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基于MATLAB实现的GROWTH2.0是一个地球物理反演工具，它支持用户在几乎全自动且非主观的模式下由观测到的重力异常数据得到地下密度异常的三维模型+使用说明文档.rar （33个子文件）

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The 3-D gravity inversion package GROWTH2.0 and its application

to Tenerife Island, Spain

Antonio G. Camacho

, Jose Ferna

ndez

, Joachim Gottsmann

Instituto de Astronomı

a y Geodesia (CSIC-UCM), Plaza de Ciencias 3, 28040 Madrid, Spain

Department of Earth Sciences, University of Bristol, Wills Memorial Building, Queens Road, Bristol BS8 1RJ, UK

article info

Article history:

Received 1 December 2009

Received in revised form

7 October 2010

Accepted 7 December 2010

Available online 25 January 2011

Keywords:

Gravity inversion

3D crustal models

Model exploration

Anomalous density contrast

Tenerife Island

abstract

We present the gravity inversion software GROWTH2.0 and its application to recently obtained gravity

data from the volcanic island of Tenerife (Canary Islands, Spain) to inform on its subsurface density

structure. GROWTH2.0 is an inversion tool which enables the user to obtain, in a nearly automatic and

nonsubjective mode, a 3D model of the subsurface density anomalies based on observed gravity

anomaly data. The package is composed of three parts: (a) GRID3D to generate a 3D partition of the

subsurface volume into parallelepiped elements, (b) GROWTH to perform the inversion routine and to

obtain a 3D anomalous density model, and (c) VIEW for visual representation of the input data, the

inversion mode l, and modeling residuals. The current version of the tool has been developed from an

earlier code (Camacho et al., 2002) and now incorporates several novelties: (1) a Graphical User

Interface (GUI), (2) an optional automated routine for determination of param eter

, which controls the

balance between model ﬁtness and smoothness, (3) optional determination of values for minimum

density contrast, (4) a robust handling of outlier data, and (5) improved automated data reduction for

terrain effects based on anticorrelation with topographic data. The new capabilities and applicability of

GROWTH2.0 for 3-D gravity inversion are demonstrated by a case example using new gravity data from

the volcanic island of Tenerife. In a nearly automatic approach, the software provides a 3-D model

informing on the location and shape of the main structural building blocks of the island. Our model

results allow us to shed light on th e low-density structure of the islands dominant Pico Viejo–Pico Teide

(PV–PT) volcanic complex and the identiﬁcation of an intrusive structure (the east bulge volcano)

embedded in Teide’s east ﬂank. A low-density body located at around 5.8 km depth beneath PT’s

summit may represent a current magma or hybrid reservoir.

1. Introduction

The gravimetric inverse problem, namely the determination

of a subsurface mass density distribution consistent with an

observed gravity anomaly, has an intrinsic nonuniqueness in its

solution (e.g., Al-Chalabi, 1971). Moreover, the available data

must be regarded as insufﬁcient and inaccurate to solve ambi-

guities. Nevertheless, particular solutions can be obtained by

including additional constraints on model parameters (geological

or mathematical hypotheses on the subsurface mass structure)

and on data parameters (statistical properties of the inexact data,

e.g., Gaussian distribution). Methods based on a simple linear

adjustment of model densities are appealing in their application,

yet they require a quantitative formulation of model ‘‘smoothing’’

to prevent artifacts from ‘‘oversmoothing’’ (Bertete-Aguirre et al.,

2002). Other inversion methods seek to adjust the geometry of

modeled anomalous bodies at depth by assuming a priori deﬁned

subsurface density contrasts (see Pedersen, 1979; Barbosa et al.,

1997). For the latter, the anomalous bodies are described by a

discrete set of parameters (e.g., vertex coordinates in Enmark,

1981), or as accretion of ﬁlled cells from a general three-dimen-

sional grid of the subsurface. These methods follow a nonlinear

inversion process, and the traditional approaches work iteratively,

for instance, by means of gradient methods (e.g., Farquharson and

Oldenburg, 1998), starting from an approximate initial solution.

