Potential-field inversion for a layer with uneven thickness: the Tyrrhenian Sea density model

Accepted Manuscript

Title: Potential-field inversion for a layer with uneven thickness: the Tyrrhenian Sea density model

Authors: F. Caratori Tontini, L. Cocchi, C. Carmisciano

PII: S0031-9201(07)00229-4

DOI: doi:10.1016/j.pepi.2007.10.007

Reference: PEPI 4876

To appear in: Physics of the Earth and Planetary Interiors Received date: 4-12-2006

Revised date: 16-10-2007 Accepted date: 17-10-2007

Please cite this article as: Tontini, F.C., Cocchi, L., Carmisciano, C., Potential-field inversion for a layer with uneven thickness: the Tyrrhenian Sea density model, Physics of the Earth and Planetary Interiors (2007), doi:10.1016/j.pepi.2007.10.007

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Accepted Manuscript

Potential-field inversion for a layer with

1

uneven thickness: the Tyrrhenian Sea density

2

model

3

F. Caratori Tontini

∗ , L. Cocchi

, C. Carmisciano

4

aIstituto Nazionale di Geofisica e Vulcanologia, Sede di Portovenere,via Pezzino

5

Basso 2, 19020, Fezzano (SP), ITALY.

6

Abstract

7

Inversion of large-scale potential-field anomalies, aimed at determining density or

8

magnetization, is usually made in the Fourier domain. The commonly adopted ge-

9

ometry is based on a layer of constant thickness, characterized by a bottom surface

10

at a fixed distance from the top surface. We propose a new method to overcome this

11

limiting geometry, by inverting in the usual iterating scheme using top and bottom

12

surfaces of differing, but known shapes. Randomly generated synthetic models will

13

be analyzed, and finally performance of this method will be tested on real gravity

14

data describing the isostatic residual anomaly of the Southern Tyrrhenian Sea in

15

Italy. The final result is a density model that shows the distribution of the oceanic

16

crust in this region, which is delimited by known structural elements and appears

17

strongly correlated with the oceanized abyssal basins of Vavilov and Marsili.

18

Key words:

Potential field, inversion, mathematical modeling, Tyrrhenian Sea

19

PACS:

91.25.Rt, 93.85.Hj, 93.85.BC, 93.85.Jk, 91.50.Kx

20

* Manuscript

Accepted Manuscript

1 Introduction

21

The inversion of potential-field anomalies, to obtain information on the density

22

or magnetization distribution generating the observed field, is characterized by

23

non-uniqueness of the solution. To reduce the level of ambiguity the user must

24

introduce some additional information about the source, known as “a priori”

25

information. Large-scale potential-field anomalies are usually inverted in the

26

2-D limit, with the horizontal extension of the source effectively larger than

27

the depth extension. In this approach, a unique solution is obtained because

28

the source is confined between two surfaces of given shapes and the density

29

or magnetization is invariant in each vertical column of the layer. This model

30

is particularly effective when the observed anomaly is mostly generated by

31

topographic features, assuming the top surface of the layer is described by

32

topography. Inversion methods using a linear approach by discrete cells pa-

33

rameterization exist, but they are particularly ineffective when dealing with a

34

large amount of data since the dimensions of the associated matrices are of the

35

same order of the squared number of data points. With increasing amounts

36

of data, the inversion is usually performed in the Fourier domain. Since the

37

pioneering work of Parker (1972), different methods have been developed. In

38

particular, Parker and Huestis (1974) have provided an iterative framework

39

for the inverse problem. This approach has been successfully used to compute

40

the field at a constant level by using an equivalent source distribution (Hansen

41

and Miyazaki, 1984; Pilkington and Urquhart, 1990). Oldenburg (1974) mod-

42

ified this method to determine the geometry of the density interface from

43

∗

Corresponding author

Email addresses: caratori@ingv.it

(F. Caratori Tontini),

(L.

Cocchi),

(C. Carmisciano).

