Notes on Physical Geodesy - Tim Jensen.pdf

Notes on Physical Geodesy Tim Jensen January 20, 2011 These notes are written during a course concerning "The Gravity Field of the Earth". They are based on the book "physical geodesy" by Bernhard Hofmann-Wellenhof and Helmut Moritz, which was used during the course. These notes are thus very similar to the content of this book. The purpose of writing these notes was to make myself reflect on the contents of the book while reading it and not in order to make a collection of notes that can match the content of the book.

Contents 1 Fundamentals of Potential Theory 1.1 Defining the Gravitational Potential . 1.2 Harmonic Functions . . . . . . . . . . 1.3 Spherical Harmonics . . . . . . . . . . 1.4 Legendre’s Functions . . . . . . . . . . 1.5 Orthogonality Relations . . . . . . . . 1.6 Fully Normalized Spherical Harmonics 2 The 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13

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Gravity Field of the Earth Introducing Gravity . . . . . . . . . . . . . . . . . . . . . . Level Surfaces and Plumb Lines . . . . . . . . . . . . . . . . Defining the Geoid and Orthometric Height . . . . . . . . . Curvature of Level Surfaces and Plumb Lines . . . . . . . . The Potential of the Earth in Terms of Spherical Harmonics The Gravity Field of the Level Ellipsoid . . . . . . . . . . . Series Expansion for the Normal Gravity Field . . . . . . . Anomalous Gravity Field, Geoidal Height and Deflections of Gravity Anomalies Outside the Earth . . . . . . . . . . . . Stoke’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . Generalization to an Arbitrary Reference Ellipsoid . . . . . Koch’s Formula and Gravity Disturbances . . . . . . . . . . The Formula of Vening Meinesz (deflections of the vertical)

3 Gravity Reduction 3.1 Free-air Reduction . . . . . . . . . . . . 3.2 Bouguer Reduction . . . . . . . . . . . . 3.3 Terrain Correction . . . . . . . . . . . . 3.4 Poincaré and Prey Reduction . . . . . . 3.5 Topographic-isostatic Reductions . . . . 3.6 The Indirect Effect . . . . . . . . . . . . 3.7 The Inversion Reduction of Rudzki . . . 3.8 The Condensation Reduction of Helmert 1

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3 3 3 4 5 6 6

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8 8 9 9 9 11 12 13 14 15 15 16 17 17

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18 18 18 19 19 19 20 20 20

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4 Heights 4.1 Geopotential Numbers . . . . . . . . . . . . . 4.2 Orthometric Height . . . . . . . . . . . . . . . 4.3 Normal Heights . . . . . . . . . . . . . . . . . 4.4 Summarizing and Comparing Height Systems

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5 Projection onto the Ellipsoid

21 21 21 22 22 23

6 Gravimetric Methods 6.1 Gravity Reductions and the Geoid . . . . . . . . . . . . . 6.2 Geodetic Boundary Value Problems . . . . . . . . . . . . 6.3 Molodensky’s Approach . . . . . . . . . . . . . . . . . . . 6.4 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The Spherical Case . . . . . . . . . . . . . . . . . . . . . . 6.6 Solution by Analytical Continuation . . . . . . . . . . . . 6.7 Computational Formulas and the Molodensky Correction 6.8 Determination of the Geoid from Ground Level Anomalies

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24 24 24 24 25 26 26 27 28

7 Statistical Methods in Physical Geodesy

29

8 Least-Squares Collocation

30

2

1 1.1

Fundamentals of Potential Theory Defining the Gravitational Potential

From Newton law of gravitation we define the potential V as m (1) l2 Where the mass of the attracted body is set to unity. This is useful since we replace the three components of F with a single function V , wherein all information is contained and can be found by differentiating the function. The total potential is now the sum of all the individual contributions, i.e. in the case of a continuous distribution like the earth, the total potential is represented by an integral ZZZ ρ dv (2) V =G l F = grad V = G

v

Here ρ is the density and dv the volume elements. The potential is continuous throughout all space and vanishes as 1/l at infinity, which is clear from the fact that as l → ∞ the body approximately acts like a point mass.

1.2

Harmonic Functions

The gravitational potential has the special property that its second derivatives satisfy the Poisson equation 4V = −4πGρ

(3)

Where 4 denotes the Laplacian operator ∂2 ∂2 ∂2 + + ∂x2 ∂y 2 ∂z 2

(4)

Since the density ρ is zero outside the attracting body, the Poisson equation becomes 4V = 0

(5)

in areas outside the attracting body. The solutions to this equation are called harmonic functions, i.e. the gravitational potential is a harmonic function in the space outside the attracting body. The gravitational potential is a harmonic function outside the attracting body, because it involves the reciprocal distance 1 1 =q l 2 2 2 (x − ξ) + (y − η) + (z − ζ)

(6)

which is a harmonic function between two points P (ξ, η, ζ) and P (x, y, z). This means that outside the attracting body, the laplacian operator acting on the potential, becomes     ZZZ ZZZ ρ 1   4V = G 4 dv = G ρ4 dv = 0 (7) l l v

v

3

1.3

Spherical Harmonics

In spherical coordinates, the Laplace equation is written as     ∂2V 1 ∂ 1 ∂ ∂V 1 2 ∂V (8) 4V = r + 2 sin ϑ + 2 2 r ∂r ∂r r sin ϑ ∂ϑ ∂ϑ r sin ϑ ∂λ2 Where r is the radius, ϑ is the polar distance and λ is the geocentric longitude. What we now want to do is find the solutions the equation 8, which we do by the method of separating the variables. We assume that V (r, ϑ, λ) = f (r)Y (ϑ, λ)

(9)

Which has the solutions known as solid spherical harmonics Yn (ϑ, λ) rn+1 Where n is an integer 0,1,2,... and that all harmonic functions can be expressed as a superposition of all these solutions V = rn Yn (ϑ, λ)

V =

∞ X

rn Yn (ϑ, λ)

and

V =

and

V =

n=0

∞ X Yn (ϑ, λ) n=0

rn+1

(10)

(11)

The part Yn (ϑ, λ) which is independent of the radius is known as the surface spherical harmonics. Solutions to this equation is again found by separating the variables Yn (ϑ, λ) = g(ϑ)h(λ)