Resultant solutions are dependent on the quality of the initial

model used to deﬁne the unknown geometrical parameters and to

assure the convergence of the nonlineal process. Moreover, for

these methods positive and negative density contrasts are not

simultaneously accepted in the model. For the fully nonlinear

treatment, the methods of exploration of the space model often

provide the best option (Tarantola, 1988). This exploration pro-

cess can be conducted randomly (Silva and Hohmann, 1983)or

systematically.

In Camacho et al. (2000, 2002), we proposed a method of 3-D

gravity inversion inspired by the method of Rene

(1986), which is

based on a ‘‘growth’’ process and the mathematical exploration of

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/cageo

Computers & Geosciences

doi:10.1016/j.cageo.2010.12.003

Code available from server at http://www.iamg.org/CGEditor/index.htm.

Corresponding author. Tel.: +34 1 394 4478; fax: +34 1 394 4615.

E-mail address: antonio_camacho@mat.ucm.es (A.G. Camacho).

Computers & Geosciences 37 (2011) 621–633

the model space (in the sense of Tarantola (1988)) for deﬁned

density contrasts. Characteristics of the method are the following:

(1) solution for 3-D model space, (2) acceptance of nongridded

nonplanar imprecise data, (3) calculation can start from an empty

body or optionally from initial ‘‘seeding’’ structure, (4) automatic

determination of a linear regional gravity trend from the data,

(5) simultaneous inversion for both positive and negative density

contrasts, (6) inversion for irregularly shaped structures com-

posed of several individual bodies, (7) inversion accounting for an

optional downward density increase, (8) semi-automated data

inversion routine. This inversion routine was previously imple-

mented and made available in the software package GROWTH

in Camacho et al. (2002), enabling the user to load and invert

gravity anomaly data and to graphically display inversion results.

Here, we present a new and updated version of this original

tool, denoted now as GROWTH2.0, which aims at an improved

user friendliness via integration of a graphical user interface (GUI)

while also incorporating methodological improvements (robust

treatment of outliers, automated routine for determination of

parameters that control the balance between model ﬁtness and

smoothness, etc.) on data manipulation and the inversion routine.

We also present results from the application of GROWTH2.0 to

new gravity data collected on the volcanic island of Tenerife

(Canary Islands).

2. Methodology

Large parts of the inversion methodology and associated

mathematical concepts are described by Camacho et al. (2000,

2002) and Gottsmann et al. (2008). We therefore just brieﬂy

summarize key concepts here.

Suppose a data set constitutes gravity values observed at n

gravity measurement points, irregularly distributed (Fig. 1). Let

), i¼1,y,n, be the planar coordinates (UTM coordinates)

and the altitudes of the gravity stations P

and let

, be the

respective gravity anomaly (Bouguer anomaly data, corrected for

terrain effects). We must consider the gravity data as imprecise

values whose uncertainty is Gaussian, characterized by a covar-

iance (n,n)-matrix, Q

. Usually, we take q

¼0, for iaj, and q

¼e

where e

, i¼1,y,n, are standard deviations of the gravity values.

The inversion process constructs a subsurface model deﬁned by

a 3-D aggregation of m parallelepiped cells, which are ﬁlled, in a

‘‘growth’’ process, by means of prescribed positive and/or negative

density contrasts (unique values



for the total model or

local values according to speciﬁc criteria). The design equation to

relate observables, i.e., the gravity anomaly

at n benchmarks

), with modeling parameters, and residuals v

j A J



reg

top

þv

, i ¼ 1, ..., n, ð1Þ

where A

is the vertical attraction for unit density for the jth

parallelepiped cell on the ith observation point (Pick et al., 1973),



are prescribed density contrasts (negative and positive)

