POLITECNICO DI MILANO
Department of Environmental and Civil Engineering
Doctoral programme in Environmental and Infrastructure Engineering
Bayesian gravity inversion
by Monte Carlo methods
Doctoral dissertation by:
Lorenzo Rossi
820437
Supervisor:Dr. Mirko Reguzzoni
Tutor:
Prof. Giovanna Venuti
The Chair of Doctoral programme:Prof. Alberto Guadagnini
Acknowledgement
I wish to express my deepest gratitude to my supervisor Dr. Mirko Reguzzoni for the encouragement, support and thoughtful guidance throughout the development of this research project. Your enthusiasm and critical attitude have stimulated my interest in the research activity and greatly improved the quality of this thesis. I will never stop thanking you for that.
I would like also to express my gratefulness to prof. Fernando Sans`o for laying the foundations of this project about five years ago. Without your experience, all the fruitful discussions and advices received during these years, this project would never have been concluded.
Another owe thanks goes to the staff of GReD s.r.l., in particular to Dr. Daniele Sampietro. We shared various activity in this project and it has been a pleasure working together.
The test case presented here cannot be realized without the precious contribu-tion of Fabio Mantovani, Virginia Strati, Marica Baldoncini and Ivan Callegari of University of Ferrara. I would like to thank you for the work done to retrieve the geological data necessary to apply the developed algorithm on a real test case.
A special thanks to the two reviewers of this research manuscript, prof. Bat-tista Benciolini and prof. Wolf-Dieter Schuh. Your thoughtful criticism and your suggestions increased a lot the quality of the final version of this manuscript.
Last, but not least, I would like to thank Greta for all the support received support during these years. It has been very important to conclude this work, especially during the last months. A special thanks also for patiently digitizing the figures of the original manuscript.
Contents
Abstract 7
Conventions 9
1 The gravity field and its observation 11
1.1 Theory of gravitation . . . 11
1.2 Earth’s gravity field . . . 15
1.3 Gravimetry . . . 17
1.3.1 Absolute gravimetry . . . 17
1.3.2 Relative gravimetry . . . 19
1.3.3 Gravimetry from moving platforms . . . 21
1.3.4 Gradiometry . . . 22 1.4 Data reduction . . . 23 1.4.1 Normal Gravity . . . 24 1.4.2 Free-air correction . . . 25 1.4.3 Bouguer correction . . . 26 1.4.4 Tidal correction . . . 28 1.4.5 Other effects . . . 28 2 Gravity interpretation 29 2.1 Geometrical approximation . . . 30 2.2 Forward methods . . . 31 2.3 Inverse methods . . . 32
2.3.1 Linear inverse problem . . . 33
2.3.2 Solution to the inverse gravimetric problem . . . 34
2.4 The Bayesian approach . . . 37
2.4.1 Prior probability . . . 38
2.4.2 The Bayesian approach in gravity inversion . . . 39
3 Optimization by Monte Carlo methods 41 3.1 Simulated annealing . . . 43
3.2 Gibbs sampler . . . 45
3.3 Markov random fields . . . 48
3.4 Deterministic optimization . . . 51
4 The Bayesian inversion algorithm 55 4.1 Problem setting . . . 55
4.2 Posterior probability formalization . . . 59
4.2.2 Prior probability distribution . . . 59
4.2.3 Posterior probability distribution . . . 65
4.3 Optimization algorithm . . . 66
4.3.1 Stochastic optimization . . . 66
4.3.2 Deterministic optimization . . . 72
4.4 Accuracy assessment . . . 73
4.5 Discussion . . . 74
5 Empirical setup of parameters 75 5.1 Physical parameters setting . . . 75
5.2 Weights tuning . . . 79
5.3 Temperature law . . . 81
6 The developed software 87 6.1 Structure of the software . . . 87
6.2 Forward modelling . . . 91
6.3 Inverse of the density correlation matrix . . . 94
6.4 Sampling algorithm . . . 98
6.4.1 Labels . . . 98
6.4.2 Densities . . . 100
7 Oil exploration test case 101 7.1 Problem setting . . . 101
7.2 Prior probability of labels . . . 105
7.3 Inverse solution . . . 109
7.4 Effect of the density correlation matrix . . . 113
8 Crustal determination test case 119 8.1 Motivation . . . 119
8.2 Gravity signal and 3D voxel model . . . 120
8.3 Geological setting of the area . . . 123
8.4 Geophysical input . . . 124
8.5 Building prior probability . . . 128
8.6 Estimated crustal model . . . 132
8.7 Comparison with a classical solution . . . 139
9 Conclusions 143 A Normal probability 145 A.1 Conditional distribution . . . 145
A.2 Product of normal distributions . . . 147
A.3 Normal likelihood distribution . . . 150
Abstract
The inverse gravimetric problem consists in the reconstruction of the Earth mass density distribution from the observation of its gravitational field. The solution to this problem is generally ill-conditioned and non-unique, but, introducing very strong constraints or numerical regularization, a unique solution can be retrieved. This could be not representative of the actual mass distribution, because of non-physical or too restrictive constraints. Moreover, a strong regularization may cause very smooth solutions in terms of estimated density. In this case, the identification of the boundaries between different materials becomes a hard task. To overcome these limitations, a possible option is to apply the Bayesian approach that easily allows to introduce prior information on the unknown parameters. Moreover, sec-ondary parameters can be easily introduced, e.g. to characterize the different types of material, allowing a sharp classification of the subsurface.
In the present work, a Bayesian gravity inversion algorithm is developed, with the aim of estimating a mass density distribution together with a classification of the different types of material. The latter allows to identify the boundaries of the different materials and is developed by borrowing image analysis techniques. Con-sequently, the investigated volume is subdivided into volume units (voxels), each of them characterized by two random variables: the label defining the type of material (discrete) and the density (continuous). The a-priori geological information is trans-lated in terms of this model, providing the mean density with the corresponding variability for each class of materials and the a-priori most probable label for each voxel with a set of neighbour rules. These rules have the aim to obtain a clustered model in terms of geometry with smooth boundary surfaces between the different materials. The final solution is retrieved by invoking the Maximum A Posteriori principle (MAP). The MAP is determined by applying Markov Chain Monte Carlo optimization algorithms. In particular, a simulated annealing performed by a Gibbs sampler is chosen to maximize the posterior distribution. The obtained result is then refined by a deterministic optimization algorithm.
The proposed method has been implemented into a hybrid Matlab/C code. The software is then applied in the frameworks of oil exploration and crustal investigation. The former application is performed by a simulated scenario, avoiding border effects in the gravity data reduction and allowing a better understanding of the meaning of the parameters that define the prior probability. The latter is related to the inversion of the crustal structure in south China, with the aim of improving its knowledge just below a detector of neutrinos and geoneutrinos. The obtained Bayesian solution is also compared with a classical solution to the inverse gravimetric problem, showing a good agreement in terms of estimated geometry (differences with about 1 km of
standard deviation). However, the Bayesian solution improve the smoothness of the estimated geometry and allows to infer horizontal and vertical density variations.
In general, all the tests show that the method is able to retrieve an estimated model that is consistent with the given prior information and fits the gravity ob-servations according to their accuracy. In this sense, one can conclude that the developed method is a sort of “artificial intelligence”, supporting the user by au-tomatizing the trial-and-error techniques. In fact, an initial guess on the model parameters is required, but differently from a trial-and-error solution these parame-ters are automatically adapted to fit the gravity observations, according to the rules introduced by the user as prior probability. If the solution is not satisfactory, this should be attributed to the weak information provided by the gravity or to the wrong or incompatible geological information supplied by the user.