(12)

g(ϑ) = Pnm (cos ϑ)

(13)

The solutions are

h(λ) = cos mλ

and

h(λ) = sin mλ

(14)

Where n and m are integers 0,1,2,... and m is smaller than or equal to n. Pnm (cos ϑ) is the Legendre function, which is described in subsection 1.4. Again all solutions can be expressed as Yn (ϑ, λ) =

n X

[anm Pnm (cos ϑ) cos mλ + bnm Pnm (cos ϑ) sin mλ]

(15)

m=0

Where anm and bnm is arbitrary constants. By combining equation 11 and 15 we get the harmonic functions Vi (r, ϑ, λ) =

∞ X

rn

n=0

Ve (r, ϑ, λ) =

n X

[anm Pnm (cos ϑ) cos mλ + bnm Pnm (cos ϑ) sin mλ]

(16)

m=0

∞ n X 1 X [anm Pnm (cos ϑ) cos mλ + bnm Pnm (cos ϑ) sin mλ] rn+1 m=0 n=0

(17)

Which is denoted by subscript i for interior and e for exterior. Meaning that equation 16 represent the harmonic function inside a certain sphere and equation 17 represents the harmonic function outside a certain sphere. Since the gravitational potential is not a harmonic function inside the earth, our main focus will be on equation 17 representing the exterior region. 4

1.4

Legendre’s Functions

The Legendre functions Pnm (cos ϑ) is solutions to a special Legendre differential equation. The subscript n denotes the degree and the subscript m the order. The substitution t = cos ϑ is often used and is represented by Pnm (t) =

n+m

n 1 2 m/2 d 1 − t t2 − 1 n n+m 2 n! dt

(18)

Legendre’s Polynomials Legendre functions of order zero, that is putting m = 0, is known as Legendre’s polynomials in that they are simply polynomials in t 1 . They are denoted by Pn (t) is is given by
n dn+m 2 (19) t −1 n+m dt From equation 17 we notice that harmonic functions involving the Legendre polynomials, i.e. m = 0 is not dependent on the longitude λ. This means that their geometrical interpretation can be divided into zones as in figure 1(a). The polynomials are of degree n, meaning that they have n zeros. Pn (t) =

1

2n n!

Associated Legendre Functions If the order is not zero, that is, for m = 1, 2, ..., n (remember that m ≤ n), Legendre’s functions are called associated Legendre functions. They can be reduced to the Legendre polynomials by the equation dm Pn (t) (20) dtm Looking at the geometrical interpretation of the harmonic functions involving the associated Legendre functions, we see that these have n − m zeros in the interval 0 ≤ ϑ ≤ π, that is along the direction of latitude. It changes sign 2m times in the interval 0 ≤ λ ≤ 2π, that is along the direction of longitude. This can be seen from equation 17, that involves both cos mλ and sin mλ. This gives a tesseral image of the sphere (like a chessboard) represented by figure 1(b). From this it is obvious that if m = n there are no zeros along the direction of latitude, meaning that the sphere will have a sectorial picture if m = n as in figure 1(c). Pnm (t) = (1 − t2 )m/2

Figure 1: Geometrical interpretation of spherical harmonics: (a) zonal, (b) tesseral, (c) sectorial 1 A polynomial is an expression of finite length constructed from variables and constants, using only addition, subtraction, multiplication and non-negative integer exponents

5

1.5

Orthogonality Relations

Introducing the abbreviations Rnm (ϑ, λ) = Pnm (cos ϑ) cos mλ Snm (ϑ, λ) = Pnm (cos ϑ) sin mλ

(21)

We can exploit the orthogonality relations of the above expressions, where the product of two different functions is zero, if integrated over the entire surface of the sphere2 . RR  Rnm (ϑ, λ) Rsr (ϑ, λ) dσ = 0 σ RR if s 6= n or r 6= m Snm (ϑ, λ) Ssr (ϑ, λ) dσ = 0  (22) σ RR Rnm (ϑ, λ) Ssr (ϑ, λ) dσ = 0 always σ

Whereas the product of two equal function, i.e. n = s and m = r gives 3 ZZ 4π 2 [Rn0 (ϑ, λ)] dσ = 2n + 1 σ ZZ ZZ 2π (n + m)! 2 2 [Rnm (ϑ, λ)] dσ = [Snm (ϑ, λ)] dσ = 2n + 1 (n − m)! σ

(23)

σ

These orthogonality relations allows us to determine the constants anm and bnm in the surface spherical harmonics f (ϑ, λ) =

∞ X

Yn (ϑ, λ) =

n ∞ X X

[anm Rnm (ϑ, λ) + bnm Snm (ϑ, λ)]

(24)

n=0m=0

n=0

and to mathematically normalize these same equations. Note that these surface spherical harmonics is just the solid spherical harmonics from equation 17, where the radial dependence has been left out.

1.6

Fully Normalized Spherical Harmonics

We want to mathematically normalize the surface spherical harmonics from equation 24, which is done by assuring that ZZ ZZ 1 1 2 2 ¯ Rnm dσ = S¯nm dσ = 1 (25) 4π 4π σ

σ

Notice that the R and S now have an overbar, indicating that they are mathematically normalized 4 . The equations are normalized by the coefficients √ √ ¯ n0 (ϑ, λ) = 2n + 1 Rn0 (ϑ, λ) = 2n + 1 Pn (cos ϑ) R s ¯ nm (ϑ, λ) = 2 (2n + 1) (n − m)! Rnm (ϑ, λ) R (n + m)! (26) s (n − m)! S¯nm (ϑ, λ) = 2 (2n + 1) Snm (ϑ, λ) (n + m)! 2

RR σ

=

R 2π R π λ=0 ϑ=0

3 Note

that the case Sn0 (ϑ, λ) is not represented since sin mλ = 0 always 4 The reason that I consequently use the term mathematically normalized, is that these equations can be normalized in a non-traditional mathematical sense

6

The spherical harmonics can now be expressed on the form f (ϑ, λ) =

∞ X n X   ¯ nm (ϑ, λ) + ¯bnm S¯nm (ϑ, λ) a ¯nm R

(27)

n=0m=0

Notice that the coefficients also have an overbar in the above equation. They are different from the ones in equation 24.

