Gerassimos Manoussakis

Abstract: This paper presents an approach to determine the gradient of curvature of the normal plumblines at a point P above the ellipsoid and introduces a new geometrical object which is the isocurvature line. The assumed facts are the coordinates of the point P and the formula for the normal gravity potential U. For the determination of the gradient of the normal plumbline curvature k at the point P we define a small circle on the meridian plane of P whose center is at the point P. The circle has the radius of ε = 1 m and interior D. In this circle we construct a function ka to approximate the curvature function k. The function ka is a quotient of polynomials hence it is easy to find its partial derivatives at the point P i.e. grad ka(P). For the construction of the function ka we make the assumption that in the interior of the circle D the first order partial derivatives of U behave linearly and the second order partial derivatives have constant values which equal their value at the point P and then we set grad k(P) = grad ka(P). An isocurvature line of the normal gravity field passing through a point P is a curve � such that the value of the function of the plumblines� curvature k is constant and equals k(P). We give a formula to find the direction of the isocurvature line on the meridian plane and we prove that there are infinitely many isocurvature lines passing through the point P and they all lie on a special surface, the isocurvature surface.

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Abstract: The orthometric height is the distance, measured positive outwards along the plumbline, from the geoid to a point of interest usually situated on the Earth�s topographic surface. According to its �classical definition�, it can be computed from the geopotential number of a point, using the mean value of the Earth�s gravity acceleration along the plumbline within the topography (i.e. between the geoid and the Earth�s surface). Hence, the main problem in the rigorous definition of an orthometric height reduces to the accurate evaluation of the mean value of the Earth's gravity acceleration along the plumbline. Alternatively, recent efforts concentrate on the determination of orthometric heights from GPS derived geodetic heights (above the ellipsoid) and geoid undulations derived from detailed local geoid models using the familiar Stokes integration or FFT techniques. In this paper, we seek to determine the orthometric height from the knowledge of the geodetic (ellipsoidal) height and a representation for the gravity field at the surface point and without any information about the topographic mass distribution. We show that a rigorous and accurate determination of the orthometric height of a point on the Earth�s surface can be made by a methodology which relates the orthometric height with the geopotential number C, the magnitude of the gravity vector , and the curvature k of the plumbline, all determined at the point of interest on the physical surface. The required geopotential number C is computed through the evaluation of the Earth�s gravity potential W from one of the available Global Geopotential Models (GGMs) in spherical harmonics, while and k are computed by suitable analytical formulae which use the first and second partial derivatives of the disturbing potential T (Eötvos components) and the normal potential U accordingly. An overview is given of the steps involved in the computational process and the assumptions made. This rigorous approach was tested using different Global Geopotential Models (e.g. EGM96, GPM98CR, and recent models from the CHAMP and GRACE missions, such the EIGEN-GL03C and EIGEN-CG04C set of harmonic coefficients) and an extensive GPS/Leveling dataset on benchmarks in the USA. Results from these comparisons are presented for the larger part of the conterminous United States in non-mountainous areas (with orthometric heights ranging from zero up to 1200 m). They demonstrate that generally the differences between the these rigorously determined orthometric heights and actual orthometric heights from geodetic leveling typically range from a few centimeters and up to 3 decimeters, thus showing the viability of the methodology and its future promise as new and continually improving geopotential models from the CHAMP/GRACE and GOCE missions become available to be used for this purpose. Plans for future work will also be given.

Abstract: A neutral direction of a gravity field is a direction along which the coordinates of the gravity vector remain locally unchanged. This research will focus on the neutral directions for the normal gravity vector. The necessary condition for the existence of neutral directions at an arbitrary point P above the ellipsoid is the determinant of the Eötvös matrix be zero. We will also show that the existence of such directions at that point depends on the values of principal curvatures. The slope of these directions depend on the value of the second principal curvature and the curvature of the plumbline. In some cases the slope is independent from the plumbline curvature and from the value of second principal curvature. In all cases the neutral directions lye on the meridian plane at P. Very interesting cases are when the neutral direction coincides with a principal direction on the tangent plane or it is a bisector. A special case is described for which the neutral directions at P, is the z � axis. This case is characterized as special case because the vertical gradient of normal gravity at point P is equal to zero. Finally in the last part of the presentation we provide an informative table relating to a classification model for these neutral directions.