Conventions
a, A Vector
a| Transpose of the vector a
aji Vector of elements from i to j, i.e. aji = [xi, xi+1, . . . , xj−1, xj]|
A Matrix
A−1 Inverse of a matrix
ai Column i of the matrix A
A−i Matrix A without the column i
Aij Element (i, j) of the matrix A
(A−1)ij Element (i, j) of the matrix A−1
I Identity matrix
ei Column i of an identity matrix I
e Vector with all the elements equal to 1 ex, ey, ez Unit vector representing the Cartesian axes
F (·) Fourier transform of (·)
F−1(·) Inverse Fourier transform of (·)
χ[a,b](·) Characteristic function
(
1 a ≤ · ≤ b 0 otherwise N[µ,C](•) Normal probability distribution function
exp12(• − µ)|C−1(· − µ) √
2π det C U[a,b](·) Uniform probability distribution function b−a1 χ[a,b](·)
d· Differential of ·
Chapter 1
The gravity field and its
observation
The measurement of the Earth’s gravity field has implications in many topics, like geophysics, geodesy, geodynamics, and ocean circulation. In the former, gravity has two big advantages compared to other geophysical prospecting measurements: it can be observed without being directly in contact with the ground, thanks to shipborne, airborne or satellite data acquisition, and it does not require any active source, as happens e.g. in seismic data acquisition. In fact, the gravity field spatial variation is directly related to the density variation inside the Earth. Therefore, measuring the gravity anomalies, i.e. the variation of the gravity with respect to a normal one, the density anomalies, namely the variation of the density with respect to the one attributed to a “normal” Earth, can be inferred.
Gravity prospecting is commonly used to investigate crustal structure at basin or regional scale and to oil and hydrocarbon exploration. In particular, in the last activ-ity gravactiv-ity prospecting is used for reconnaissance survey, thanks to its homogeneous accuracy and regular spacing during the acquisition, high benefits-costs ratio, and fast measurement time, or, whit a higher resolution, when seismic observations are not effective. That is why this method was used since 19th century (Krynski, 2012), but only starting from the 1980s it had an increasing diffusion, correspondingly to the advent of airborne and shipborne gravimetry, and later of satellite gravimetry. In fact, these measurement techniques allow to reduce surveying time and costs and to acquire accurate, homogeneous and regularly distributed observations even in inaccessible areas (Reynolds, 1997).
The aim of this chapter is to understand the principles of the gravitation theory, the Earth gravity field, and the basic principle of its observation.
1.1
Theory of gravitation
The fundamental equation related to gravity observation is the universal law of gravitation by Newton: “The magnitude of the gravitational force between two masses is proportional to each mass and inversely proportional to the square of their
separation” (Newton, 1687), shown in Equation 1.1: fP Q= −GmPmQ rQP r3 QP (1.1)
where fP Q is the attractive force generated by the point mass mQ on the point
mass mP, according to Figure 1.1, G is the universal gravitational constant which
approximately assumes the value G = 6.67 × 10−11m3
/s2kg, and rQP is the vector
joining points P and Q, rP Q magnitude. Using Cartesian coordinates, the position
vector can be expressed as:
rQP = (xP − xQ) ex+ (yP − yQ) ey + (zP − zQ) ez (1.2)
where ex, ey, ez represents the unit vector defining the three axes, and
x y z P mmPP fQP rP Q Q mQ fP Q
Figure 1.1: Gravitational attraction between two point masses.
x y z rQ Q(rQ) rP Q P (rP) rP ρ(rQ) B O
Figure 1.2: Gravitational field due to a generic mass distribution.
Dividing Equation 1.1 by mP the Newtonian gravitational attraction gN(P )
ex-erted by Q on P is retrieved, i.e. the acceleration felt by P from Q is:
gN(P ) = −GmQ
rQP
r3 QP
(1.3)
The three components of the gravitational attraction in a Cartesian coordinate sys-tem can be computed by introducing Equation 1.2 into 1.3:
gN,x(P ) = gN(P ) · ex = −GmQ xP − xQ r3 QP gN,y(P ) = gN(P ) · ey = −GmQ yP − yQ r3 QP gN,z(P ) = gN(P ) · ez = −GmQ zP − zQ r3 QP (1.4)
CHAPTER 1. THE GRAVITY FIELD AND ITS OBSERVATION
Another important property of the gravitational attraction is that it is a vectorial field which can be described by a scalar potential. In other words the vectorial gravitational field can be expressed as the gradient of a scalar field, the so-called gravitational potential V (Heiskanen and Moritz, 1967):
gN(P ) = ∇V (P ) (1.5)
The expression of the potential field V generated by the mass mQ at a generic point
P is:
V (P ) = GmQ rQP
(1.6) It is worth to notice that the expression of the gravitational acceleration through the potential V , as shown in Equation 1.5, allows to express the three components of g(P ) by means of a scalar field.
The potential generated by a set o masses mq, where q = 1, 2, 3, . . . , n, can
be computed by applying the superposition principle, as the sum of the individual contribution of each point mass (Kellogg, 1929):
V (P ) = G n X q=1 mq rqP (1.7)
Assuming that masses are distributed continuously inside a volume B, as shown in Figure 1.2, their density becomes:
ρ = dm
dv (1.8)
where dv is the infinitesimal element of volume and dm the infinitesimal element of mass. Now, defining Q as a generic point inside the body B, the sum shown in Equation 1.7 becomes an integral:
V (P ) = G Z B dm |rQP| = G Z B ρ(Q) |rQP| dv (1.9)
Assuming xP, yP, and zP the coordinates of the observation point and x, y, and z
the coordinates of the “running” point inside B, the infinitesimal volume dv is equal to dx · dy · dz. Introducing this assumption, Equation 1.9 turns in:
V (xP, yP, zP) = G Z Z Z B ρ(x, y, z) q (xP − x)2+ (yP − y)2+ (zP − z)2 dxdydz (1.10)
Applying Equation 1.5, we can compute the value of the three components of the gravity acceleration of the continuous body B. For example, the component along the z coordinate becomes:
gN,z(xP, yP, zP) = ∂V ∂zP = = G ∂ ∂zP Z Z Z B ρ(x, y, z) q (xP − x) 2 + (yP − y) 2 + (zP − z) 2 dxdydz =
= G Z Z Z B ρ(x, y, z) ∂ ∂zP 1 q (xP − x)2+ (yP − y)2 + (zP − z)2 dxdydz = = −G Z Z Z B ρ(x, y, z) zP − z (xP − x) 2 + (yP − y) 2 + (zP − z) 23/2dxdydz (1.11)
By properly setting the domain of the integral of Equation 1.10 or 1.11 according to the shape of the body B, the potential or the gravity acceleration contribution gen-erated by simple bodies with different shapes, e.g. a sphere, a cylinder, a rectangular prism, etc., can be computed.
In the case of a rectangular prism, assuming a Cartesian coordinate system with the origin in the point P , such that xP = 0, yP = 0, zP = 0 as depicted in Figure
1.3, and a mass density that is constant inside the prism itself, the potential can be computed starting from Equation 1.10:
V (P ) = Gρ Z x2 x1 Z y2 y1 Z z2 z1 dxdydz px2+ y2+ z2 (1.12)
Finally, integrating Equation 1.12 the potential of the rectangular prism is retrieved as (Nagy et al, 2000): V (P ) = Gρ
xy log(z + r) + yz log(x + r) + zx log(y + r)
−x 2 2 arctan yz xr − y2 2 arctan zx yr − z2 2 arctan xy zr x2 x1 y2 y1 z2 z1 (1.13)
where r =px2+ y2+ z2 is the modulus of the position vector of the “running” point
inside the prism. Deriving Equation 1.13, the gravitational acceleration generated
x y z x1 x2 y1 y2 z1 z2 P
CHAPTER 1. THE GRAVITY FIELD AND ITS OBSERVATION
by the prism can be computed. For example, along the z direction, the Newtonian attraction of the prism becomes (Nagy et al, 2000):
gN,z(P ) = ∂V (P ) ∂z = Gρ
x log(y + r) + y log(x + r) − z arctan xy zr x2 x1 y2 y1 z2 z1 (1.14)
According to the Syst`eme Internationale (International System, abbreviated SI) the unit of measurement of gravitational acceleration is m/s2. Nevertheless, a unit of
measurement that is very often used in gravimetry and geodesy fields is the Gal, that derives from the CGS. In particular, 1 Gal = 1 cm/s2 = 10−2 m/s2. Furthermore, according to the order of magnitude of the observed anomalies, usually sub-multiples of the Gal are used in gravimetric application, namely mGal and µGal.