for the jth cell, J

, J



are sets of indexes corresponding to the cells

ﬁlled with positive or negative density values, and

reg

top

are

optional terms for data adjustment (regional trend and additional

topographical corrections) given by

reg

¼ g

þg

ðx

x

Þþg

ðy

y

Þ, i ¼ 1, ..., n, ð2Þ

top

: ð3Þ

Here g

, g

, and g

are parameters for the linear regional

gravity trend, x

and y

are averaged coordinates; C

is the usual

coefﬁcients for the terrain correction, and

is an optional

additional value with respect to the initially adopted terrain

density. In Eqs. (1)–(3),



are assumed (a priori deﬁned)

anomalous density values (constant extreme values, positive

and negative



for the entire model, or extreme values

according to a prescribed variation, for instance, for the case of

stratiﬁcation as described below). g

, g

, and

(regional

parameters) are the main unknowns to be determined by the

inversion. Sets J

and J



design cells ﬁlled with positive and

negative density contrast, which determine the geometry of the

anomalous bodies in a nonlinear relationship.

There are two major problems to be solved: (1) the general

nonuniqueness of the solution, requiring some additional (math-

ematical) constrains and (2) the nonlinearity of the design system

with a very large number of degrees of freedom (the quite inﬁnite

number of possibilities of constructing the model by aggregation

of thousands of cells).

The problem of nonuniqueness is usually approached by

numerical techniques in the form of a generalized matrix inver-

sion including some form of ‘‘perturbation’’ scheme such as the

Levenberg–Marquardt method (e.g., Enmark, 1981; Garcia-

Abdeslem, 2000; Nagihara and Hall, 2001). Tarantola (1988) gives

a general treatment of the least-squares inversion methods by

including them into a general theory, which represents the

inversion problem as a problem of combination of inaccurate

information (inaccurate data, inaccurate statistical hypothesis,

inaccurate initial models, etc.). The least-squares inversion

approach is presented as the combination of observable data

with Gaussian uncertainties (given by a covariance matrix Q

)

and information on the model, given by a previous model m

which is also subject to Gaussian uncertainties (given by a

covariance matrix Q

). For the case where the previous model

is exactly zero (m

0, no particular previous information on

the model is available) the formulas of the general methodology

of Tarantola are close to the numerical methods pointed out

before. For the particular case of gravity inversions aiming at

deriving a unique model of density distribution, the general

condition of l

-minimization of the model parameters becomes

a minimization condition for the total anomalous mass; i.e., a

model involving anomalous bodies with minimum total mass

(and, then, simple, and smoothed geometries) is uniquely deter-

mined (within the data error level). This is a basic idea to evaluate

and understand the models obtained by this methodology in

practical applications. Accordingly, in order to select a valuable

particular solution (a sets, named J

and J



, of cells ﬁlled with

positive or negative density contrast) among the inﬁnite set of

possible solutions from our data inversion, we adopt a mixed

minimization condition, based on model ‘‘ﬁtness’’ (least-square

Fig. 1. Partition of the subsurface volume into parallelepiped cells under the

survey benchmarks.

A.G. Camacho et al. / Computers & Geosciences 37 (2011) 621–633622

ﬁtness) and ‘‘smoothness’’ (total anomalous mass)

1

vþ

1

m ¼ min, ð4Þ

where m ¼ð

, ...,

are prescribed density contrast values

for the m cells of the model (superscript T denotes transpose),

v ¼ðv

, ..., v

are the vector of residual values for the n data

points, Q

is an a priori covariance matrix for uncertainties of the

gravity data, Q

is an a priori covariance matrix for uncertainties of

the model parameters, and

is a dimensionless factor (assuming

that matrices Q

and Q

include normalization terms for data and

mass) for selected balance between ﬁtness and smoothness of the

model. In traditional numerical inversion methods based on the

Levenberg–Marquardt smoothing approach, an identity matrix, I,is

used in place of Q

. However, taking into account that the cells of

the model have different sensitivities (lower for deep or peripheral

cells; see below), using a covariance matrix to obtain similar values

of weighted cell sensitivity based on its geometry and location, will

produce more balanced and coherent minimization conditions for

the model parameters, and will thus produce more consistent

solutions. For a problem without previous information about the

model structure (Camacho et al., 1997), we suggest to take a model

matrix Q

1

given by a diagonal normalizing matrix of non-null

elements that are the same as the diagonal elements of A

1

with matrix A of elements A

from (1).