Accepted Manuscript

the gravity anomaly, assuming a known density distribution. More recently,

44

Hussenhoeder et al. (1995) have inverted magnetic data directly in cases of un-

45

even tracks. However, when dealing with inversion aimed at determination of

46

density or magnetization distribution, these methods often assume a constant

47

thickness for the layer. In some geological settings this could be a straightfor-

48

ward assumption, for example in the analysis of marine magnetic anomalies,

49

where the magnetized portion of the crust generating the observed anomaly

50

can be considered uniform in thickness (Macdonald et al., 1983). However, in

51

general, the constant thickness of the layer may be a limiting assumption, and

52

thus the density or susceptibility obtained under this hypothesis may be signif-

53

icantly different than in cases of uneven thickness. The choices of the limiting

54

top and bottom surfaces fully characterize the non-uniqueness domain of this

55

inversion method, and thus the solution strongly depends on the particular

56

shapes of these enclosing surfaces (Blakely, 1995). This problem is particularly

57

evident in the case of gravity, in which the isostatic setting is characterized

58

by a deep root that sustains the topography. Thus, we have modified this

59

method and its theoretical framework in order to deal with differing top and

60

bottom surfaces of the layer, as in cases of uneven thickness. This has been

61

obtained by a simple mathematical modification of the method by Parker and

62

Huestis (1974). We will demonstrate the capability of the method with some

63

synthetic tests, both for gravity and magnetic data. Finally we will develop

64

the density model of the Southern Tyrrhenian Sea (Italy) starting from the

65

isostatic residual anomaly, which allowed us to obtain important information

66

about the distribution of the oceanic crust in this region.

67

Accepted Manuscript

2 Theoretical framework

68

Parker (1972) has demonstrated that the total-intensity magnetic anomaly of

69

the layer in the Fourier domain can be written as

70

F [ΔT ] = μ

2 Θ

Θ

e

n=1

( −| k | )

n! F [M (z

− z

)], (1)

71

while the equivalent gravitational expression is

72

F [g

] = 2πγe

n=1

( −| k | )

n! F [ρ(z

− z

)], (2)

73

where

74

(1) ΔT and g

are the magnetic and gravity anomalies;

75

(2) μ

and γ are, respectively, the magnetic and gravity constants;

76

(3) F indicates the Fourier operator;

77

(4) Θ

and Θ

are phase factors depending on the directions of the magne-

78

tization and the ambient field (Schouten and McCamy, 1972);

79

(5) | k | =

k

+ k

is the wavenumber obtained by the spatial frequencies k

80

and k

;

81

(6) z

and z

are the top and bottom surfaces;

82

(7) M and ρ are the 2D distribution of magnetization and density confined

83

between z

and z

.

84

Though these expressions have been successfully used to evaluate anoma-

85

lies generated by topographic sources (Macdonald et al., 1983; Parker, 1972),

86

they can be easily introduced in the inverse model by an iterative procedure.

87

In the case of a layer with constant thickness t, Parker and Huestis (1974)

88

have demonstrated that, by isolating the n = 1 term of the summation of

89

Accepted Manuscript

eqs.(1),(2), we obtain a self-consistent equation:

90

F [M ] = 2 F [ΔT ]

μ

Θ

Θ

e

(1 − e

) −

n=2

( −| k | )

n! F [Mz

], (3)

91

for the magnetic case, and

92

F [ρ] = F [g

]

2πγe

(1 − e

) −

n=2

( −| k | )

n! F [ρz

], (4)

93

for the gravity anomaly. These equations can be solved iteratively by making

94

an initial guess and then solving for a new magnetization or density distri-

95

bution which is then reintroduced into eqs.(3) and (4) until the method nu-

96

merically converges. This procedure is subject to some instability (Schouten

97

and McCamy, 1972; Blakely and Schouten, 1974), such that the result should

98

be low-pass filtered at each iteration. A detailed discussion on the calibration

99

of the filtering procedure is given in Phipps Morgan and Blackman (1993).

100

They posed this problem as an inverse problem and determined the optimal

101

filter, according to a weighted combination of a minimum slope or maximal

102

smoothness criteria, in order to find the crustal thickness variation. The order

103

of expansion can be stopped at n = 5; this is a good compromise according to

104

the resolution of marine gravity data (Marks and Smith, 2007). The original

105

approach by Parker and Huestis (1974) can be modified in order to deal with

106

a layer characterized by an uneven thickness. It simply states from eq.(1), that

107

we can isolate again the n = 1 term of the summation as follows

108

F [M (z

− z

)] = − 2 F [ΔT ]

μ

Θ

Θ

| k | e

−

n=2

( −| k | )

n! F [M(z

− z

)]. (5)

109

Since z

= z

is a root of the polynomial (z

− z

), it is also clear that (z

− z

)

110

is a divisor of this polynomial. This also means that the term (z

− z

) can be

111

factorized as (z

− z

)P

(z

, z

), where P

(z

, z

) is a polynomial of degree

112

n − 1. At this level, we can use surface magnetization M

≡ M (z

− z

), to

113

Accepted Manuscript

obtain a new self-consistent equation:

114

F [M

] = − 2 F [ΔT ]

μ

Θ

Θ

| k | e

−

n=2

( −| k | )

n! F [M

P

(z

, z

)], (6)

115

where it is straightforward to demonstrate that

116

P

(x, y) = x

+ x

y + ... + xy

+ y

(7)

117

is the complete polynomial of order n with unitary coefficients. The gravita-

118

tional equivalent expression is:

119

F [ρ

] = 2 F [g

]

2πγe

−

n=2

( −| k | )

n! F [ρ

P

(z

, z

)], (8)

120

where the surface density takes on the equivalent expression ρ

≡ ρ(z

− z

).

121

Eqs. (6) and (8) can be solved iteratively as in the original formulation by

122

Parker and Huestis (1974) for surface magnetization or density. When the

123

method converges, true magnetization or density is determined by dividing for

124

(z

− z

). Again, low-pass filtering at each iteration is needed to provide con-

125

vergence because of the instability in the high-frequency sector of the Fourier

126

spectrum. To properly define the density or magnetization distribution, we

127

adopt a cosine roll-off filter H, defined in the following equation:

128

H = 1 | k | < k

129

H = 0.5

1 + cos

π( | k | − k

) (k

− k

)

k

≤ | k | ≤ k

(9)

130

H = 0 | k | > k

131132

where the interval [k

, k

] defines the thickness of the filter. A good compro-

133

mise can be chosen as k

= 0.5k

. We are thus left with the choice of the

134

optimal value of k

. To this aim, we adopt an empirical criterion. The conver-

135

gence of the method is achieved when the iterated distribution of density or

136

magnetization no longer exhibits any significant change, namely < 0.1%, with

137

Accepted Manuscript

respect to the previous distribution. This happens only if the filter is properly

138

determined; otherwise the differences between the distributions at following

139

iterations diverge or oscillate. A set of decreasing values of k

is thus used,

140

and the optimal value is determined as soon as the differences between the

141

distributions at each iteration assumes the correct decreasing trend. Iterations

142

are thus stopped when the difference between the distribution is < 0.1%. This

143

procedure slightly increases the execution times, but avoids over-filtering of

144

the solutions.

145

Fig.1 Approximately here.

146

Some synthetic tests that highlight the good performance of this method are

147

shown in Fig.1, both for magnetic and gravity data. The density, magneti-

148

zation and topography models were generated according to a fractal noise

149

distribution, which is believed to be a good statistical model to describe these

150

phenomena (Pilkington and Todoeschuck, 1993; Maus and Dimri, 1994; Tur-

151

cotte, 1997). Practically we generated a square grid with N × N equally spaced

152

points, giving a random value to each point according to a Gaussian distribu-

153

tion. A 2D Fourier transformation of these value is taken to obtain complex

154

coefficients C(k

, k

). A scaling coefficient, β, is used to define the Fourier

155

transformed distribution as:

156

C

(k

, k

) = C(k

, k

)/k

, (10)

157

and, performing the Fourier inverse transformation, we obtain the density or

158

magnetization and the top and bottom surfaces. We show a test made of 256

159

× 256 data points, in which we used a value β = 3.5 for the top surfaces,

160

β = 4.0 for the bottom surfaces and β = 4.5 for the magnetization or density

161

distribution. These values are in good agreement with experimental informa-

162

Accepted Manuscript

tion (Pilkington and Todoeschuck, 1993; Maus and Dimri, 1994; Turcotte,

163

1997). Random Gaussian noise with a standard deviation of 3% of the data

164

magnitude was added to simulate a real survey. Fig. 1 shows that the recov-

165

ered models are in good agreement with the true models, even if the bottom

166

surface is significantly different in shape than the top surface.

167

3 The Southern Tyrrhenian Sea density model

168

Fig.2 Approximately here.

169

In this section we invert the isostatic residual anomaly of the Tyrrhenian Sea

170

in Italy. The Tyrrhenian Sea is a young extensional basin of triangular shape,

171

developed in the hanging-wall of the westward Apennine subduction (Mal-

172

inverno and Ryan, 1986; Patacca et al., 1990; Doglioni, 1991). The opening

173

process of this basin is related to a westward migration of the subduction

174

system between the African and European plates. The entire Tyrrhenian Sea

175

represents the easternmost oceanic back-arc basin of the Mediterranean Sea.