1.2
Earth’s gravity field
The gravitation theory, explained in the previous section, is valid only when an inertial reference system is considered. However, and Earth-fixed system cannot be considered inertial because there are at least two important non-uniform motions: the revolution of the earth around the sun and the rotation of earth around its own axis.
The former effect can be neglected even if its contribution is quite large (of the order of 0.6 Gal) because the Earth is in a free fall motion with respect to the sun, namely the centrifugal force (revolution) and the centripetal one (heliocentric attraction) are balanced at the Earth barycentre. Consequently, a point on the Earth surface suffers only of a small residual attraction, i.e. the tidal potential, of the order of 0.025 mGal (Sans`o and Sideris, 2012). Therefore, an Earth centred reference system with the ZQI axis along its rotation axis and XQI and YQI pointing
always in the same direction with respect to the fixed stars can be considered quasi-inertial, i.e. Newton law’s holds with a quite good approximation.
Now, we can define also an Earth-fixed reference system, where the Z axis is the same of the previous one, i.e. Z = ZQI, while X and Y rotate around the Z axis with
a constant angular speed with respect to XQI and YQI. This assumption modifies
the fundamental law of dynamics of a point P with mass m. In fact, starting from the second law of dynamics:
m ¨rQI(P ) = f (P ) + mgN(P ) (1.15)
where the subscript QI denotes that the acceleration ¨rQI is expressed in the
quasi-inertial reference frame, gN(P ) is the Newtonian attraction acting on P and f (P ) is
the sum of the other forces acting on m. Recalling Coriolis theorem, the acceleration ¨
rQI present in Equation 1.15 can be expressed in terms of the Earth-fixed reference
frame as (Arnold, 1978):
m[ ¨r + 2ω × ˙r + ω × (ω × r) + ˙ω × r] = f (P ) + mgN(P ) (1.16)
where ω is the angular velocity of the Earth. If the measurement point has no velocity with respect to the Earth, i.e. ˙r = 0, the terms 2ω × ˙r and ¨r vanish.
Furthermore, the angular velocity of the Earth can be considered a constant in time, thus also the term ˙ω × r assumes null value. Solving Equation 1.16 under the assumption that the Z axis of the Earth-fixed reference system is parallel to the earth rotation axis, the force f (P ) required to keep the point P clamped to the Earth can be computed as (Sans`o and Sideris, 2012):
f (P ) = m[ω × (ω × r) − gN(P )] = m−ω2(xeX + yeY) − gN(P )
(1.17) where eX and eY are the unit vector representing the Cartesian axes of the
Earth-fixed system. If the point mass is Earth-fixed to the Earth, it feels an acceleration equal to −m1f (P ), that is the so-called gravity acceleration g(P ):
g(P ) = −1
mf (P ) = gN(P ) + ω
2(xe
x+ yey) = gN(P ) + gC(P ) (1.18)
where gC(P ) is the centrifugal acceleration. It is worth to notice that also the
centrifugal acceleration can be expressed as the gradient of a potential:
gC(P ) = ω2(xex+ yey) = ∇ 1 2ω 2 x2+ y2 = ∇VC (1.19)
where VC is the centrifugal potential. This turns in the fact that the Earth’s gravity
field is a potential field too, allowing to derive the gravity field potential W (P ) as:
W (P ) = V (P ) + VC(P ) (1.20)
The gravity acceleration vector g represents the total acceleration felt by a body. Its magnitude g = |g| is called gravity and the vector is directed in the direction of the plumb line (Heiskanen and Moritz, 1967). As it will be shown in the next section, gravity is a quantity that is directly observable.
It is important to remark that the previously neglected Coriolis acceleration plays a role when the observation point P is not Earth-fixed. In fact when ˙r and ¨r are not equal to 0 the term m[2ω × ˙r], i.e the so-called Coriolis force present in Equation 1.16, has to be considered when the gravity acceleration is observed from a moving body, e.g. satellite or airborne platform. This term is proportional to the relative velocity between the moving body and the Earth (Heiskanen and Moritz, 1967).
An approximate value of the gravity potential is the so called normal potential U (P ), obtained by approximating the Earth with an ellipsoid of revolution:
U (P ) = VE(P ) + VC(P ) (1.21)
where VE(P ) is the ellipsoidal gravity potential that is a function of two ellipsoidal
parameters, e.g. major and minor semiaxes a and b, or major semiaxis a and eccen-tricity e, and of the total mass of the Earth M . The difference between the potential W (P ) and the normal potential U (P ) is the so called anomalous potential T (P ):
T (P ) = W (P ) − U (P ) = V (P ) − VE(P ) (1.22)
The anomalous potential highlights the deviation of the real gravity field with respect to the normal one, i.e. it shows mass anomalies with respect to a reference density model consistent with the normal potential.
CHAPTER 1. THE GRAVITY FIELD AND ITS OBSERVATION
The gravitational potential is harmonic outside the masses. Therefore, in case of the anomalous potential, solving the Laplace equation:
∇2T = 0 (1.23)
by separation of variables in an outer spherical domain, lead to the fact that the anomalous potential can be expressed as a spherical harmonic series (Heiskanen and Moritz, 1967; Blackely, 1996; Sans`o and Sideris, 2012).
1.3
Gravimetry
As we have already introduced in the previous section, gravity is an accelera-tion. Therefore, its measurement should involve determinations of length and time. However, such an apparently simple task is not easily achievable at the accuracy required in gravity surveying.
Gravimetry can be subdivided in two main classes: absolute or relative. As sug-gested by the name, the former deals with the modulus of the gravity that is directly observed, while the latter deals with gravity variations at different stations. Absolute gravimetry usually requires long period of observation with bulky and very expensive instruments, while relative gravimetry is featured with faster measurement time and lighter and cheaper instruments. Furthermore, to interpret gravity anomalies there is no need to know the absolute value of gravity, as it will shown later. Nevertheless, thanks to networks of gravity stations covering all over the world, the absolute value of gravity can be retrieved through relative measurement when at least one differ-ence is observed with respect to one of this station. The most important network is the International Gravity Standardisation Net (IGSN 71) that provides the value of the absolute gravity at each station (Morelli et al, 1974; Krynski, 2012).
The instruments used to measure gravity are called gravimeters or gravity meters and rely on one of the following physical laws (Torge, 1989; Fedi and Rapolla, 1993): pendulum oscillation, free-falling body or Hooke’s law describing spring elongation. Gravity was traditionally measured from the ground, but starting from the 1980s (Schwarz and Li, 1997) measurements from moving platforms became feasible. This turned in a wider use of gravimetry, since it became possible to observe the gravity even in inaccessible areas with uniform data distribution, e.g. by using an airborne platform.
In the next subsections absolute and relative gravimetry will be discussed, then the principle of airborne and shipborne gravimetry will be introduced. In particular, the focus will be put on the physical principles behind the observations and on the achievable accuracy. Finally, also gradiometry, a kind of measurement also used in satellite missions, are briefly explained.
1.3.1
Absolute gravimetry
Absolute gravimeters are able to observe the modulus of the gravity vector. The two main physical principles used are: the free-fall body and the pendulum.