7

2 2.1

The Gravity Field of the Earth Introducing Gravity

Besides Newton gravitational force, there is a centrifugal force f acting on a body at rest on the surface of the earth f = ω2 p p Here ω is the angular velocity of earth and p = x2 + y2 is the distance from the rotational axis. The centrifugal force can also be described by a potential

(28)


1 2 2 ω x + y2 (29) 2 so that f = gradΦ. The term gravity is now introduced as the sum of the gravitational force F and the centrifugal force f and their total potential W is given by ZZZ
ρ 1 W = W (x, y, z) = V + Φ = G dv + ω 2 x2 + y 2 (30) l 2 Φ=

v

Here the integration over v denotes integration over the entire earth. Applying the Laplace operator 4 to the total potential 30 we have the generalized Poisson equation 4W = −4πGρ + 2ω 2 From the total potential 30 we can also derive the gravity vector g as RRR x−ξ  2  −G  ∂W  l3 ρ dv + ω x v   RRR ∂x x−η 2  = g = grad =  ∂W l3 ρ dv + ω y  −G ∂y v RRR   ∂W x−ζ −G ρ dv ∂z 3 l

(31)

(32)

v

The direction of the gravity vector g is by definition the same direction as the plumb line, which is explained in the next subsection.

8

2.2

Level Surfaces and Plumb Lines

The level surfaces or equipotential surfaces are defined as surfaces where the total potential is constant W (x, y, z) = constant

(33)

Applying the chain rule to the above expression we arrive at dW =

∂W ∂W ∂W dx + dy + dz = gradW · dx = g · dx ∂x ∂y ∂z

(34)

From this we arrive to the definition of plumb lines (the direction of the gravity vector), since the product g · dx is equal to the change in total potential (from the above equation). We notice that the change in potential dW is zero if we move along the level surface, implying that g · dx = 0 along this direction and thus the gravity vector g is orthogonal to the equipotential surface. It is important to remark that plumb lines are not straight lines, but curve through space.

2.3

Defining the Geoid and Orthometric Height

The geoid is by definition a level surface (denoted W0 ), but is used as a reference for the orthometric height H. The geoid is defined as going through the mean surface of the ocean (which ideally is a level surface). Thus the orthometric height H is at the same time the height above the surface of the ocean and above the geoid. The orthometric height H is taken along the plumb line and is thus not a straight line. It is measured positive upwards, meaning that it has the opposite direction of the gravity vector ∂W (35) ∂H The geoid W0 and the orthometric height H make out the foundation of height measuring, giving the convention of level surfaces an essential role in physical geodesy. g · dx = −g dH = dW

2.4

⇒

g=−

Curvature of Level Surfaces and Plumb Lines

The curvature of a curve y = f (x) is κ=

1 = %

d2 y/dx2 2

1 + (dy/dx)

3/2

(36)

If we place the x-axis, so that it is parallel to the tangent of the top point P of the curve, we have that dy/dx = 0 and κ=

1 d2 y = 2 % dx

Figure 2: The curvature of a curve

9

(37)

Level Surfaces The curvature of a level surface in a point P can be split into a curvature of a curve in two directions (x and y direction), with the z-axis directed vertical. The two curvatures can be written in terms of the total potential as K1 =

∂ 2 W/∂x2 g

K2 =

∂ 2 W/∂y 2 g

(38)

Here K1 is the curvature in the xz-plane (y=0) and K2 is the curvature in the yz-plane (x=0). We now define the mean curvature of the level surface as the arithmetic mean of the two curvatures 1 J = − (K1 + K2 ) = − 2

∂2W ∂x2

+

∂2W ∂y 2

2g

(39)

Exploiting the generalized Poisson equation 4W =

∂2W ∂2W ∂2W + + = −4πGρ + 2ω 2 2 2 ∂x ∂y ∂z 2

(40)

We can obtain a very important Bruns equation relating the vertical gradient of gravity to the mean curvature of the level surface. ∂g = −2gJ + 4πGρ − 2ω 2 (41) ∂H Which is a beautiful example of relating physical quantities to geometrical quantities. Plumb Lines In a similar way at the curvature of the level surfaces, we have two curvatures for the plumb line κ1 =

1 ∂g g ∂x

κ2 =

1 ∂g g ∂y

In this case the mean curvature is defined by Pythagoras’ theorem s   2 q 2 ∂g 1 ∂g κ = κ21 + κ22 = + g ∂x ∂y

(42)

(43)

The Gradient of the Gravity Combining this with equation 41 we get the generalized Bruns equation
gradg = −2gJ + 4πGρ − 2ω 2 n + gκn1

(44)

Here the vectors n and n1 are defined as n = [0, 0, 1] n1 = [cos α, sin α, 0] and α is the angle between the principal normal n1 and the x-axis.

10

(45)

2.5

The Potential of the Earth in Terms of Spherical Harmonics

In the quest for determining the gravity potential W , we must conclude from equation 30 that this is a difficult task, since we need the mass density ρ everywhere inside the earth, in order to solve the integral for the gravitational potential V . However, we discovered in section 1 that outside the attracting masses, the potential can be expanded into a series of spherical harmonics. The trick is that we can expand the inverse distance l as 0 ∞ 1 X rn Pn (cos ψ) or = l rn+1 n=0  ¯ ∞ n 1 XX 1 S¯nm (ϑ, λ) 0 n ¯ Rnm (ϑ, λ) 0 n ¯ 0 0 0 0 r Rnm (ϑ , λ ) + r Snm (ϑ , λ ) = l 2n + 1 rn+1 rn+1 n=0m=0

(46)

Leaving the gravitational potential V as a series of spherical harmonics V =

∞ X Yn (ϑ, λ) n=0

rn+1

(47)

Which is equivalent to equation 17. We can also express the potential through the fully normalized spherical harmonics as V =

 n  ∞ X X ¯ nm (ϑ, λ) ¯ R ¯nm Snm (ϑ, λ) A¯nm + B rn+1 rn+1 n=0m=0

(48)

Here the coefficients are integrals, which is impossible for us to determine ZZZ 0 ¯ nm (ϑ0 , λ0 ) dM ¯ r nR (2n + 1) Anm = G ¯nm = G (2n + 1) B

Zearth ZZ

(49) 0 r n S¯nm (ϑ0 , λ0 ) dM

earth

But luckily these coefficients can be determined from the boundary values of gravity at the earths surface. Practically these equations can always be regarded as convergent, but keep in mind that they are only applicable to the gravity field outside the attracting masses.