176

It is well-known from previous literature that the oceanic crust emplacement

177

took place in the Southern portion of the Tyrrhenian Sea, where the high-

178

est velocity of the roll-back movement was recorded (Patacca et al., 1990;

179

Doglioni, 1991; Doglioni et al., 2004; Nicolosi et al., 2006). Under this hypoth-

180

esis, the Tyrrhenian Sea can be divided in two distinct domains: The Northern

181

and the Southern. The boundary between the two domains is represented by

182

a peculiar crustal portion known as the 41

parallel zone. Around this dis-

183

continuity, the magmatic products change their petrological affinity due to a

184

differentiation of magmatic sources (Serri, 1990). In the Southern Tyrrhenian

185

Sea, the evolution of the opening process started during the Tortonian, with

186

Accepted Manuscript

an E-W roll-back movement (Patacca eta al., 1990, Royden, 1988, Doglioni et

187

al., 2004, Rosenbaum and Lister, 2004). The starting point of the continental

188

crust is represented by a NE-SW faults system called the Selli Line (Man-

189

tovani et al., 1996). This structure divides the two crustal domains into the

190

Sardinia passive margin (Cornaglia Terrace) and the oceanic Magnaghi and

191

Vavilov basins (Fig. 2). During the Pliocene-Pleistocene the extension moved

192

from the E-W to NW-SE direction forming the Marsili basin (Doglioni, 1991,

193

Marani and Trua, 2002, Doglioni et al., 2004). The change in the direction of

194

extension may be fixed between the Marsili and Vavilov basins along the Issel

195

Bridge, which represents a NE-SW crustal portion of supposed continental na-

196

ture. From a morphological point of view, along this zone the bathymetric level

197

up-rises with a sequence of structural highs. Instead, the Issel Bridge shows

198

an acoustic basement linked to an emplacement of different carbonate-type

199

units covering a metamorphic portion (Doglioni et al., 2004). The Southern

200

Tyrrhenian Sea is thus dominated by an oceanic floored basin where sporadic

201

continental crust blocks outcrop, even if the true distribution of the oceanic

202

crust is still under discussion. The change of rheologic characteristics from

203

continental and oceanic crust may be considerable in terms of crustal density.

204

The continental crust is not homogeneous and is represented by the superpo-

205

sition of shallow sedimentary units over crystalline/metamorphic rocks with

206

high fracturement. The density of this crust ranges from 2.5 to 2.7 g/cm

. The

207

oceanic crust is formed by a gabbroic basement with a thick cover of basaltic

208

lava and pillow lava. The average density of this material is comparable to the

209

density of the tholeiitic basalt (3-3.1 g/cm

). In these terms, a reconstruction

210

of the density distribution of the Tyrrhenian Sea can be used to define the

211

oceanic continental portion of the basin.

212

Accepted Manuscript

Fig.3 Approximately here.

213

In Fig. 3a we can see the bathymetric setting of the Tyrrhenian Sea. The

214

crustal-mantle interface known as the Moho is characterized by a zone of dif-

215

ferentiation of seismic velocity. In this sense the Moho may be interpreted as a

216

point of variations in rheologic properties. To this aim, we have used this sur-

217

face in order to define the bottom level of the model for the Southern Tyrrhe-

218

nian Sea. The Moho depth distribution was obtained by the recent compilation

219

by Sartori et al. (2004). It consists of the integration and reprocessing of multi-

220

channel reflection seismic profiles, coming from different datasets acquired by

221

various institutions. The final seismic dataset represents the results of differ-

222

ent corrections such as velocity analysis, CDP stacking, spherical divergence

223

correction and time variation filtering (Sartori et al., 2004). The surface of the

224

Moho depth is shown in Fig. 3b. The free-air gravity data used in this pa-

225

per comes from satellite GEOSAT and ERS-1 missions (Sandwell and Smith,

226

1997). We performed a complete Bouguer correction using a reference density

227

of 2.67 g/cm

, and then subtract the gravitational effect of the Moho discon-

228

tinuity in Fig. 3b to obtain the isostatic residual anomaly (Simpson et al.,

229

1986). The final result, which is more representative of the crustal density lat-

230

eral variations, is shown in Fig. 3c. Fig. 3d shows the recovered density model

231

of the Southern Tyrrhenian Sea. The bold lines MB and VB represent, respec-

232

tively, the borders of the Marsili and Vavilov Basins, while the westernmost

233

SL fault system is the Selli Line. The recovered density, obtained by adding

234

the reference value of 2.67 g/cm

used for the Bouguer reduction, ranges from

235

2.56 g/cm

to 3.0 g/cm

; these values are in agreement with the theoretical

236

densities described above. The Southern Tyrrhenian basin shows a peculiar

237

distribution of the recovered density. The highest values (3.0 g/cm

) take place

238

Accepted Manuscript

on the oceanized abyssal basins of Marsili and Vavilov. This high density area

239

is clearly confined by several structural elements (Fig. 2, Fig. 3d).The Selli Line

240

represents the northwestern margin whereas the 41

is the uppermost limit.