The free-fall body method is based on the equation of motion, considering that only the gravity field is acting:
m¨z = mg(z) (1.24)
where ¨z = d2z/dt2, z is the vertical coordinate and t is the time. Considering the gravity field constant along the vertical direction Equation 1.24 can be integrated two times as follows:
˙z = ˙z0+ gt (1.25)
z = z0+ ˙z0t +
g 2t
2 (1.26)
Since the initial position z0 and velocity ˙z0 are not well known, it is required to
measure the position and the time at least three times during the falling path as shown in Figure 1.4. This allows the estimation of the initial conditions together with the value of gravity acceleration. Modern instruments, called multiposition experiments, allows to measure more than three positions, thus introducing redun-dancy in the estimation of gravity and initial conditions that can be reckoned by using a least squares approach. Position and time are measured with laser inter-ferometers and atomic clocks, respectively. Those instruments are able to reach accuracies of the order of ±0.2 nm (interferometers) and ±0.1 ns (clocks). Propa-gating these values, a final accuracy of the order of the µGal in the observed gravity can be obtained, considering a common falling distance of 0.2 m and a consequently falling time of about 0.2 s (Torge, 1989). However, to reach such a level of accuracy, a sequence of multiposition experiments distributed over different days is required. In fact, the repetition allows reducing random errors, like errors in the resolution of the interference-fringe-signal and time, errors in associating the signal with time impulses, and microseismic effects. Moreover, there are other systematic errors that have to be modelled in order to avoid biases in the final estimation, e.g. length and time standards used, light path, electronic time measurements and non-gravitational forces (residual atmospheric pressure effects, magnetic disturbing forces, electrical currents, etc.).
Figure 1.4: Distance-time diagram
representing proof-mass position in the
free-falling experiment. The three
points (at least) where position and time are observed are shown over the graph. z x ϕ ϕ0 g m ` g sin ϕ
Figure 1.5: Scheme of the forces act-ing on an absolute pendulum.
CHAPTER 1. THE GRAVITY FIELD AND ITS OBSERVATION
The pendulum method is based on the measurement of the oscillation period T and of the pendulum length ` of a freely swinging pendulum. Starting from the equation of oscillation, according to Figure 1.5, the following relationship holds:
m` ¨ϕ(t) + mg sin ϕ(t) = 0 (1.27)
where ϕ(t) is the phase angle and ¨ϕ(t) is the angular acceleration. Integrating Equation 1.27 over a full period leads to an elliptical integral. After expansion into a series, the period T can be evaluated as (Torge, 1989):
T = 2π s ` g 1 + ϕ 2 0 16 + . . . (1.28)
If the amplitude ϕ0 is kept small, the absolute value of g can be derived by measuring
T and `. It is worth to notice that Equation 1.28 is valid in case of an ideal pendulum, while in case of a real pendulum the mass distribution and the shape of the pendulum itself has to be taken into account for retrieving the gravity.
The pendulum method is no longer applied today due to its poor accuracy that is of about 0.1 mGal. The main errors affecting the observations are related to temperature variations that causes length variations, amplitude reductions, and de-formations that happen during the oscillation process. Nevertheless, this kind of instruments governed gravimetry for about 300 years, therefore it has a fundamen-tal importance in the gravity observation. Moreover, recent results of pendulum observations are still contained in some gravity networks (Torge, 2001).
1.3.2
Relative gravimetry
A relative gravity measurement deals with the gravity variation between two stations or with gravity variation in time. This requires to measure only the time or the length, keeping the other quantity as fixed. As a consequence, relative mea-surements can be performed more easily and more accurately than absolute ones. Relative gravimeters can rely on dynamic or static methods. In the first case, the period of a pendulum oscillation is observed, while in the second one the observation is a length, usually the length variation of a spring.
The dynamic method is based on the pendulum principle. In fact, by using the same pendulum at two stations P1 and P2, then measuring the oscillation period
and, finally, applying twice Equation 1.28, the gravity difference can be reckoned as: g2 g1 = T 2 1 T2 2 → g2− g1 = −2g1 T2− T1 T2 + g1 (T2− T1)2 T2 2 (1.29)
Also in relative gravimetry, this method is no more used because of its accuracy. In fact, by using relative pendulums only an accuracy of the order of 0.1 mGal can be achieved. Therefore, starting from the 1930’s they were superseded by elastic spring gravimeters. However, they were still used until 1960’s for gravimeter cali-bration lines. In fact, they directly retrieve gravity in its own measurement units without requiring a calibration, as it happens in case of static gravimeters, as shown afterwards.
The static gravimeters use a counterforce to keep a test mass in equilibrium with the gravity force. Measuring the counterforce, the gravity variations can be detected by applying a transformation, derived from a calibration function. Usually, an elas-tic counterforce is used, but also a magneelas-tic counterforce could be an alternative. Elastic spring gravimeters are based on the principle of the spring balance. In fact, by changing the applied gravity force, the elongation of a spring changes follow-ing Hooke’s law. Sprfollow-ing balance gravimeters can be subdivided into translational systems and rotational systems, as shown in Figure 1.6.
For a translational system, also called vertical spring balance gravimeter, see Figure 1.6(a), the equilibrium condition is (Torge, 1989):
mg − k(` − `0) = 0 (1.30)
where k is the elastic spring constant and ` and `0 the spring length with or without
the force applied, respectively. Observing two different lengths `1 and `2 at two
dif-ferent stations P1 and P2, their gravity difference can be reckoned applying Equation
1.30 at the two points:
g2− g1 =
k
m(`2− `1) (1.31)
The calibration consists into determining the value of the ratio k/m used to convert the observed length difference into the gravity variation.
z m mg `0 ` k(` − `0) (a) z τ (α0+ α) α a m mg O (b) z δ d h ` k( ` − `0 ) m mg a b α O (c)
Figure 1.6: Elastic spring gravimeter principle: a) vertical spring balance, b) lever torsion spring balance c) lever spring balance.
Rotational systems use a lever to support the proof mass m, able to rotate around an axis O. Equilibrium is reached through a torsion spring, see Figure 1.6(b), or through a helical spring, see Figure 1.6(c). According to Figure 1.6(b), in torsion spring gravimeters the equilibrium condition is given by:
mga sin α − τ (α0+ α) = 0 (1.32)
where a is the lever length, α the angle between vertical and the lever, α0 and τ
the pretension angle and the torsion constant of the spring, respectively. Gravity differences generate angular changes in Equation 1.32, thus solving this non-linear system it is possible to retrieve gravity variation between the stations. Furthermore,
CHAPTER 1. THE GRAVITY FIELD AND ITS OBSERVATION
thanks to this non-linearity, the sensitivity of the instruments can be increased by properly choosing the system parameters (Torge, 1989).
The torsion spring can be substituted by a helical spring, as shown in Figure 1.6(c). In this case the equilibrium condition becomes:
mga sin(α + δ) − kbd` − `0
` sin α = 0 (1.33)
As it happens in torsion lever gravimeters, by properly setting the parameters of the system, its sensitivity can be increased.
Comparing lever spring with vertical spring gravimeters, the sensibility increases of about 2000 times. That is the reason why the lever spring is the widest used principle when building relative gravimeters.
It is worth to notice that in order to reach an accuracy of the order of 0.01 mGal a high precision reading system and a high stability of the counterforce in time are required. In other words, the elasticity of the spring should be stable while moving the instrument between different stations. Another limitation of spring gravimeters is that they have a range of measurement related to the maximum extension of the spring. Modern instruments, e.g. LaCoste and Romberg model G (LaCoste, 1934), presents a range of about 7000 mGal, while in the 1950’s maximum gravity range was about 2000 mGal.
Despite all the efforts to protect instruments from environmental disturbances, spring gravimeters suffers of drift effects. They are caused by ageing of the spring material and short-time changes that occurs during field survey. In order to evaluate the systematic effects of the drift, usually repeated occupations of one or more points during the same survey are performed. Therefore, under the assumption that gravity is constant at the same location, the drift function is interpolated experimentally from observations and then it is removed from the measurements taken at stations surveyed only once.