A second problem arises from the very large number of

degrees of freedom for describing the model as aggregation of

small parallelepiped cells ﬁlled with anomalous mass in a non-

linear ﬁt. In fact, such a model would suppose one parameter for

each cell, in a model composed of several thousand cells. There-

fore, a general exploratory inversion approach, simultaneously for

every cell, would be ineffective. An interesting idea was proposed

by Rene

(1986). He applied (in a more restrictive context) an

exploratory method, not to test every possible density distribu-

tion for the complete model (too computationally demanding),

but to test only the different options for ﬁlling one cell for each

step of the growth process for building the model from initial

‘‘seeds.’’ In the present methodology the bodies are formed by a

nearly homogeneous growth by cell addition, from previously

adjusted ‘‘skeletal’’ structures, until the bodies attain a suitably

developed size. At this stage, rather than continuing the explora-

tion of the entire model space, the routine explores the possibi-

lities of cell growth (cell by cell) for each step of the anomalous

body expansion. Thus, for each step, the prismatic empty cells are

systematically tested with each prescribed density contrast. The

best ﬁtting solution is selected. It corresponds to a particular cell

that is deﬁnitively ﬁlled (with the best ﬁtting contrast) and

aggregated to let the anomalous bodies growth. The minimization

ﬁt conditions are applied for each growth step, and include a scale

factor f which relates the immature model to the global condi-

tions concerning gravity ﬁt and model size (mass and volume).

In practice, for an arbitrary (k+1)th step, k prisms have been

deﬁnitively ﬁlled with positive or negative contrasts. The resul-

tant modeled gravity values are



, ð5Þ

where J

and J



are the index sets corresponding to the

previously modiﬁed cells. Now, the process seeks, throughout

the m–k empty prisms, for one new prism to be ﬁlled and

aggregated to the model (sets J

, J

–

). For that, for each jth empty

prism, jeJ



, the following equation system is considered:

ð

þA

Þf ðg

þg

ðx

x

Þþg

ðy

y

ÞÞ

¼ v

i ¼ 1, ..., n,

ð6Þ

where

can take values

and



, and fZ 1 is an unknown

scale factor for ﬁtting the modeled anomalies ð

þA

Þ to the

observed anomalies (

g). The positive and the negative pre-

scribed values are successively tested for the additional density

contrast

to ﬁt a minimization condition similar to (4) but now

including the scale factor f.

Next, the unknown parameters f, g

, g

in (6) are

adjusted for a mixed minimization criterion corresponding to the

(k+1)th step

1

vþ

1

m ¼ min, ð7Þ

where the vector m of solutions now includes the values for the

previously ﬁlled cells and the value

that is being tested:

m ¼ðm

, with: m

, m

if lA J



if lA J



, otherwise m

¼ 0

(see Camacho et al. (2007) for details).

Once the former linear equations have been solved, we can

calculate the corresponding residual values v

in terms of the

selected

. Then, we take the global misﬁt value e

deﬁned by

¼ v

1

vþ

1

m ð8Þ

as the parameter for the suitability of jth prism and the adopted

density contrast (positive

or negative



). In essence, for

step (k+1), the method explores model possibilities for each of

the unchanged prisms and for either a negative or positive density

contrast. Then, the jth prism with a density contrast producing a

minimum value of e

(value that we rename as E(k+1)) is selected

to let the anomalous bodies growth. Its effect is added to the

modeled

values.