Abstract: In everyday life we use several instruments like for example a knife, a pair of scissors, or a screwdriver to make some jobs easier to be done. The essence of this idea can be applied to mathematics - i.e. to introduce �useful geometrical instruments� - so that to solve various problems. In this presentation we will introduce the term �spanner surface�. The term is based on the well known tool called �spanner�. A spanner surface SS related to a surface S is a surface such that quantities of the surface S (components of surface vectors, vertical vectors, curvatures etc) can be expressed with the quantities of SS. The problem which we are going to solve is the following �Knowing the gradient of the curvature function k of the plumblines of the Earth�s normal gravity field at a point P (i.e. gradk at P) find the components of gradk at an arbitrary point Q in an small area around point P such that k(P) = k(Q) without using the curvature function twice and without using Taylor series of the curvature function�. In this application we will show that with the help of a spanner surface it is possible to introduce an alternative way of estimating gradk without the use of Taylor series. This is quite an advantage since it will not be necessary to involve high order partial derivatives of the normal potential (second order, third order) in our calculations. In addition we will show that only the first order partial derivatives of the normal potential are necessary for the estimation of gradk in a small area of the isocurvature surface. In this problem we have two unknowns - which are the components of gradk expressed in a local Cartesian system � therefore we need two equations to relate these components with the components of gradU. The desired system of algebraic equations is formulated with the aid of a chosen spanner surface related to the original isocurvature surface. The spanner surface is chosen to be an equipotential surface of the normal gravity field. The solution of this algebraic system will show that the components of gradk at point Q are expressed as a combination of the components of gradU. This theoretical problem may be proven very hard or impossible to be solved without the use of a spanner surface.

Abstract: An isocurvature surface of a gravity field is a surface on which the value of the plumblines� curvature is constant. Here we are going to study the isocurvature surfaces of the Earth�s normal gravity field. The normal gravity field is a symmetric gravity field therefore the isocurvature surfaces are surfaces of revolution. But even in this case the necessary relations for their study are not simple at all. Therefore to study an isocurvature surface we make special assumptions to form a vector equation which will hold only for a small coordinate patch of the isocurvature surface. Yet from the definition of the isocurvature surface and the properties of the normal gravity field is possible to express very interesting global geometrical properties of these surfaces without mixing surface differential calculus. The gradient of the plumblines� curvature function is vertical to an isocurvature surface. If P is a point of an isocurvature surface and �ΦN� is the angle of the gradient of the plumblines� curvature with the equatorial plane then this direction points to the direction along which the curvature of the plumbline decreases / increases the most, and therefore is related to the strength of the normal gravity field. We will show that this direction is constant along a line of curvature of the isocurvature surface and this line is an isocurvature circle. In addition we will show that at each isocurvature surface there is at least one isocurvature circle along which the direction of the maximum variation of the plumblines� curvature function is parallel to the equatorial plane of the ellipsoid of revolution. This circle is defined as a Special Isocurvature Circle (SIC). Finally we shall prove that all these SIC lye on a special surface of revolution, the so � called SIC surface. That is to say, a SIC is not an isolated curve in the three dimensional space.

Abstract: In this paper we form a relation between the Ricci curvature of the equipotential surface of the normal gravity field and the Ricci curvature of the equipotential surface of the actual equipotential surface at a point P. For the point P we suppose that it is on a G.O.C.E. satellite track. The G.O.C.E. satellite provides us the necessary Eötvos components for which we suppose that � after some reductions � they referred to an Earth Fixed Reference Frame (X, Y, Z). The final formula is described in Cartesian coordinates (x, y, z) such that along the x � axis the longitude is constant, along the y � axis the latitude is constant and the z � axis is vertical to the tangent plane of the normal equipotential surface at the point P. This system is chosen because the final formula takes its simplest form. We prove that the final formula between the actual and normal Ricci curvature contains also the measure of the normal gravity vector, the curvature of the plumbline, the three Eötvos components Txx, Tyy and Txy , and the functions εij which are functions of the astronomical latitude, astronomical longitude, normal astronomical latitude, and geodetic longitude. All the previous mentioned quantities are referred to the point P. The separation between the �normal� part from the �disturbing� part of the actual Ricci curvature is not possible. The main reason is that the angle between the normal gravity vector and actual gravity vector at the point P can be well greater than a few seconds of arc. Therefore this fact permits us to make approximations only for the Eötvos components Txx, Tyy and Txy and not for the second order partial derivatives of the normal potential.