Figure 3: Expansion into spherical harmonics

11

2.6

The Gravity Field of the Level Ellipsoid

To a second approximation the earth can be considered as an ellipsoid of revolution. This means, that by making the assumption, that the equipotential surface of the gravity field is ellipsoidal, we can mathematically create a "normal" gravity field, which is completely determined by the total mass M and the shape of the ellipsoid. Because the earth is not a perfect ellipsoid, practically there will be a small "disturbing" field. But because the ellipsoid is a very good approximation, these deviations from the normal field can be considered linear. This normal gravity field U is given by    
E 1 2 2q 1 1 GM 2 −1 tan + ω a sin β − + ω 2 u2 + E 2 cos2 β (50) U (u, β) = E u 2 q0 3 2 Notice that the only constants in the above expression is a, b, GM and ω. This means that the normal gravity field is completely determined by • The shape of the ellipsoid of revolution (a and b) • The total mass M • The angular velocity ω Some final remarks on equation 50. Notice that we are using ellipsoidal coordinates u (distance), β (latitude) and λ (longitude) and that the longitude λ is absent in the expression because of rotational symmetry. We have also introduced the abbreviations      1 E u2 u q= 1 + 3 2 tan−1 −3 2 E u E (51)      2 b E b 1 1 + 3 2 tan−1 −3 q0 = 2 E b E The normal gravity vector γ is given by γ = gradU

(52)

The version of Brun’s formula corresponding to 41 is in ellipsoidal coordinates, with hight h and ρ = 0 given by ∂γ = −2γJ − 2ω 2 ∂h Here the mean curvature of the ellipsoid is given by   1 1 1 J= + 2 M N

(53)

(54)

where M and N denote the principal radii of curvature. M is the radius in the direction of the meridian and N is the radius in the direction of curvature, taken in the direction of the prime vertical. M=

a2 /b 02

(1 + e cos2 ϕ)

N=

3/2

12

a2 /b 02

(1 + e cos2 ϕ)

1/2

(55)

2.7

Series Expansion for the Normal Gravity Field

Since the earth is nearly spherical, the quantities characterizing the ellipsoidal appearance are small. p linear eccentricity E = a2 − b2 , E e= , first eccentricity a (56) E second eccentricity e0 = , b a−b f= , flattening a As a result it can be proven useful to perform a series expansion of these terms. We start by defining the gravity flattening f ∗ f∗ =

γb − γa γa

(57)

From this we have a first order approximation called Clairaut’s theorem f + f∗ =

5 m 2

(58)

Where ω2 a centrifugal force at equator = γa gravity at equator

m=

(59)

Notice that Clairaut’s theorem is only a first order approximation and that additional higher-order terms in f is needed for better accuracy (usually second order terms are included for the flattening f and fourth order terms for the second eccentricity e0 ). By including these higher order terms, we arrive at   3 9 2 3 GM = abγa 1 + m + f m + m 2 7 4   2 11 1 2 4 11 2 U0 = aγa 1 − f + m − f − f m + m (60) 3 6 5 7 4     5 1 26 15 γ = γa 1 + −f + m + f 2 − f m + sin4 ϕ 2 2 7 4 Here U0 is the normal potential on an equipotential surface corresponding to the ellipsoid of reference S0 , γa is the gravity at the equator and γb is the gravity at the pole   GM 3 3 0 γa = 1 − m − e 2m ab 2 14   (61) 0 GM 3 γb = 2 1 + m e 2m a 7 Gravity above the ellipsoid The normal gravity γ as a function of ellipsoidal height h can be given as an expansion series γh = γ +

∂γ 1 ∂2γ 2 h+ h + ... ∂h 2 ∂h2

13

(62)

From this, expanding until second order, we obtain  
2 3 γh = γ 1 − 1 + f + m − 2f sin2 ϕ h + 2 h2 a a

(63)

Here γh denotes the normal gravity for a point at latitude ϕ, in height h above the ellipsoid and γ is the normal gravity on the ellipsoid itself.

2.8

Anomalous Gravity Field, Geoidal Height and Deflections of the Vertical

The small difference between the actual gravity potential W and the normal gravity potential U is called the anomalous potential T , so that W (x, y, z) = U (x, y, z) + T (x, y, z)

(64)

The geoid and reference ellipsoid are defined as having the same potential values, so that W0 (x, y, z) = U0 (x, y, z)

(65)

A point P on the geoid is projected onto the point Q on the ellipsoid along the ellipsoidal normal. The distance P Q is called the geoidal height or geoidal undulation and is denoted by N . The gravity anomaly vector 4g is defined as the difference between the gravity vector g at P and the normal gravity vector γ at Q 4g = gP − γQ

(66)

Their difference in magnitude is denoted the gravity anomaly and their difference in direction is denoted deflection of the vertical. The deflection of the vertical has two components, one in the direction of north-south line, denoted ξ, and one in the east-west direction, denoted η. These components are given by ξ =Φ−ϕ η = (Λ − λ) cos ϕ

(67)

Here (Φ, Λ) is astronomical coordinates and (ϕ, λ) is ellipsoidal coordinates. Gravity Disturbance At the same point P , the difference between the actual gravity vector gP and the normal gravity vector γP is the gravity disturbance vector δg = gP − γP

(68)

Their difference in magnitude is denoted the gravity disturbance and their difference in direction is the same as before, the deflection of the vertical (this direction is the same as before, since γP and γQ coincide virtually). Relations The famous Bruns formula relates the geoidal height to the disturbing potential N=

T γ

(69)

Here is some additional equations relating various quantities of the anomalous gravity field

14

δg = −

∂T ∂γ ∂γ T = 4g − N = 4g − ∂h ∂h ∂h γ

(70)

From these relations we can also express the fundamental equation of physical geodesy 1 ∂γ ∂T − + 4g = 0 ∂h γ ∂h

2.9

(71)

Gravity Anomalies Outside the Earth

If a harmonic function H is given on the surface of the earth, we can determine the values of H at points outside the surface of the earth by Poisson’s integral formula ZZ 2 r − R2 R H dσ (72) HP = 4π l3 σ

Which is actually just solving a boundary value problem. Notice that in order to determine H in a single point, you need to integrate over the entire surface of the earth, i.e. knowing the boundary values everywhere on the surface of the earth.