This process is repeated in a stepwise manner obtaining a best

ﬁt model for each growth step (see Eq. (4)). For each successive

step, the scale value f decreases and the additional parameters g

, g

reach nearly stable values. The process stops when f

approaches 1 (there is no empty cell whose changing, thus ﬁlling

with positive or negative difference density, would make the

value of the functional lower), resulting in the modeled 3-D

structure for anomalous density distribution, a ﬁnal regional

trend and a value (or a corrective term) for the terrain density.

Camacho et al. (2000, 2002) and Gottsmann et al. (2008) give

some simulation examples showing the suitability of this 3D

inversion approach.

3. Methodological improvements in GROWTH2.0

Version 2.0 of the GROWTH software includes several new

methodological improvements. They are the following:

(a) Automatic selection of the balance factor

for practical

applications: For low values of

the inversion results in the

generation of a (potentially overly) complex model of the

subsurface structure with very good ﬁt to the data (potentially

even to data noise). For high values

the inversion produces a

simple model but with poor ﬁt to the data. Then, a suitable

criterion to select the optimal

value is desired. For practical

applications, we propose the criterion of choosing the

value

as that producing uncorrelated inversion residuals. In fact, for

too high a value of

, model residuals are large and auto-

correlated with respect to their mutual distance. For too low a

value of

, model residuals show negative values of autocor-

relation for small distances due to accounting for noise. The

optimal value satisﬁes the condition of a zero autocorrelation

of residuals with respect to the mutual distance between

benchmarks. See Camacho et al. (2007) for more details.

(b) Detection and treatment of outlier gravity values in the data

set: A small number of observed gravity values can contain

large errors, not normally distributed, due to sharp local

A.G. Camacho et al. / Computers & Geosciences 37 (2011) 621–633 623

effects, observer carelessness, transcription errors, etc. They

can distort the inversion model based on Gaussian assump-

tion. To detect and minimize the effects of these outlier

values Camacho et al. (2007) developed a robust approach

which works across the growth process by applying weight-

ing coefﬁcients corresponding to the successive residual

values and to the scale factor f. The present version of the

GROWTH software includes this robust approach. Once the

errors are detected and isolated, they can be deleted for a

further running of the process.

previous works, we considered the anomalous bodies as

constructed independently of the characteristics of the sur-

rounding medium (Camacho et al., 2007). Therefore, the

modeled structures must be regarded as additional structures

within an undeﬁned ‘‘nonanomalous’’ average medium. In

some cases, variations in crustal properties with depth as

derived from other geophysical data need to be accounted for.

Thus, background media density variation about an average

value

ðzÞ as a function of z is introduced in GROWTH2.0.

Using the methodology of modeling anomalous bodies as

described before, we have now also included the option of

solving for anomalous bodies within a density stratiﬁed

medium. In this way, solutions are obtained by prescribing

density contrasts



and

(cf. Eq. (1)) for each cell

located at a mean depth z



ðz

Þ,



ðz

Þ, ð9Þ

where



are shallow density contrasts at surface

levels and incremental value

ðzÞ depending on the depth z

of the corresponding cell. Consequently, when these (inver-

sely stratiﬁed) anomalous bodies are superimposed on the

(normally stratiﬁed) medium, the resulting structure gives

nearly homogeneous bodies inserted in a stratiﬁed back-

ground volume.

Generally, density increases with depth. If a clear distribution

into deﬁned layers is not known for the average medium, we

provide the option of accounting for a simple continuous density

increase: (i) a linear increase and (ii) an exponential increase.