1.3.3
Gravimetry from moving platforms
The usage of moving platform is a very interesting task for rapid and high reso-lution gravimetric survey, especially facing with inacessible areas, e.g. oceans, polar regions, high mountains, tropical forest, etc. The most used platforms are airplanes and ships, but also helicopters or land vehicle could be used as an alternative.
The principle used in kinematic platform is based on the equation of motion by Newton in an inertial reference system:
¨
r = f + g (1.34)
where r is the position vector, ¨r is the acceleration, f is the force acting on a unitary proof mass, i.e. the spring counterforce in case of a spring gravimeter, and g is the gravitational effect acting on the proof mass. Thanks to Global Navigation Satellite System (GNSS) sensors and using an Inertial Measurement Unit (IMU) or a modified land gravimeter mounted on a damped two-axes stabilized platform the acceleration ¨r and the force f can be measured, respectively.
Equation 1.34 can be expressed in a local system, with the z axis oriented as the vertical and the x and y axes towards the eastern and northern direction respectively, in order to retrieve the gravity at a certain point:
gL = ¨rL− RL
BfB+ 2ωLIE+ ωEBL × ˙rL (1.35)
where L denotes quantities expressed in the local frame, RL
B is the rotation matrix
between body and local frames, ωIEL and ωEBL the angular velocity of the Earth-fixed reference frame E with respect to the inertial reference frame I and the angular veloc-ity of the platform B with respect to the Earth-fixed reference frame E, respectively. Equation 1.35 can be applied to retrieve the scalar value of the gravity, or the full gravity vector.
The damped platforms use two gyroscope or accelerometer pairs operating in a feed-back mode, to realize the local system and to observe the magnitude of the gravity vector. The latter can be determined starting from 1.35 that, projecting the vectors on the vertical direction, becomes (Torge, 2001):
g = fz− ¨z + 2ωcosϕ sin αv +
v2
r (1.36)
where fz is the vertical acceleration observed by the gravimeter, ¨z is the vertical
component of the acceleration of the platform, ω is the Earth angular speed, ϕ the geodetic latitude, α the geodetic azimuth, v the platform velocity with respect to the Earth, and r the distance to the centre of the Earth. The velocity dependent term in Equation 1.36 represents the so-called E¨otv¨os correction which increases (for a west-east directed course) the angular velocity of the Earth rotation and the centrifugal acceleration arising from the angular velocity v/r of the platform around the centre of the Earth (Torge, 1989).
The kind of disturbing acceleration, namely fz, varies depending on the used
platform, e.g. in shipborne measurements with stabilized platforms they have a pe-riod between 2 and 20 s, while in airborne measurements their pepe-riod is between 1 and 300 s. These specific platform dependent characteristics depends on the different vehicle velocity (i.e. 10 ÷ 20 km/h for a ship, 250 ÷ 450 km/h for an airplane) and has an inpact on the kind of data filtering to be applied at the observations. The platform velocity also influences the final resolution of the observed data. Typically, resolutions from 0.1 to 2 km in case of shipborne acquisition and of about some 10 km in case of airborne acquisition can be achieved (Torge, 2001).
The accuracy of the observed gravity signal is generally comprised between 1 and 6 mGal, depending on the survey condition (e.g. sea state, air turbulence, velocity of the platform, etc.) and on the quality of the separation between gravity and disturb-ing accelerations, mainly influenced by the effectiveness of the dampdisturb-ing system, the filtering techniques, and the accuracy of velocity observation (Torge, 2001; Schwarz and Li, 1997).
1.3.4
Gradiometry
Gradiometry is the measurement of the second derivatives of the Earth gravity potential, that represents the curvature of the field. It is represented by a second
CHAPTER 1. THE GRAVITY FIELD AND ITS OBSERVATION
order tensor, the so called gravity gradient tensor (Marussi, 1985):
W = grad(g) = Wxx Wxy Wxz Wyx Wyy Wyz Wzx Wzy Wzz (1.37)
The tensor contains local gravity field information and it is interesting for high-resolution gravity field determination. Its units of measurement is s−2, but it is generally expressed in 10−9s−2 = ns−2 traditionally called E¨otv¨os unit (E). It can be determined by means of gravity gradiometer, by observing the different reaction of neighbour proof masses. So, a gradiometer is composed by a couple of accelerometers rigidly connected. The observed quantity is the force difference at the centre of mass C of the system, namely:
f2− f1 = WC(r2− r1) (1.38)
where r1 and r1 are the position vector of the two sensors. In order to recover the
full gravity gradient tensor, a series of couple of accelerometers disposed on different axes is used.
This technology was also used in satellite measurements. In fact, thanks to force differentiation, no GNSS observations are required to retrieve gravity gradients, thus allowing the usage of GNSS only for Satellite to Satellite Tracking (SST) technique and with the aim of giving a position to the observations. A recent satellite gradiom-etry missions was GOCE by ESA launched in 2009, designed with a nearly polar circular orbit at altitudes of about 250 km. The GOCE mission was able to estimate the gravity field with about 1 mGal of accuracy at about 70 km × 70 km resolution.
1.4
Data reduction
Sections 1.2 and 1.3 describe the Earth gravity field and its observations. Nev-ertheless, to interpret observed signal in terms of local density variations, the data have to be to reduced with the aim to isolate only the signal due to the investigated body. In fact, these bodies usually cause variations in the gravity field of the order of 100 mGal, as shown in figure 1.7, thus making crucial the data reduction in their identification.
Figure 1.7: Main gravitational constituent of terrestrial gravity field (ESA, 2008).
In order to isolate the effects of the investigated bodies, it is required to model all the other components of the gravity signal, which in general can be thought as the sum of the following effects (Blackely, 1996):
1. attraction of the reference ellipsoid and effect of the Earth rotation (normal gravity);
2. effect of observation point elevation above sea level (free-air); 3. effect of “normal” masses above sea level (Bouguer and terrain);
4. time-dependent variation of the effect of the attraction of the sun and of the moon (tidal correction);
5. effect of density variation with respect to the “normal” density considered before. The last effect is the one generated by local crustal structures that usually are the target of the gravity inversion. Therefore, to isolate this contribution, all the others has to be modelled and the observations has to be reduced by removing them. In the following, the various reduction applied to the observed data will be described.
1.4.1
Normal Gravity
The so-called normal gravity is the gravity contribution generated by a reference ellipsoid. Starting from the normal potential, shown in Equation 1.20, and applying the gradient operator, as discussed in Equation 1.5, the normal gravity γ can be computed. The outcome is a series that, truncated at the second order, assumes the shape (Blackely, 1996):
γ(ϕ) = γ0 1 + α sin2ϕ + β sin22ϕ
(1.39)
where α and β are constants depending on the parameters of the chosen reference ellipsoid, e.g. the total mass M , the flattening f , the major semi-axis a, the angular velocity of the Earth ω, on the gravity at the Equator γ0, i.e. γ(ϕ = 0), and on the
latitude of the computational point ϕ. Nevertheless, also a closed-form of the normal gravity equation exists, the so-called Somigliana equation (Somigliana, 1929):
γ(ϕ) = γ0 1 + k sin2ϕ p 1 − e2sin2ϕ ! (1.40)
where k is a constant depending on the ellipsoid parameters and e is the ellipsoid eccentricity. Again the ellipsoid parameters used in Equation 1.40 depend on the considered reference ellipsoid. The set of parameters in case of different standard reference ellipsoid can be found in Heiskanen and Moritz (1967), Blackely (1996), and Sans`o and Sideris (2012).