2.10

Stoke’s Formula

In deriving Stoke’s formula, we first arrive at Pizzetti’s Formula ZZ R T (r, ϑ, λ) = S (r, ψ) 4g dσ 4π

(73)

σ

Which determines the anomalous potential T everywhere in space, from integration over the entire earth. This is possible only because T has known boundary conditions and satisfies the Laplace equation 4T = 0. In the equation    2R R Rl R2 r − R cos ψ + l S (r, ψ) = + − 3 2 − 2 cos ψ 5 + 3 ln (74) l r r r 2r On the geoid itself, equation 73 reduces Stoke’s formula ZZ R T = 4gS (ψ) dσ 4π

(75)

σ

Where S denotes Stokes function S (ψ) =

  ψ ψ ψ 1 − 6 sin + 1 − 5 cos ψ − 3 cos ψ ln sin + sin2 sin(ψ/2) 2 2 2

This gives us the famous Stoke’s formula or Stoke’s integral ZZ R N= 4gS (ψ) dσ 4πγ0

(76)

(77)

σ

Here γ0 is the normal gravity on the geoid. This equation is very important in physical geodesy because it determines the geoid from gravity data. These formulas are based on spherical approximations, meaning that quantities of the order 3 · 10−3 N are neglected. Stokes formula only applies if the following conditions are satisfied • The reference ellipsoid has the same potential as the geoid U0 = W0

15

• The reference ellipsoid encloses a mass that i numerically equal to the mass of the earth • The reference ellipsoid has its center at the center of gravity of the earth These conditions assure that the spherical harmonics of degree zero and one is left out in the equations. We can also write Stoke’s integral in terms of polar coordinates, which has origin in a local point P and uses the spherical distance 5 ψ and the azimuth α N=

R 4πγ0

2π

Z

Z

π

4g (ψ, α) S (ψ) sin ψ dψ dα

(78)

ψ=0

α=0

And in terms of ellipsoidal coordinates (ϕ, λ) N (ϕ, λ) =

R 4πγ0

Z

2π

λ0 =0

Z

π/2

4g (ϕ0 , λ0 ) S (ψ) cos ϕ0 dϕ0 dλ0

(79)

ϕ0 =−π/2

Stoke’s function can also be written in terms of spherical harmonics, in this case the Legendre polynomials (zonal harmonics) S (ψ) =

∞ X 2n + 1 n=2

2.11

n−1

Pn (cos ψ)

(80)

Generalization to an Arbitrary Reference Ellipsoid

If the conditions in the former chapter are not fulfilled, this means that the zero and first order terms need to be included in the equation 77, so that T (ϑ, λ) = T0 + T1 (ϑ, λ) + T 0 (ϑ, λ)

(81)

Where T 0 (ϑ, λ) =

inf Xty

Tn (ϑ, λ)

(82)

n=2

Is equal to Stoke’s integral. Equation 81 is valid if the reference ellipsoid is sufficiently close to the geoid, so that the deviations can be treated as linear. Now the zero order term is GδM , δM = M − M 0 (83) R Here M is the mass of the earth and M 0 is the mass of the ellipsoid. It turns out that the first order term can always be assumed zero as long as the center of the ellipsoid coincides the center of earth, so that 81 can be expressed by ZZ GδM R T = + 4gS (ψ) dσ (84) R 4π T0 =

σ

Which holds for any arbitrary reference ellipsoid whose center coincides with the center of the earth. 5 cos ψ

= sin ϕ sin ψ 0 + cos ψ cos ψ 0 cos (λ0 − λ)

16

2.12

Koch’s Formula and Gravity Disturbances

Koch’s formula is an alternative to Stoke’s formula, expressing the same results, but in terms of the gravity disturbance δg. From Koch’s formula we have the anomalous potential on the geoid ZZ R T (r, ϑ, λ) = K (ψ) δg dσ (85) 4π σ

Where   1 1 K (ψ) = − ln 1 + sin(ψ/2) sin(ψ/2)

(86)

The same result is expressed by N=

R 4πγ0

ZZ K (ψ) δg dσ

(87)

σ

2.13

The Formula of Vening Meinesz (deflections of the vertical)

The figure in this subsection shows the intersection of the geoid and the reference ellipsoid, with the vertical plane. From this it is clear that the deflection of the vertical ε can be written in terms of dN (88) ds We now split this deflection into a north-south direction ε = ξ and an east-west direction ε = η, which can be written dN = −ε ds

ε=−

or

dN 1 ∂N =− R dϕ R ∂ϕ dN 1 ∂N η=− =− R cos ϕ dλ R cos ϕ ∂λ ξ=−

(89)

These are the equations we need to integrate in order to derive an expression for the deflection of the vertical at the surface of the earth. Doing this we arrive at the formulas of Vening Meinesz 1 ξ (ϕ, λ) = 4πγ0

Z

1 η (ϕ, λ) = 4πγ0

Z

2π

λ0 =0 2π

λ0 =0

Z

π/2

4g (ϕ0 , λ0 )

ϕ0 =−π/2

Z

π/2

dS(ψ) cos α cos ϕ0 dϕ0 dλ0 dψ

(90)

dS(ψ) 4g (ϕ , λ ) sin α cos ϕ0 dϕ0 dλ0 dψ 0 ϕ =−π/2 0

0

Where the differentiation of Stoke’s function is   dS(ψ) cos(ψ/2) 1 − sin(ψ/2) =− +8 sin ψ−6 cos(ψ/2)−3 +3 sin ψ ln sin(ψ/2) + sin2 (ψ/2) 2 dψ sin ψ 2 sin (ψ/2) (91)

