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3.01 Overview T.A. Herring, Massachusetts Institute of Technology, Cambridge, MA, USA © 2007 Elsevier B.V. All rights reserved. 3.01.1 Introduction Modern geodesy as discussed in this volume started with the development of distance measurement using propagating electromagnetic signals and the launch of Earth-orbiting satellites. With these developments, space-based geodesy allowed global measurements of positions, changes in the rotation of the Earth, and the Earth’s gravity field. These three areas (positioning, Earth rotation, and gravity field) are considered the three pillars of geodesy. The accuracy of current measurement systems allows time variations to be observed in all three areas. Also, the complexity of problems is such that each of the pillars interacts with each other and also with many other branches of Earth Science. This interaction is most apparent in the role that water plays in modern geodetic measurements. Every chapter in this volume mentions the role of water. It is critical because it can move rapidly and over large distances; it can exist in all three phases, gas, fluid, and solid; and modern geodetic methods are accurate enough that their measurements are sensitive to its effects. In its vapor form, its refractive properties delay microwave signals propagating through the Earth’s atmosphere. For geodetic positioning, this is a noise source but it is a signal for metrological applications. In the liquid form, it forms oceans that affect both the tidal signal and the rotation of the Earth. Also in liquid form, its mass changes the gravity field as it is moved through the hydrologic cycle. In solid form, it has a gravitational and deformation signal that changes if melting of the ice unloads the surface of the Earth. The interactions between the pillars include the elastic loading effects of changing mass loads that can be seen in the gravity field and in the positions of ground stations. The movement of water to and from the oceans can be seen with altimeter satellites whose orbital information is derived from measurements from ground stations whose positions are affected by the changing mass load. In modern, time-dependent geodetic data analysis these interactions need to be accounted for. The common interface between the geodetic methods is the coordinate systems and reference frames used to analyze data. 3.01.2 Coordinate Systems A common aspect of geodesy that permeates the literature on the subject is the coordinate system definition. With the development of space-based methods that allow global measurements to be made, the subject has been, in some senses, greatly simplified in recent years while at the same time complicated by the increased accuracy needs. The simplification comes from being able to use a Cartesian coordinate system with an origin at the center of mass of the Earth and axes aligned in a well-defined manner to the outer surface of the Earth. The complications arise from the need for a coordinate system definition that accounts for deformations from plate tectonics, tides, tectonic events, and other time-variable deformations. Historically, coordinate systems in geodesy are divided into two parts: horizontal coordinates such as latitude and longitude; and a vertical coordinate called height. These two systems are fundamentally different in that the horizontal one is geometric and based on the direction of the normal to an ellipsoidal body that represents the average shape of the Earth. The height system, called orthometric height, is based on the gravitational potential field of the Earth. In this system, surfaces of constant height are equipotential surfaces and hence fluid will not flow along these surfaces. One special height surface is called the geoid and is associated with the surface representing mean sea level. In Chapters 3.02 and 3.05, the subtle problems with these definitions – from defining an equipotential surface to the meaning of mean sea level in an ocean with currents – are discussed. In conventional geodetic systems there is a blend of potential-based and geometric-based systems. The determination of the geodetic latitude and longitude is complicated by measurements being made in the Earth’s gravity field. Specifically, with the development of electronic distance measurements, the distance measurement itself does not depend on the gravity field (except for small relativistic effects) but the projection of the direct measurement to a horizontal measurement does depend on the slope of the measured line. The angle between measured direction and local vertical (which depends on the local direction of gravity) can be easily measured but is strictly not the correct measurement. The angle to the local normal to the ellipsoid is needed. The difference between these directions is called the ‘deflection of the vertical’ and as discussed in Chapter 3.02 can be determined from measurements of gravity and solving the appropriate boundary-value problem. Deflection of vertical is also the difference between geodetic and astronomic latitude and longitude. The blend of potential-based and geometry-based systems in geodetic coordinate systems is a consequence of the methods available for making measurements. The local nature of these measurements resulted in large differences between systems adopted by different countries. The reference ellipsoidal shape for the Earth could have differences in the semimajor axis of over a kilometer (e.g., Clarke1866 6378206.0 m, and Everest 6377276.0 m (Smith, 1996)) and it was not uncommon at boundaries of countries to have coordinate differences of several hundred meters. The advent of space-based measurements allows the determination of a best-fit ellipsoidal shape for the Earth based on global data. There are simple relationships between the gravity of the Earth, expressed in spherical harmonics; the moments of inertia of the Earth; and an appropriate flattening of the Earth. After the central force term in the gravity field, the flattening term is the largest, being at least 1000 times larger than any other terms in the field. The leading terms in the gravitational potential field, V, of the Earth can be written in spherical harmonics as (e.g., Stacey, 1992): \[ V = -\frac{GM}{r} + \sum_{n=2}^{\infty} J_n a^n P_n(\cos x) \] where G is the gravitational constant; M is the adopted mass of the Earth; \(x\) and r are colatitude and radial distances to the point where the potential is determined; a is the equatorial radius of the Earth; \(J_0\), \(J_1\), and \(J_2\) are coefficients of the gravity field; and \(P_n\) are Legendre functions. When the adopted mass of the Earth matches that of the Earth, the \(J_0 P_0\) term is 1, and if the center of mass of the Earth corresponds to the origin of the coordinate system used to measure latitude, \(J_1\) is zero. When only the largest terms in the gravity field are considered, the Earth is axially symmetric and thus there is no longitude dependence. The second-degree harmonic term, \(J_2\), is related to both the moments of inertia of the Earth, which will control the Earth’s rotational behavior, and the flattening of the shape of the Earth. The moments of inertia of the Earth can be expressed in terms of \(J_2\) through MacGullagh’s formula yielding: \[ J_2 = \frac{C - A}{\frac{1}{2} M a^2} \] where A and C are the minimum and maximum moments of inertia of the axially symmetric Earth. By setting the potential to be the same at the equator and the pole, the semimajor and semiminor axes of an ellipsoid, \(a\) and \(c\), that match these values allows the flattening of the ellipsoid to be derived from the gravity field. Neglecting higher-order terms in the gravity field, we have (Stacey, 1992): \[ f = \frac{a - c}{c} \] where \(f\) is the flattening, and the centrifugal component of acceleration from the rotation of the Earth (rotation rate !) has been included. The ratio of the equatorial gravitational to rotational force is given by \(m\). From eqns 1–3, we see the first-order relationships between the gravity field of the Earth, the moments of inertia of the Earth, and the shape of the Earth. With modern space-based measurements, all of these quantities can be measured with great precision. Chapter 3.02 details the theory of gravitational potential; Chapters 3.10 and 3.09 examine the rotational consequences of moments on inertia; and Chapter 3.05 examines the detailed shape of the geoid based on altimeter measurements from space. Ключевые слова: e, r, o