The observed data reduced for the normal the gravity are the so-called gravity anomaly ∆g, whose expression in a point P becomes:
∆g(P ) = go(P ) − γ(ϕP) (1.41)
where go(P ) is the observed value of gravity at point P and γ(ϕ
P) is the above
CHAPTER 1. THE GRAVITY FIELD AND ITS OBSERVATION
1.4.2
Free-air correction
Equations 1.39 and 1.40 do not take into account the elevation hP of the point
P above the reference ellipsoid. In other words, they consider elevation as a null quantity. Therefore, to correctly take into account the decay of the gravity signal for the distance to the reference ellipsoid, the so-called free-air correction has to be applied. As a first approximation, the free-air correction can be recovered by a Taylor expansion of the gravity as (Heiskanen and Moritz, 1967):
g(P ) = γ(ϕP) + ∂g ∂hhP + 1 2! ∂g2 ∂2hh 2 P + . . . (1.42)
where the zero order term is exactly the normal gravity and the series starting from the first order term is what usually is called free-air correction to the normal gravity. Truncating the expansion at the first order the free-air correction assumes the following value (Blackely, 1996):
γfa(P ) = ∂g ∂hhP ∼ = −0.3086mGal/m h P (1.43)
Therefore, applying the correction to the observed gravity, the so-called free-air gravity anomaly is retrieved as:
∆gfa(P ) = go(P ) − γ(ϕP) − γfa(hP) (1.44)
Summarizing, the free-air gravity anomaly represents the gravitational effect at point P of the local density variations with respect to the density of the reference
Topography Moho Density anomalies Crust Core Mantle Crust Mantle Core
observed gravity normal gravity gravity anomaly
−
=
−
=
Figure 1.8: Free-air gravity anomaly computation: the masses generating the observed gravity signal minus the masses generating the normal gravity results in the masses generating the free-air gravity anomaly. They are represented together with their signals in terms of gravitational acceleration. It is possible to see how the free-air gravity anomaly highlights effect of local density variations.
ellipsoid. It is worth to notice that the gravitational attraction of the masses outside the ellipsoid, namely the masses composing the topography, is not taken into account into the free-air anomaly. In fact, with the normal gravity principle those masses are considered anomalies with respect to the reference ellipsoid, with a contribution, in terms of density contrast, equal to their density, since the background density outside ellipsoid is null. The worldwide resulting free-air gravity anomaly and a scheme of its causative masses are graphically shown in Figure 1.8. In practice, the free-air gravity anomaly highlights the effects of local density variations with respect to a layered ellipsoidal Earth, thus making possible to distinguish them from the global effect due to Earth’s mean shape and rotation.
1.4.3
Bouguer correction
As previously stated, the free-air gravity anomaly does not take into account the effect due to the masses outside the reference ellipsoid, namely the masses of the topography, bathymetry, ices, etc.
As a first approximation, when the observation is directly taken at a ground level, namely the height above the ellipsoid of the observation point corresponds to the height of the topography, the masses of topography can be considered an infinitely extended slab, i.e. the so-called Bouguer slab, with a thickness corresponding to the elevation of the observation point. Therefore, according to the gravitational effect of an infinite slab, the so-called simple Bouguer correction is given by (Heiskanen and Moritz, 1967):
gsb(P ) = 2πGρChP (1.45)
where ρC is the crustal density, typically assumed to be ρC = 2670 kg/m3.
Intro-ducing this value into Equation 1.45, it becomes (Blackely, 1996):
gsb(P ) = 0.1119mGal/m hP (1.46)
It is worth to recall that density equal to ρC = 2670 kg/m3 is an approximation that
is valid only when the point P is located on the ground. However, it is possible to take also observations in contact with the surface of the sea, e.g. by shipborne gravity survey. In that case the density anomaly to be considered in Equation 1.45 is the difference between the “true” density, namely the one of the water ρW = 1030 kg/m3,
and the density of the crust present in the reference ellipsoid, whose mean value is usually assumed to be ρC = 2670 kg/m3, as recalled above; the thickness of the slab
is assumed to be the bathymetric depth under the point P . This last reduction can be thought as an ideal filling of the bathymetry with crust.
A remark is that the choice of crustal density at ρC = 2670 kg/m3 is done
according to its mean value all over the Earth. This translates in the fact that applying the simple Bouguer correction, the crustal density anomalies with respect to the chosen reducing density ρCare reflected in the final reduced signal. Consequently,
if more information about the local geological characteristics of the crust is available, this reference density value has to be chosen according to them, e.g. in case of oceanic crust a good choice of the reference density could be ρC = 2900 kg/m3.
As stated before the simple Bouguer correction is an approximation. In fact, it takes into account only the shape of the terrain just above the observation points,
CHAPTER 1. THE GRAVITY FIELD AND ITS OBSERVATION observed gravity ×105 ρ = 2670 kg/m3 ρ = 3070 kg/m3 ρ = 2970 kg/m3 60 40 20 0 9.790 9.795 9.800 −60 −40 −20 0 20 40 60 Distance [km] Dep th [km] [mGal] (a) observed gravity normal gravity ρ = 2970 kg/m3 ρ = 2670 kg/m3 ∆ρ = −400 kg/m3 ∆ρ = 300 kg/m3 60 40 20 0 −1000 −500 0 −60 −40 −20 0 20 40 60 Distance [km] Dep th [km] [mGal] (b) free-air anomaly ρ = 2970 kg/m3 ρ = 2670 kg/m3 ∆ρ = −400 kg/m3 ∆ρ = 300 kg/m3 60 40 20 0 −200 0 200 400 −60 −40 −20 0 20 40 60 Distance [km] Dep th [km] [mGal] (c) complete Bouguer simple Bouguer ∆ρ = −400 kg/m3 ∆ρ = 300 kg/m3 60 40 20 0 −150 −100 −50 0 −60 −40 −20 0 20 40 60 Distance [km] Dep th [km] [mGal] (d)
Figure 1.9: Example of gravity data reduction (Blackely, 1996). Starting from (a) the observed signal, the effect of the various corrections is shown: (b) normal gravity reduction, (c) free-air correction, and (d) Bouguer correction.
without considering variations in the elevation around the point P . This translates into the fact that valleys or mountains near the points are not correctly modelled, thus tending to overcompensate the terrain effect, as shown in Figure 1.9. To avoid
overcompensation, simple Bouguer correction should be replaced by a proper terrain correction, also called complete Bouguer correction.
Terrain correction is obtained by modelling the gravitation signal of the topo-graphic masses gtcof the area surrounding observation points. The topography could
be usually approximated with a digital elevation model, which gravitational attrac-tion is computed by means of forward modelling, e.g. by means of a set of regular prisms (Equation 1.14), point masses (Equation 1.4), tesseroids (Uieda et al, 2016), etc. In order to speed-up the computation parallel computing and/or Fourier tech-nique are often used (Godson and Plouff, 1988; Tscherning et al, 1992; Biagi and Sans`o, 2001; Sampietro et al, 2016) to perform this task.
The final reduced value of gravity is called complete Bouguer gravity anomaly and is derived as follows:
∆gb(P ) = go(P ) − γ(ϕP) − γfa(hP) − gtc(P ) (1.47)
1.4.4
Tidal correction
The tidal correction has to be applied to compensate the time-varying effect of the attraction caused by the sun and the moon, that can be detected by gravimeters. The magnitude of this variation depends on the latitude of the station and is always smaller than 0.3 mGal. In particular, it decreases with the latitude and it has a strong periodic component, with a period of the order of 12 hours (Blackely, 1996). Despite formulas to calculate the tidal effect at any time and at any place on the Earth exist (Longman, 1959), a common technique (especially for less accurate sur-veys) is to empirically remove this effect together with other time-dependent effects (e.g. the instrument drift) by means of repeated occupation of a point (Blackely, 1996).
1.4.5
Other effects
Up to now we discussed effects that should be removed during the data reduction independently from the measurement technique and interpretation target. In fact, according to these two factors also other reduction could be added to the reduc-tion workflow shown in Figure 1.9. In particular, when the goal of the interpretative problem is a specific crustal features or buried bodies (e.g. a salt dome), effects of the known surrounding density anomalies should be removed from the observed signal, thus leaving into the observation only the gravitational contribution of the investi-gated structure. Also this further reduction is performed by forward modelling, as already discussed for terrain correction.