17

3

Gravity Reduction

In order to use Stoke’s formula for determination of the geoid, it is required that the gravity anomalies 4g represent boundary values at the geoid. This means that a gravity measurement need to been reduced to sea level. A second criteria is that there must be no masses outside the geoid, which means that a gravity reduction in principle does two things • the topographic masses outside the geoid are completely removed or shifted below sea level • the gravity station is lowered from the earth’s surface P to the geoid P0

3.1

Free-air Reduction

Suppose we have measured gravity gP at the gravity station and want to reduce this measurement to the geoid g0 . Here we assume that all masses above the geoid have been mathematically removed, so we just want to move the measurement from P to P0 through free air. This can be obtained by the taylor expansion ∂g H + ... ∂H Neglecting all terms higher that 1 order we have that g0 = g −

∂g H ∂H =g+F

g0 = g −

(92)

(93)

Here F is the free air reduction. For many practical purposes it is sufficient to use the normal gradient of gravity, so that F ≈−

∂γ H ≈ 0.3086H [mgal] ∂h

(94)

Here H is in meters.

3.2

Bouguer Reduction

Assume that the area around the gravity station P is completely flat and horizontal (this can be made so mathematically by terrain correction). We then want to mathematically remove this plate, so there is nothing but free air between P and P0 . This incomplete version of the Bouguer reduction is AB = 4πGρH

(95)

3

With standard density ρ = 2.67 g/m this becomes AB = 0.1119H [mgal]

(96)

The complete version of the Bouguer reduction is by doing both the Bouguer plate reduction and the free air reduction 94, so that gB = g − AB + F = g + 0.1967H [mgal]

(97)

Now the gravity measurement g is reduced to the geoid, so that the gravity anomaly can be obtained by subtracting the normal gravity value γ. 4gB = gB − γ These are called the Bouguer anomalies. 18

(98)

3.3

Terrain Correction

This correction is made in order reduce the surrounding topographic to a flat Bouguer plate. This is done by dividing the surrounding area into templates, that are assumed having constant density and height. The effects of all these templates re them summarized to a total correction At X At = 4A (99) Adding this term to our total reduction, we arrive at the refined Bouguer gravity gB = g − AB + At + F

(100)

Which has a corresponding refined Bouguer anomaly that can be applied in Stoke’s integration and which sums up all the required corrections in order to reduce the gravity measurement to the geoid.

3.4

Poincaré and Prey Reduction

The Poincaré and Prey reduction is for reducing a gravity measurement to a point inside the earth, i.e. by comparing to the Bouguer anomaly, we need to do the same procedure, but after the free air reduction, we add the topography on top of the point. Often the terrain correction is left out in this procedure. Reducing the gravity gP at point P to a point Q inside the earth, is done by the integration Z

P

gQ = gP − Q

∂g dH ∂H

(101)

This can be obtained by Brun’s formula 69 if the mean curvature J of the geopotential surfaces and the density ρ between P and Q is known. This is not yet possible in physical geodesy, instead we apply the approximation gJ ≈ γJ

(102)

And then use Brun’s formula for the normal gravity field 53, to get ∂g ∂γ = + 4πGρ ∂H ∂h Which, by ignoring the variation of ∂γ/∂h with latitude and using the density ρ = 2.67 g/cm3 , we get the numerical approximation ∂g = −0.0848 gal/km ⇒ gQ = gP + 0.0848 (HP − HQ ) ∂H Which is a very crude formula, but is often used in practice.

3.5

(103)

(104)

Topographic-isostatic Reductions

In topographic-isostatic reductions we are trying to make the earth’s crust as homogenous as possible. The topographic masses are not removed as in Bouguer corrections, but are shifted below the geoid in order to compensate the "lack" of mass under the continents (according to isostatic theory). In the Pratt-Hayford model, this is done by standardizing the density of the crustal layer to ρ0 = 2.67 g/cm3 . In the Airy-Heiskanen model we need change the density of the roots, under the continents, which is done by changing the mass of the roots from ρ0 to ρ1 = 3.27 g/cm3 . Basically the method follows three step, from which two is known from Bouguer reduction, the difference comes from taking isostatic compensation into account 19

• removal of topography • removal of compensation • free-air reduction to the geoid

3.6

The Indirect Effect

As an effect of gravity reductions, you actually change the geoid, which is a result of removing or shifting some mass. Thus the surface computed by Stoke’s integral is not the geoid, but a slightly different geoid called the cogeoid. If we let N c denote the undulation of the cogeoid, the undulation of the geoid will be N = Nc +

δW = N c + δN γ

(105)

Where δW is the change in potential at the geoid and δN is an application of Brun’s formula 69. The indirect effect of Bouguer anomalies is very large, because the earth is generally topographic-isostatic compensated, thus the Bouguer anomalies cannot be used for determination of the geoid. With topographic-isostatic gravity anomalies, the effects are much smaller. Before applying Stoke’s theorem to these anomalies, we need to reduce the anomalies from the geoid to the cogeoid, which is done by free-air reduction (see subsection 3.1) δ = 0.3086 δN [mgal]

(106)

Now the topographic-isostatic gravity anomalies refers strictly to the cogeoid. This means that the application of Stoke’s theorem gives N c , which needs to be corrected by the indirect effect δN in order to give the undulation N of the actual geoid. Notice that these indirect effects also affects the deflections of the vertical.

3.7

The Inversion Reduction of Rudzki

The procedure suggested by Rudzki is a way of reducing gravity, while assuring that the indirect effect is zero.

3.8

The Condensation Reduction of Helmert

Here the topography is condensed to form a surface layer on the geoid, as a thin but heavy layer. This means that the mass is shifted along the vertical.

20

4

Heights

There is no direct way of measuring heights on the surface of the earth, because the equipotential surfaces is not parallel. Thus it is convenient to use potential differences as a measure of heights. This is useful because the potential difference is given as an integral between two points and this integral is independent of the path chosen between these points B

Z WB − WA = −

B X g dn = − g δn

A

(107)

A

In practical applications this integral is often transformed into a sum of small increments δn. The actual height difference can be expressed as 4nAB =

B X

B

Z δn =

dn

(108)

A

A

But is not independent of the path chosen, which is exactly the problem of height measurement on earth.