Abstract: In this paper we form relations for the determination of the elements of the Eötvos matrix of the Earth�s normal gravity field. For this purpose we use a global Cartesian system (X, Y, Z) and use the variables X, and Y to form a local parameterization a normal equipotential surface to describe its fundamental forms and the plumbline curvature. The first and second order partial derivatives of the normal potential can be determined from suitable matrix transformations between the global Cartesian coordinates and the ellipsoidal coordinates. Due to the symmetry of the field the directions of the local system (x, y, z) are principal directions hence the first two diagonal elements of the Eötvos matrix with the magnitude of the normal gravity vector are sufficient to describe the Gauss curvature of the normal equipotential surface and this aspect gives us the opportunity to insert into the elements of the Eötvos matrix the Gauss curvature. In addition this gives us the opportunity to form a relation between the Gauss curvature of the normal equipotential surface and the Gauss curvature of the actual equipotential surface both passing through the point P.

Abstract: The curvature k of a plumbline of the Earth�s normal gravity field U passing through a point P is a function which contains the first and second order partial derivatives of the normal potential U (referring to a Cartesian system). To determine the gradient of curvature at P the third order partial derivatives of the normal potential are also needed. However the determination of these high order partial derivatives demands too many complicated and tedious calculations. Here we describe a method to determine the gradient of curvature without using the third order partial derivatives of U. As a first step we express the partial derivatives of normal potential U in a global Cartesian system (X, Y , Z) such that the Z-axis is the Earth�s mean axis of rotation, the X-axis is the intersection of the equator�s plane and the plane of the Greenwich meridian and the Y -axis makes the system right-handed. For the problem at hand, we first introduce a local Cartesian (x, y, h) system such that a) the x � axis is tangent to the parallel circle at � = �(at P) , b) the y � axis is tangent to the meridian λ = λ(at P) and c) the h � axis is the vertical to the ellipsoid passing through the point P. Subsequently we introduce a local Cartesian system (x1, y1, h1) whose center is the point P and the transformation equations are x1 = x, y1 = y, and h1 = hP - h. Now in the interior of a circle of radius δ (δ is less than a meter) which has as a center the point P and lies on the meridian plane of P we assume that the coordinates of the gradU change linearly and the second order partial derivatives of U practically do not change. In the interior of the circle � we name it D � we construct a function ka = ka(y1, h1) with the use of which we determine the curvature of a plumbline at a specific point in the set D. The function ka is a quotient of polynomial functions and it is a good approximation of the function k in the set D. Hence it is easy to determine the gradka in terms of the (x1, y1, h1) coordinates in D and consequently at the point P. Finally using the coordinate transformations we express the gradka in the global Cartesian system (X, Y , Z). The isocurvature lines are curves such that if k is the function which describes the curvature of the plumblines then at each point it holds that k (X, Y , Z) = ko = const. We prove that there are at least two isocurvature lines which pass through a point P, they are orthogonal to each other and both of them are plane curves. Next we prove that these two curves lie on a special surface which is the isocurvature surface passing through the point Pand finally we prove that the isocurvature surfaces are surfaces of revolution. The study of these new geometrical entities may reveal more properties of the normal gravity field.

Abstract: Abstract: Generally given a smooth surface, a ridge is a curve such that at each of its points, the maximum (minimum) principal curvature has an extremum along its curvature line. Ridges are curves of extremal curvature and therefore encode important information used in surface analysis or segmentation. We are interested in the case where the surface of interest is an equipotential surface of the earth�s gravity field, under the assumption that there are no umbilical points, i.e. points at which both principal curvatures are equal, and every tangent vector is a principal direction. In this case, if P is a point of a specific ridge, then the angle between the vertical line at P and the corresponding vertical line at a neighboring point P� along the principal direction at distance ds apart, is a supreme/infimum. For this reason, we call them �curves of supreme and infimum variation of the equipotential surface�s vertical line�. Furthermore, if the point P lies on the geoid then the corresponding differential geoid undulation dNPP� is also a supreme/infimum. In this paper, we carried out a study of the local geoid ridges with the help of algebraic equations. At first, we derived the condition which must be satisfied in order that a point P on an equipotential surface of the earth�s gravity field lies on a local ridge. Then we outline the necessary equations that allow determining the direction of the ridge at each of its points. We show that locating and reporting them requires manipulating third, forth and fifth order partial derivatives of the gravitational potential � whence imposes numerical difficulties.