According to Figure 1.9, when investigating local crustal structure there is also the big contribution of the Moho to be removed, namely the density contrast of crust and mantle with respect to the normal reduction. In this case, the reduction can be performed by taking into account an isostatic compensating Moho (Heiskanen and Moritz, 1967) or by introducing a global crustal model, e.g. CRUST1.0 (Laske et al, 2013) or GEMMA (Reguzzoni and Sampietro, 2015).
An other contribution that should be removed is the effect of moving platforms in data acquisition (E¨otv¨os effect), explained in Section 1.3.3.
Chapter 2
Gravity interpretation
Gravity interpretation consists in using observed gravity anomalies to determine one or some characteristics of the crustal density anomalies or buried bodies (e.g. a salt dome). Examples of the characteristics that can be retrieved are: the depth, the shape or the (anomalous) density distribution. In other words, starting from the link between the parameters and the data:
y = F (x) (2.1)
the aim of the gravity interpretation is to obtain an estimate of the parameters x for which the corresponding estimated data y agree as best as possible with the observations. The relationship F (x) is called forward function and, in case of gravity interpretation, it can be expressed by the well known Newton’s law (see Section 1.1). In practice, the forward function maps the model space into the data space. Therefore, by means of this function, the external gravity field can be uniquely determined for every admissible model and this operation is called forward modelling. Nevertheless, different models can generate the same gravity effect. This implies that the solution to the gravity interpretation problem is inherently non-unique.
The gravity interpretation problem can be faced by means of different approaches. They can be mainly classified into: forward methods or inverse methods. As sug-gested by the name, the former are based on the forward modelling, that is a trial-and-error procedure is adopted to retrieve the best values of the unknown parameters according to the observed gravity and to the experience of the operator. On the other hand, the latter retrieve the unknown parameters by analytically inverting the for-ward function. In this case the non-uniqueness has to be formally solved. This task can be in turn performed by different approaches, all of them trying to additionally exploit the available geological knowledge of the region.
The aim of this chapter is to briefly describe the different approaches to the gravity interpretation. In particular, the first section will be dedicated to the geo-metrical approximation used to describe the model parameters, the second to a basic description of the forward methods. As for the inverse methods, they will be treated more in details, dedicating a section to the general concept and another section to the Bayesian approach to inverse problems, thus introducing the fundamental idea of the present work.
2.1
Geometrical approximation
The geometrical approximation used to describe the mass model is assumed independently from the type of method used to solve the interpretative problem. This choice affects the shape of the forward function and the number and the kind of parameters present in the vector x. There are three possible approximations (Fedi and Rapolla, 1993): 3D, 2D, and 21/2D.
The 3D approximation is the one that better describes the real world. Using this approximation, the volume of the model is finite and its geometry is variable along the three dimensions of the space, as shown in Figure 2.1(a). Therefore, the forward function is derived by integrating Equation 1.9 over a three-dimensional domain and considering the density as a function of all the three coordinates, namely ρ = ρ(x, y, z) in a Cartesian coordinate system.
As for the 2D and 21/2D approximations, they are very similar. In both cases the
body parameters are a function only of two coordinates, namely a horizontal and a vertical ones, and they are used to interpret gravity profiles, namely a set of gravity observations acquired along the straight line defining the horizontal axis. Therefore, the assumptions performed on the model parameters are:
- the density is considered as a constant in the direction perpendicular to the vertical plane, namely ρ = ρ(x, z), assuming that the coordinate x identifies the main horizontal direction;
- the geometry of the body is described by a surface in the x-z plane and does not vary in the y direction, as show in Figure 2.1(b). This allows to simplify the domain over which the integral of Equation 1.9 is computed, since it depends on two coordinates only.
The difference between 2D and 21/2D is in the way in which the domain of this
integral is defined along the y direction. In fact, in the first case the integral domain is assumed infinite in that direction, while it is finite in the second case. These two geometrical approximations simplify the problem from the computational point of view, but have to be carefully managed. In fact, the choice of the main horizontal axis direction is crucial to get a realistic approximation. A good choice is to set the
x y z (a) 3D approximation x y z (b) 2D approximation
Figure 2.1: Geometrical approximation used to describe the parameters of the investigated body in the forward function.
CHAPTER 2. GRAVITY INTERPRETATION
x axis laying on the shorter symmetry axis of the investigated structure when using the 2D approximation, and on one of the symmetry axes of the structure when using the 21/2D approximation (Fedi and Rapolla, 1993).
The choice of the kind of modellization influences the number of parameters to be estimated in the interpretative method. Usually, the 3D modellization requires a huge number of unknown parameters that are instead significantly reduced in the 2D and 21/2D cases. On the other hand, when reducing parameters additional
approxi-mations have to be introduced, thus reducing the quality of the solution too. That is why the 21/2D case could be a good compromise between model simplification and
number of unknown parameters, even though the 3D modellization is increasingly chosen thanks to the increasing computational power.
2.2
Forward methods
Using forward methods the solution to the interpretative problem is found by following a trial-and-error approach. In practice, the operator iteratively builds a model of the investigated crustal feature or buried body(ies) and verifies the consis-tency between the modelled forward signal and the observed gravity anomaly. The procedure is repeated until a satisfactory fitting of the observed gravity anomaly is reached, as shown in Figure 2.2. The starting model is derived from the geologi-cal information available into the investigated region, and the operator, driven by the observed gravity, manually modifies it until a satisfactory solution is reached. Therefore, the key point in this kind of methods is the experience and sensibility of the operator to both geology and gravity modelling.
Gravity observations Compare the model
anomaly with the observed anomaly Guess the initial
model parameters (from geology) Estimated model Compute the model gravity anomaly Do they match? Update the model
parameters
YES NO
Figure 2.2: Flow chart of a trial-and-error procedure.
In order to increase the number of possible trials, it is required the usage of fast forward techniques. Examples of forward techniques can be found in literature: Talwani and Ewing (1960) introduce an algorithm that is able to compute the effect of any body by approximating its shape with a horizontal lamina with the boundary of an irregular polygon, while Goetze et al (1982) and G¨otze and Lahmeyer (1988) use polyhedra to represents bodies. This last approach is more versatile, since it allows a larger number of possible shapes. Furthermore, in the last years it becomes
reliable the usage of a set of simple shapes in building complex geometry, like point masses, prisms (Nagy et al, 2000; Zhang and Jiang, 2017), spheres, or tesseroids (Uieda et al, 2016), especially thanks to the available computational power.
Anyway, the usage of these methods is very time-consuming, because the space of possible solutions is explored by using trials, and the final result cannot be evaluated from a statistical point of view. In fact, the obtained solution is data driven but it largely depends on the operator choices, thus giving the best solution according to the “experience of the operator”. Some results of trial-and-error solutions coherent with the geological a-priori information and the gravitational signal can be found in Oezsen (2004); Caratori Tontini et al (2009); Gordon et al (2012).
2.3
Inverse methods
Inverse methods solve the gravity interpretation problem by analytically inverting the forward function. An important feature of this kind of solution is that they can be also evaluated from a statistical point of view, filling a gap present in forward methods.
The inverse theory is a general theory that can be applied in different physical science fields. In fact, the aim is to make inferences about physical parameters from a set of observed data. In some ideal circumstances there is an exact theory that explains how the data should be treated in order to recover the correct model. For instance, if the seismic velocity inside the Earth depends only on depth, the velocity model can be reconstructed exactly from the measurement of the arrival time as a function of the distance of seismic waves using an Abel transform (Herglotz, 1907; Wiechert, 1907).
Nevertheless, these exact inversion schemes holds only for few limited cases (Snieder and Trampert, 2000). In fact, they are usually applicable only for ide-alistic situations that may not hold in practice. and they are often very unstable. Moreover, in many inverse problems the model that one aims to determine is a con-tinuous function of the space variables. This means that the model has infinitely many degrees of freedom, but in a realistic experiment the amount of data that can be used for the determination of the model is usually finite. A simple count of variables shows that the data cannot carry sufficient information to determine the model uniquely. Therefore, the model obtained from the inversion of the data does not necessarily correspond to the true model that one seeks.