4.1

Geopotential Numbers

The geopotential number C is defined as the potential difference between two heights, one of these being on the geoid (at sea level) Z

A

C = W0 − WA =

g dn

(109)

0

This is called the geopotential number of point A. Formerly the concept of dynamic heights were used. This is just the geopotential number, divided by the normal gravity γ0 for a standard latitude H dyn =

C γ0

(110)

Basically this is just the same as the geopotential number 109, but deviates by a scaling factor that has the property of giving the dimensions of length. This does not mean that it represents the actual length.

4.2

Orthometric Height

The orthometric height H relates to the geopotential number in a way, that the integration is taken along the plumb line, so that Z

H

C = W0 − W =

g dH

(111)

0

Mathematically this means that the orthometric height can be obtained by the integration Z

W

H=− W0

dW = g

Z 0

C

dC g

(112)

But for practical use, we use the formula H= Where 21

C g¯

(113)

H

Z

1 g¯ = H

g dH

(114)

0

Is the mean value of gravity along the plumb line. Since g is not strongly dependent on H you can approximate the value by using simplified Prey reduction 3.4.

4.3

Normal Heights

The normal height corresponds to the orthometric heights, if the actual gravity field were the normal gravity field. This means that H∗

Z

γ dH ∗

C = W0 − W =

(115)

0 ∗

C

Z

H = 0

1 γ¯ = ∗ H

Z

dC γ

(116)

γ dH ∗

(117)

H∗

0

This geometrically means that if point P has orthometric height H there is a point Q along the plumb line, where WP = UQ , with normal height H ∗ above the ellipsoid. Notice that H is height above the geoid, while H ∗ is height above the ellipsoid.

4.4

Summarizing and Comparing Height Systems

The different kinds of heights can be written in means of the geopotential number Z

point

C = W0 − W =

g dn

(118)

geoid

The different heights are then height =

C G0

(119)

Where the height system differ according to the gravity value G0 , we have that • Dynamic height: G0 = γ0 = constant • Orthometric height: G0 = g¯ • Normal height: G0 = γ¯ The dynamic height has no geometrical interpretation, but has the dimension of height, and has the intuitive advantage that points on the same level surface has the same height value. The orthometric height is the height above the geoid, which gives it geometrical and physical significance, but because the equipotential surfaces are not parallel, different points on the same surface does not have the same height. The normal height does not have a direct geometrical interpretation, but is of great importance in modern geodesy.

22

5

Projection onto the Ellipsoid

Pizzetti’s Projection A point P has natural coordinates Φ, Λ, H and can be projected to the geoid along the plumb line. The point P0 on the geoid is then projected once again to the ellipsoid in point Q0 along the straight ellipsoidal normal. Pizzetti’s projection from the surface point P to the ellipsoidal point Q0 is thus a double projection of two heights H = P P0 and N = P0 Q0 . Helmert’s Projection A point P on the surface of the earth, has ellipsoidal coordinates ϕ, λ, h and is projected directly to the ellipsoidal point Q along the ellipsoidal normal. The projection is thus of height h = P Q.

23

6

Gravimetric Methods

6.1

Gravity Reductions and the Geoid

The integrals of Stokes (77) and Vening Meinesz (89) presupposes that the disturbing potential T is harmonic outside the geoid, which means that there is no masses outside the geoid. Thus it is needed to move or completely remove all masses outside the geoid in order to treat the potential problem as a boundary value problem. By exploiting the various reducing techniques introduced in section 3 we need to consider the indirect effect, that we change the geopotential surfaces in the process. This means that we change the actual geoid to a cogeoid. Furthermore gravity measurements are made on the surface of the earth, and need to be moved to sea level (the geoid), this is done by using the principle of free air reduction 3.1. This method is described by the following 6 steps: • The masses outside the geoid are either moved inside the geoid or completely removed. The indirect effect of this procedure on gravity g at the station need to be considered • The gravity station is moved from P down to the geoid P0 . Again, the effect on gravity need to be considered • The indirect effect, meaning the distance δN = P0 P c , is obtained by applying Brun’s theorem δW (120) δN = γ Here δW is the change in potential at the geoid • The gravity station is moved from the geoid P0 to the cogeoid P c . By doing this we obtain the boundary value of gravity at the cogeoid g c • The shape of the geoid is computed from the reduced gravity anomalies 4g c = g c − γ

(121)

by using Stokes formula. This gives us the undulation N c = Q0 P c . • Finally, the actual geoid is determined by considering the indirect effect. The geoidal undulation N is obtained by N = N c + δN (122)

6.2

Geodetic Boundary Value Problems

6.3

Molodensky’s Approach

Molodensky introduced a way performing the geodetic computations without having to make assumptions about the density of the masses above the geoid. From the surface point P a projection is made to the ellipsoid according to Helmert, see section 5. The difference between the ellipsoidal height h and the normal height H ∗ is denoted the height anomaly ζ = h − H∗

(123)

The normal height H ∗ is the height of the telluroid above the ellipsoid. The telluroid is a surface where the normal potential UQ is equal to the gravity potential WP of the surface point. Notice that this is not an equipotential surface, as the surface points P is not an equipotential surface. The height anomaly ζ is the difference between the telluroid and the surface and thus corresponds to the geoidal undulation N = h − H. The telluroid can be

24

determined by leveling combined with gravity measurements. This is done by determining the geopotential number Z

P

C=

g dn

(124)

0

Where g is the measured gravity at the surface and dn is the leveling increment. From this we can determine the normal height H ∗ by the relation "  2 #
C C C H∗ = 1 + 1 + f + m − 2f sin2 ϕ (125) + γQ0 aγQ0 aγQ0 Where γQ0 is the normal gravity at the ellipsoidal point Q0 . The normal height H ∗ of the surface point P corresponds to the ellipsoidal height h of the point Q, which is the ellipsoidal height of the telluroid ζP = hP − HP∗ = hP − hQ

(126)

We now redefine the gravity anomaly into being the difference between the actual gravity measurement gP at the the surface and the normal gravity γQ on the telluroid 4g = gP − γQ

(127)

The normal gravity γQ on the telluroid is computed from the normal gravity γQ0 on the ellipsoid, by upwards free-air reduction, which is done directly by "  ∗ 2 #
H∗ H 2 +3 (128) γ = γQ0 1 − 2 1 + f + m − 2f sin ϕ a a For this reason the gravity anomalies introduced by Molodensky is called free-air anomalies, but they have nothing in common with a free-air reduction of actual gravity. The height anomaly ζ can also be seen as the distance between the geopotential surface W = WP = constant and the corresponding spheropotential surface U = WP = constant. Hence, we can apply Brun’s theorem ζ=

T γQ

(129)

where T = WP − UP is the disturbing potential at ground level and γQ is the normal gravity at the telluroid. The surface given by the distance ζ above the ellipsoid is called the quasigeoid, but is not a level surface and has no physical meaning. The normal height HP∗ of a point P is its elevation above the quasigeoid, just as the orthometric height is its elevation above the geoid.

