Vertical Motion Geoffrey Blewitt Mackay School of Earth Sciences and Engineering, University of Nevada, Reno, USA. Abstract. Vertical motion is generally described as 2 What is Vertical Motion? motion normal to some defined horizontal surface or family of surfaces. Such a surface can be con- Vertical motion is generally defined as motion that structed geometrically or gravitationally. In the is normal to some defined horizontal surface. Thus case of a geometrical surface, the ellipsoid is the it is entirely defined by the horizontal surface itself. shape of choice. In the case of a gravitational sur- face, it needs to be specified which masses are to be 2.1. Physical Model included in the definition of the center-of-mass ori- gin, and whether the surface evolves in time along In a physical model, the horizontal surface is gener- with the configuration of mass (as in a dynamic ally taken as an equipotential surface. More cor- equipotential) or whether it is static (as in a refer- rectly, an infinite family of equipotentials define the ence geoid). Vertical motion is therefore not an ab- direction of vertical motions everywhere. The di- solute concept, but rather one of convenience and rection of vertical motion follows a “plumb-line” convention. Even if properly defined, the ability to trajectory, which is generally curved. realize vertical displacement coordinates may be Despite its physical nature, there are many ways limited by uncertainty in the relationship between to define vertical motion within this class of model. observed quantities and the frame definition. Thus Most fundamentally, “equipotential” is not an abso- discrepancies between two analysis groups’ results lute concept, as it depends on the local relativistic on vertical station velocities may not be a result of frame. The equivalence principle states that accel- systematic error, but rather may simply reflect in- eration cannot be distinguished from gravity, hence ability to realize the same reference frame. equipotential surfaces be different in frames that accelerate relative to one another. Keywords. Vertical, reference frame, ellipsoid, One obvious cause of acceleration is rotation. gravity, gravitation, geoid, equipotential, sea level. Co-rotating frames therefore “see” different equipo- tential surfaces. Since Earth rotation is far from constant, this presents a problem. A conventional 1 Issues for Discussion solution is to adopt some defined constant angular velocity for Earth rotation, thus enforcing a conven- Vertical motion is an ambiguous term, in the sense tional centrifugal contribution to the equipotential. that vertical displacement coordinates depend on the In this case, it is typical to define “gravity” as being adopted conventions of the reference frame and equal to “gravitation” plus the conventional (rather coordinate system, and the ability to realize such a than actual) “centrifugal” term. Gravitation is here reference frame in practice. This paper addresses understood to be in a quasi-inertial local-Earth the roots of this ambiguity by posing the question, frame, co-orbiting the Sun along with the Earth sys- “what is vertical motion?” We identify different tem’s center of mass, and so is the Earth system’s classes of defined vertical motion, which are based gravity field as sensed by orbiting satellites, after on physical, geometric, and hybrid models. accounting for solar system gravitational forces. By analyzing this ambiguity, we investigate to However this is only one possibility. what extent differences between these models are Note that other choices of origin and rotation predictable. Part of the answer of course lies with create additional fictitious (non-inertial) forces and reference frames and reference system conventions. so have different equipotential surfaces. Other pos- Finally, we address the implications for ambigu- sibilities are frames that co-rotate with no-net rota- ity in vertical motion, including its effect in inter- tion with respect to the Earth’s surface, which raises comparison of vertical motion results, and on scien- the further issue of whether this condition is im- tific interpretation of vertical motions. posed as a long term average, or instantaneously as Earth rotation varies. The dynamic interpretation of vertical motion therefore strictly requires accounting same technique by two different analysis groups). for the choice of frame in this respect. In most cases the choice may not matter for practical pur- 2.3. Hybrid Model poses, however the theoretical distinctions must be kept in mind if we are to catch those few instances In a hybrid model, the horizontal surface is gener- of scientific interpretation where it does actually ally taken as a conventional ellipsoid, however the matter. center of the ellipsoid is typically taken as the Earth center of mass (as a function of time). Defining the 2.2. Geometric Model center in a physical way helps to resolve at least one source of ambiguity, although in practice different In a geometric model, the horizontal surface is