In general there are two reasons why the estimated model differs from the true model. The first reason is the non-uniqueness of the inverse problem, that causes several (usually infinitely many) models to fit the data. The second reason is that real data are always contaminated with errors that are consequently propagated into the estimated model.
Depending on the shape of the forward function F (x) the inverse problem can be linear or non-linear with respect to the parameters. Nevertheless, non-linear inverse problems are significantly more difficult than linear ones and presently no satisfactory theory exists for them (Snieder and Trampert, 2000). However, a possible solution is to linearize them around an approximate value of the parameters, reducing them
CHAPTER 2. GRAVITY INTERPRETATION
to the inverse linear problem theory. That is why in the following we briefly analyse only the solution of the linear inverse problem.
2.3.1
Linear inverse problem
The inverse problem is formalized considering the set of equations linking the data y ∈ Rm to the set of parameters x ∈ Rn, applying the forward function F (x) : Rn 7→ Rm, as shown in Equation 2.1. When the relation between data
and parameters is linear, Equation 2.1 can be written by a matrix product in the following way:
y = Fx + ν (2.2)
where F is the so-called forward matrix of dimensions m × n and ν is the observation noise. Notice that a certain arbitrariness can be used to choose the parameters composing the vector x. Taking as an example the inverse gravimetric problem, a possible choice is to describe the density of the investigated volume as a linear function with respect to the depth, while another possible alternative is to discretize the continuous density function with a given resolution. Obviously, there are other many possibilities, but these examples show that the same problem can be modelled with different parameter vectors x and consequently also with different forward matrices F. This means that the parameters contained in x do not necessarily describe the real word in a perfect way, but they already introduce some restrictions on the class of models that can be reconstructed. In the following, we assume that the true model is represented by the ideal value of these parameters, as if they completely represent the truth.
The inverse solution consists of retrieving an estimate ˆx of the model parameters, starting from the observed data y. In literature, there are different ways to define the inverse operators (Menke, 1984; Parker, 1994; Tarantola, 2005). However, the most general linear mapping from data to the estimated model can be written as:
ˆ
x = F−gy (2.3)
where the operator F−g is the generalized inverse of the matrix F. This definition is given due to the fact that the number of data is usually different from the number of parameters and therefore a formal inverse of F does not exist (Snieder and Trampert, 2000). However, for the general purposes of this section this definition is sufficient, without entering into the details of the different methods used to compute the inverse operator.
The estimated model and the true model can be put in relationship by introduc-ing Equation 2.2 into Equation 2.3, obtainintroduc-ing:
ˆ x = F−gFx + F−gν = x + F−gF − I x | {z } limited resolution + F−gν | {z } error propagation (2.4)
where the operator F−gF = R is called resolution kernel.
Since the true and the estimated models have to be equal in an ideal case, namely ˆ
introduced in the by the estimation process. The former, namely (R − I) x, ac-counts for the fact that the components of the estimated parameter vector are a linear combination of the true model parameters. In practice, only an averaging of the true parameters is retrieved, thus obtaining a blurred solution. The blurring occurs because the mean is a smoothing operator and therefore the finest details cannot be described. To avoid the blurring effect the resolution kernel has to be equal to an identity matrix, namely R = I that causes the vanishing of the limited resolution term. This condition means that the inverse problem has a perfect res-olution. Consequently, the resolution kernel can be used to understand how much the estimated model parameters are independent from the estimation process itself. However, a remark is that this resolution matrix does not completely explain the relation between the estimated model and the real physical model, because it does not take into account how the initial choice of the parameters reduces the possible models that can be reconstructed by the inversion. In practice, it is not possible to know how much the chosen model class is really representative of the truth.
The last term in Equation 2.4, namely F−gν, describes how the errors present into the data are mapped into the estimated model. Since these errors are not deter-ministically know, a statistical analysis is required, e.g. the errors in the estimated model caused by errors in the data are evaluated through the well known covariance propagation law. In a general case, the covariance matrix of the estimated model Cx ˆˆx can be retrieved starting from the covariance matrix of the observation noise
Cνν as:
Cx ˆˆx = F−gCνν F−g
|
(2.5) Equation 2.5 shows that a model without errors is obtained only if the generalized inverse matrix is a null matrix, leading to the (absurd) estimated model ˆx = 0. On the other hand, this error-free solution also implies that the resolution kernel is equal to zero. This condition is far from the ideal condition of the resolution kernel equal to an identity matrix. Hence, a perfect resolution and no errors in the estimated model cannot be simultaneously obtained and, in practice, an acceptable trade-off between error-propagation and limitations in the resolution has to be found.
A final remark is that each element of the data vector is a weighted average of all the parameters. This lead to the fact that the matrix F act as a smoothing operator, although the observation points are outside the causative body. Therefore, the inverse operator F−g is an “unsmoothing” operator and may cause numerical instability, namely small variations in the data can cause large variations in the solution (Blackely, 1996).
2.3.2
Solution to the inverse gravimetric problem
We have already seen that inverse problems can suffer of non-uniqueness and instability. The former is caused by the fact that the same gravitational signal can be generated by different bodies (Roy, 1962), while the latter is due to the shape of the forward operator, that is a smoothing function.
These problems can be overcome mainly by two approaches: by introducing very restrictive a-priori assumptions or by introducing numerical regularization. In practice, the former acts on the choice of the parameters reducing the class of the
CHAPTER 2. GRAVITY INTERPRETATION
estimated models and allowing a perfect resolution of the system. The latter, instead of reducing the number of parameters, introduces additional conditions among the parameters of the model such that an almost perfect resolution is guaranteed.
A common restrictive hypothesis is to reduce the geometry of the model to the presence of two layers, as shown in Figure 2.3. The two layer approximation may be a valid approach for example when the crustal-mantle discontinuity (i.e. the Moho) has to be inferred from the gravity observations. The notation of Figure 2.3 reflects this case, denoting the density of the crust and the mantle with the symbols ρC and
ρM, respectively, and the geometry of the discontinuity surface and the topography
with D(ξ) and H(ξ), respectively.
z 0 ξ D(ξ) H(ξ) ρM ρC
Figure 2.3: Two layer model of crust and mantle. The surface D(ξ) divides the two layers of density ρC(crust) and ρM (mantle), while the surface H(ξ) represents the topography.
Nevertheless, the restrictions in geometry are not sufficient and further restrictive hypotheses have to be introduced to make the solution unique. Sampietro and Sans`o (2012) demonstrated that the solution is unique at least in the following three cases: - the only unknown of the problem is the density of one of the two layers depending only on the planar coordinates, e.g. ρC(ξ), once the geometry of the discontinuity
D(ξ), the topography H(ξ) and the density of the other one, e.g. ρM(ξ, z), are
given;
- the only unknown is the geometry of the surface between the two layers D(ξ), once the density of the two layers ρC(ξ, z) and ρM(ξ, z) and the topography H(ξ)
are fixed to a known value;
- the density of the crust ρC(ξ, z) is modelled as a linearly varying function with
the depth and the only unknown is its vertical gradient, once the geometry of the surface between the layers D(ξ), the topography H(ξ), the density of the mantle ρM(ξ, z) and density of the crust at the topography ρC(ξ, z = H(ξ)) are given.
Another classical formulation of the inverse problem that guarantees the unique-ness of the solution is represented by the so-called theory of ideal bodies (Parker, 1974, 1975; Dumrongchai, 2007). Here, finite bodies with known density contrast are used and some parameters of these bodies are estimated uniquely.
However, both the theory of ideal bodies and the theorems on the two layer model above presented may be too restrictive to produce good quality solutions, especially in areas where complex geological structures are present. To overcome the above