6.4

Linearization

The ellipsoidal height h can be determined by GPS and can be decomposed into h = H∗ + ζ

(130)

But we still need to determine the telluroid Σ and the height anomalies ζ in order to formulate the boundary conditions. The definitions of the gravity anomaly 4g and the gravity disturbance δg on the surface of the earth, has the same form as at the geoid, equation 127 and equation 68, but can be rewritten as 1 ∂γ ∂T − T + 4g = 0 ∂h γ ∂h 25

(131)

∂T + δg = 0 (132) ∂h The spherical approximation is then done by disregarding the flattening f in the equations. This reduces the above equations to 2 ∂T − T + 4g = 0 ∂h r

(133)

∂T + δg = 0 (134) ∂h Notice that the flattening f is a small quantity, because the ellipsoidal approximation is nearly spherical and that this does not mean abandoning the reference ellipsoid.

6.5

The Spherical Case

Continuing with the linearization assumption of the former subsection, we now work with a sphere with constant radius. Furthermore we now assume that we assume that the surface of the earth S is a level surface (which is not the case!). This means that we can expand T and 4g into a series of Laplace spherical harmonics and exploit the boundary conditions, equation 133 and equation 134, of the former subsection, obtaining ZZ R S(ψ)4g dσ (135) T = 4π σ

where S(ψ) =

∞ X 2n + 1 n=2

n−1

Pn (cos ψ)

(136)

where Pn (cos ψ) are Legendre polynomials. Here ψ denotes the spherical distance from the point where T is to be computed. We also obtain the formula ZZ R T = K(ψ)δg dσ (137) 4π σ

where   1 1 K(ψ) = − ln 1 + sin (ψ/2) sin (ψ/2)

(138)

Notice that this formula is completely analogous to the formula of Stokes 135.

6.6

Solution by Analytical Continuation

The idea is that our observations 4g or δg, given on the surface of the earth, are reduced or "analytically continued" to a level surface (or normal level surface). Because of our spherical approximation the level surfaces U = U0 or U = UP are spheres. For analytical continuation we use the Taylor series 4g = 4g ∗ +

∞ X 1 n ∂ n 4g ∗ z n! ∂z n n=1

where

26

(139)

z = h − hP

(140)

Is the elevation difference between the surface and the computation point P . If we only use the taylor expansion until first order, together with Stokes formula 135 and Bruns formula ζ ≈ γT0 (γ0 is some chosen mean value of γ) we have  ZZ  R ∂4g ∂ζ ζ= 4g − h S(ψ) dσ + h (141) 4πγ0 ∂h ∂h σ ∂4g ∂h h

Notice that the use of 4g − ensures that the height anomalies are calculated at sea ∂ζ level, while the last part ∂h h moves the value back up to the surface. Notice that the computations can be performed at different reference levels h0 , by replacing h by h − h0 (as indicated in figure).

6.7

Computational Formulas and the Molodensky Correction

From the previous subsection, we that that if we want to compute at point level, equation 141 becomes  ZZ  ∂4g R (h − hP ) S(ψ) dσ (142) 4g − ζ= 4πγ0 ∂h σ

This is our computational equation. Still, only using taylor expansion until first order, we have that 4g ∗ = 4g −

∂4g (h − hP ) = 4g + g1 ∂h

(143)

where ∂4g ∂4g (h − hP ) = − (h − hP ) (144) ∂h ∂r is the Molodensky correction. From the spherical approximation we can replace h by r and exploit that ZZ ∂4g R2 4g − 4gQ = dσ (145) ∂r 2π l03 g1 = −

σ

The from our computational equation 142, we have that ZZ ZZ R R 4gS(ψ) dσ + g1 S(ψ) dσ ζ= 4πγ0 4πγ0 σ

σ

(146)

= ζ0 + ζ1 Notice that all this can be expanded to higher orders for better accuracy. This analytic or harmonic continuation is not a gravity reduction which removes mass, but is instead an analytical operation performed on the external potential field directly at ground level, which ensures that the harmonic properties are preserved (it still satisfies the Laplace equation). In a sense we continue the external potential field at ground level 4g down to the sea level, where we can exploit 4g harmonic . In theory the method of analytical continuation presupposes that all masses outside the ellipsoid can be shifted inside the ellipsoid, in a way that the potential outside remains unchanged. In theory this is impossible for a continuous distribution of points, but in practical applications our data is discrete and the error is very small. 27

6.8

Determination of the Geoid from Ground Level Anomalies

From combining the conventional expression for the height h above the ellipsoid with the modern theory, we have N − ζ = H∗ − H

(147)

Since ζ is the undulation of the quasigeoid, the difference N − ζ is the distance between the geoid and quasigeoid. From this (including small steps) we obtain g¯ − γ¯ H γ¯ ZZ ZZ R g¯ − γ¯ R 4gS(ψ) dσ + g1 S(ψ) dσ + H = 4πγ0 4πγ0 γ¯

N =ζ+

σ

(148)

σ

Where Molodensky’s formula 146 has been used. The three terms of this equation can be explained as • First term: Stoke’s integral applied to free-air anomalies at ground level • Second term: The effects of topography • Third term: The distance between the geoid and quasigeoid

28

7

Statistical Methods in Physical Geodesy

29

8

Least-Squares Collocation

30