Abstract: The orthometric height is the distance, measured positive outwards along the plumbline, from the geoid to a point of interest, usually on the Earth's topographic surface. According to its "classical definition", it can be computed from the geopotential number of a point, using the mean value of the Earth's gravity acceleration along the plumbline within the topography (i.e. between the geoid and the Earth's surface). Hence, the main problem in the rigorous definition of orthometric height reduces to the accu! rate evaluation of the mean value of the Earth's gravity acceleration along the plumbline. Alternatively, recent efforts concentrate on the determination of orthometric heights from GPS derived geometric heights (above the ellipsoid) and geoid undulations derived from detailed local geoid models using the familiar Stokes integration or FFT techniques. In this paper, we show that a rigorous and yet practical and accurate determination of the orthometric height of points on the Earth's surface can be made by a methodology which relates the orthometric height with the geopotential number C, the magnitude of the gravity vector g and the curvature k of the plumb line, all determined at the point of interest on the physical surface. The required geopotential number C is computed through the evaluation of the Earth's gravity potential W from one of the available Global Geopotential Models in spherical harmonics, while g and k are computed by suitable formulae which use the firs! t and second partial derivatives of the disturbing potential T (Eötvos components) and the normal potential U accordingly. An overview is given on our project structure, the computation strategy, the test data sets we used, the expected accuracies, and the work done so far. This approach was tested using different Global Geopotential Models (e.g. EGM96, GPM98CR, and recent models from the CHAMP and GRACE missions) and extensive leveling benchmark datasets from USA and Europe. Initial results from these comparisons will be presented for various areas and for various classes of orthometric heights (i.e. 0-250 m, 250-500 m, 500-1000 m, 1000-1500 m, and >1500 m), demonstrating that generally the differences between the these rigorously determined orthometric heights and actual orthometric heights typically range from a few centimeters and up to 4 decimeters, thus showing the future promise of this methodology as new and continually improving geopotential models from the CHAMP and GRACE missions become available to be used for this purpose. Plans ! for future work will also be given.

Abstract: The normal Eötvös matrix is a three dimensional symmetric matrix. Its elements are the second order partial derivatives of the normal potential, expressed in a local Cartesian coordinate system. With the help of the Eötvös matrix it is possible to determine the variation of the normal gravity vector locally around a point P. In addition the Eötvös matrix contains information about equipotential surfaces and the curvature of the plumbline. In this presentation we will show a way to estimate the elements of the Eötvös matrix at an arbitrary point P above the ellipsoid whose geometrical height is less than 1000 m. Under this condition it is shown that the second order partial derivatives of the normal potential vary linearly along the line vertical to the ellipsoid and passing through P. It is worth mentioning that the proposed method to estimate the elements of the Eötvös matrix is much simpler than following the definition of the matrix. Finally it will be shown that if Q is the projection of P along the vertical to the ellipsoid then the values of the Eötvös� matrix elements at point P are strongly related with plumbline curvature and the principal curvatures at Q.

Abstract: The Earth�s normal gravity field is generated by a suitable ellipsoid of revolution which is assumed that is enclosing the Earth�s mass and rotates with the same angular velocity as the real Earth. This so-defined normal figure of the Earth, when used as a geodetic reference system should guarantee a good fit to the earth�s surface and to the external gravity field. The problem in hand is that given the coordinates of a point P (initial conditions) the vector equation of a plumbline passing through P is the solution of a highly non linear system of O.D. equations. With this work we seek to formulate a vector equation (0, y(h), h) for the plumbline in a local Cartesian system (x, y, h) without integration. We take advantage of the fact that the plumbline has very small curvature, no torsion and the separation between the points of the plumb line and the points of the vertical line to the ellipsoid passing through P is also very small. As a first step the partial derivatives of the normal potential U are described with suitable products of functions of the type f(h)y + c. The second step is to formulate a suitable function to describe the curvature of the plumblines as a quotient of sums of functions of the type g(h)y + c. Making the assumption that the curvature varies linearly between points with coordinates of the form (0, y, h=const.) we subsequently arrive to an algebraic equation which under certain conditions gives the desirable solution y = y(h).

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