(d) Automatic estimation of minimum density contrast: Optimal

values



and

for bounding negative and positive

values of density contrast are difﬁcult to derive from the

gravity data set. In the absence of knowledge of positive or

negative bounds, initial results from the inversion can be used

as ﬁrst-order approximations of contrast values: skeletal and

disintegrated bodies are modeled as a consequence of a too

large prescribed density contrast (positive and negative),

while interconnected and ‘‘inﬂated’’ bodies are an artifact of

choosing a too small range of contrast. There is, however, an

ideal minimum density contrast: the smallest value for

homogeneous bodies able to reproduce the full anomaly size,

and then allow reaching f¼1. This contrast can be optionally

explored in GROWTH2.0 (see, for instance, Parker (1974) for

details on how to derive this density bound).

These methodological novelties are designed to provide,

within a new friendly environment of dialog box and windows,

a more automatic tool GROWTH2.0, suitable to obtain useful

results for a general nonexpert user.

4. Organization of the tool package GROWTH2.0

The former methodology for gravity inversion is translated

into a software package composed by three main programs

written in Fortran able to be compiled with Intel Visual Fortran

(plus some auxiliary ﬁles, see GROWTH.TXT for details of compi-

lation and running). The process of data inversion and evaluation

starts with the loading of gravity anomaly data and ends with the

output of graphical displays of the resultant 3D anomalous

density contrast model. A GUI enables a straightforward data

loading procedure and execution of inversion routine. Some

inversion parameters are preset by default values, which we

deem appropriate for most applications. This allows seeding of

the routine without the need to load an initial model. The method

does not involve inversion of large data matrices, and is not overly

exigent on memory size. Wherever possible, the vectors are

reduced to Integer

2 form. The ﬁnal memory requirements

depend on the application parameters: number of gravity stations

(limited to 900 in the present version) and desired resolution

level of the model (limited to 80,000 cells in the present version).

The process of gravity data inversion is started via program,

GRID3D. This program loads an ASCII ﬁle of the Bouguer gravity

anomaly data (saved in GRA.DAT) and determines a 3D partition

of the subsurface volume into parallelepiped cells with a geome-

trical distribution corresponding to the sensitivity of the bench-

marks distribution. The resulting 3D grid is written to an ASCII ﬁle

GRI.DAT (see Fig. 2) and contains a set of empty cells. The second

program, named GROWTH, is the main program for gravity

inversion. It reads the 3D grid and applies the inversion metho-

dology to construct the model by step-by-step ﬁlling cells with

positive or negative density contrasts. The model output is saved

in ASCII ﬁle MOD.DAT with each line of content representing

parameters for each model cell. A secondary output ﬁle FIL.DAT

contains values for regional trend, observed local gravity anom-

aly, predicted local anomaly, residuals, etc. The resulting model

MOD.DAT can be optionally used via a feedback mode to con-

struct a new 3D grid with a higher resolution seeded by the

former. The third program, named VIEW, is a program for model

visualization and export of numerical model data. Reading ﬁles

GRA.DAT (observed anomaly data ﬁle), MOD.DAT (3D anomalous

density model), FIL.DAT (observed, predicted, residual values),

and MAP.BLN (a base map ﬁle to be included in ﬁgures), it

generates color-coded images of the data, intermediate results,

or the adjusted model. The user can base his/her needs for

additional model runs on the evaluation of the output data

(discontinue lines in Fig. 2). File naming is based on default

GRA.DAT:Bouguer gravity data (geographically referenced)

GRI.DAT: 3D grid of the subsurface volume

GRI3D.EXE

GROWTH.EXE

MOD.DAT: 3D model of anomalous density

VIEW.EXE

Graphical output of solution

Files for additional manipulation

FIL.DAT:trend and residual values

MAP.BLN:line map for figures

Fig. 2. Flow diagram highlighting order of execution of the different codes

composing GROWTH2.0 software. Boxes with double contour line indicate execu-

table ﬁles and boxes with simple contour line indicate ASCII ﬁles or ﬁgures. Solid

lines represent forward process, while broken lines depict feedback process.

A.G. Camacho et al. / Computers & Geosciences 37 (2011) 621–633624

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