gen- techniques produce slightly different solutions for erally taken as a conventional ellipsoid (sometimes the location of the Earth’s center of mass. Another referred to as a “datum”). In this system, vertical problem here is that the origin is typically taken as motion is defined as motion along a straight line that the center of mass of the whole Earth system (to intersects the ellipsoid at right angles. This is very which satellite orbits are sensitive), however the different than in the physical model, where pure solid Earth itself (approximately an ellipsoid) must vertical motion does not generally follow a straight move with respect to this point as mass is redistrib- line. uted in its fluid envelope. So we have a situation There are as many ways to define vertical motion where in reality, the ellipsoid should really be mov- within this class of model as there are possible sets ing, but in a typical CTRF this is not modeled (or of ellipsoids. The typical convention is to adopt a not adequately modeled). The definition of vertical perfectly circular equator, in which case the ellip- motion in the hybrid model will depend on whether soid’s internal geometry is defined by two parame- or not this “geocenter motion” is modeled, or ters: the radius of the equator, and the flattening whether or not models specify time evolution of the factor. The external geometry (defining the location solid Earth’s deforming surface. and orientation of the ellipsoid in space) is defined by an additional six parameters: three for the origin, 3 Analyzing the Ambiguity two to define the orientation of the polar axis of symmetry, and one to define the origin of longitude From the above discussion, it is clear that “vertical (the prime meridian). Note that the ellipoid’s exter- motion” is not an absolute concept. From this con- nal geometry, including its center, is in practice de- clusion we can logically deduce the following corol- fined implicitly by a defined set of station coordi- laries. Since vertical and horizontal motion as de- nates and their evolution in time. This is called the fined above span 3-dimensional space, therefore “conventional terrestrial reference frame”, or CTRF. horizontal motion too is not absolute. From this we Even if we can all agree on a convention for the can infer that vertical motion in one reference sys- ellipsoid’s internal geometry, it is more difficult to tem can appear horizontal in another, and vice be consistent in the definition of the external ge- versa. ometry, because this requires consistency in the various CTRFs. For example, different techniques 3.1. Reference Frame Issues use different stations, and so convention would re- quire the adoption of accurate “local ties” between One might at this point naturally ask whether or (nearby) co-located stations. There would also need not these apparent differences are predictable, and to be consistency in the time evolution of the station in what cases are the differences accentuated or ex- coordinates. treme. Logically, the most extreme case is where In any situation where two groups have adopted a pure vertical motion in one system appears as pure different set of stations defining a CTRF, the defini- horizontal motion in another. As ridiculous as this tion of vertical motion will of course be different. may appear, there is actually such a case that corre- Differences in vertical motion between two different sponds to a very real physical phenomenon! frames are typically significant (much larger than An explanation of this phenomenon requires a inherent measurement precision), especially if there fundamental consideration of reference frame the- is relative motion between the origins of the two ory, as explained in Blewitt [2003], with further frames. This is major source of confusion when discussion in Blewitt and Clarke [2003], where it comparing vertical velocities of individual sites be- was shown how a perfect sphere deforms to another tween two different techniques (and even within the perfect sphere when subjected to a degree-1 spheri- cal harmonic surface load. Moreover, the sphere is loading models that use a proportional relationship displaced in inertial space due to conservation of between vertical displacement and barometric pres- momentum, such that the center of mass of the solid sure). In that case, the CL frame is implicitly as- Earth moves in response to a change in center of sumed. Another model might assume a priori that mass of the surface load. Depending on the choice surface motions are entirely horizontal (e.g., rigid of reference frame, the resulting vector field of sur- plate rotations). In this case, the CH frame is im- face displacements can appear either purely vertical, plicitly assumed. or purely horizontal. The is illustrated in Figure 1, Given that there are so many possible frames that which appears in Blewitt [2003]: are being used in either modeling or in space geo- detic analysis, it is imperative that care be taken to account for those frame differences when attempting to interpret station motions.