241

The maximum density is located over the Vavilov structure, which is clearly

242

a submerged oceanic body as confirmed by the ODP-drilled data (Kastens et

243

al., 1988). In the southernmost portion of the Tyrrhenian Sea, the highest val-

244

ues of recovered density belong to the area of the Marsili basin. In this region,

245

the density distribution is characterized by a N-S elongation similar to the

246

shape of the basin. This suggests a direct connection between the rheologic

247

feature and the structural setting. The oceanic nature of the Marsili basin is

248

also supported by results of the drilling process (Kastens et al., 1988). Instead,

249

a recent analysis based on magnetic data has suggested a Pleistocene-Present

250

evolution of the Marsili basin as an ultra-fast spreading ridge with a rate of

251

oceanic crust formation around 18-20 cm/y (Nicolosi et al., 2006). The west-

252

ward portion of this area shows a low density (2.7 g/cm

) which may represent

253

a continental band located among the main abyssal structures. Indeed, in the

254

opposite direction the Marsili basin is confined by a low density area probably

255

connected with the Sicily-Calabrian margin, where volcanic structures from

256

the Aeolian Arc are located. In this area, the westward subduction generates

257

an increase in crustal thickness with related low values for gravity anomalies.

258

In conclusion, the recovered model suggests a distribution of density strongly

259

correlated with the structural setting of the Southern Tyrrhenian basin. The

260

oceanic crustal portion represents the main components of the basin, which

261

is locally characterized by the outcropping of several blocks and bands of a

262

continental nature. These pieces of evidence are probably due to different ex-

263

tensional rates which generated a variation in the crustal stretch. Obviously,

264

our model is based on the assumption of a uniform vertical density of the

265

Accepted Manuscript

layer used for the inversion, and this may be a limiting factor. This may also

266

explain why the recovered oceanic crust distribution shows a slightly lower

267

density than the standard of 3.0-3.1 g/cm

, since it is vertically averaged with

268

sediments or other lighter rocks. Nevertheless, we suppose that the large-scale

269

density distribution so far recovered may outline the crustal characteristics of

270

the Southern Tyrrhenian Sea in terms of its oceanization.

271

Acknowledgments

272

We thank the Institutions of Geological Data Center-Scripps Institution of

273

Oceanography and Naval Oceanographic Office of US Navy that have made

274

available the data used in this paper at the following web-sites:

275

http://www.ngdc.nooa.gov/seg/topo/globe.shtml

276

http://www.mel.navo.navy.mil/data/DBDBV/dbdbv def.html

277

http://topex.ucsd.edu/WWW html/mar grav.html.

278

We thank Editor Gerald Schubert, Gregory Neumann and an unknown referee

279

for their important comments which improved the level of this paper. F.C.T.

280

wishes to dedicate this paper to his little Lorenzo.

281

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Accepted Manuscript

Fig. 1. Synthetic models for magnetic and gravity inversion. In each column from top to bottom, we show the top surface of the layer, its bottom, the true model and the recovered model from the inversion, from (a) to (d) for magnetic data and from (e) to (h) for gravity data. The data was contaminated by random Gaussian noise and generated by a fractal distribution for topography interfaces and densities. The recovered models are in good agreement with the true models. The units on the horizontal axes are in km.

Fig. 2. Geologic setting of the Tyrrhenian Sea.

Fig. 3. Tyrrhenian Sea density model. (a) Bathymetry of the Tyrrhenian Sea; (b) Depth of the Moho interface; (c) Isostatic residual anomaly; (d) Density model obtained by inversion of the anomaly in subplot (c). The density values are obtained by adding the value of 2.67 g/cm

that was used to perform the Bouguer reduction.

The bold lines MB and VB in subplot (d) represent, respectively, the borders of the

Marsili and Vavilov Basins, while the westernmost SL fault system is the Selli Line.

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Figure1

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Figure2

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Figure3