Kusumoto Progress in Earth and Planetary Science (2017) 4:15
DOI 10.1186/s40645-017-0130-0

RESEARCH ARTICLE

Progress in Earth andPlanetary Science

Open Access

Eigenvector of gravity gradient tensor for
estimating fault dips considering fault type
Shigekazu Kusumoto

Abstract
The dips of boundaries in faults and caldera walls play an important role in understanding their formation mechanisms.
The fault dip is a particularly important parameter in numerical simulations for hazard map creation as the fault dip
affects estimations of the area of disaster occurrence. In this study, I introduce a technique for estimating the fault dip
using the eigenvector of the observed or calculated gravity gradient tensor on a profile and investigating its properties
through numerical simulations. From numerical simulations, it was found that the maximum eigenvector of the tensor
points to the high-density causative body, and the dip of the maximum eigenvector closely follows the dip of thenormal fault. It was also found that the minimum eigenvector of the tensor points to the low-density causative bodyand that the dip of the minimum eigenvector closely follows the dip of the reverse fault. It was shown that the
eigenvector of the gravity gradient tensor for estimating fault dips is determined by fault type. As an applicationof this technique, I estimated the dip of the Kurehayama Fault located in Toyama, Japan, and obtained a resultthat corresponded to conventional fault dip estimations by geology and geomorphology. Because the gravity
gradient tensor is required for this analysis, I present a technique that estimates the gravity gradient tensor fromthe gravity anomaly on a profile.
Keywords: Fault dip, Gravity gradient tensor, Eigenvector, Normal fault, Reverse fault, Kurehayama Fault

Introduction
In recent years, gravity gradiometry surveys have been
widely conducted to obtain detailed subsurface structure
data (e.g., Jekeli 1988; Dransfield 2010; Chowdhury andCevallos 2013; Braga et al. 2014). Data collected by thesesurveys is the gravity gradient tensor defined by second
derivatives of the gravity potential, and its response to
subsurface structures is more sensitive than the gravity
anomaly. At present, gravity gradiometry surveys havemainly been performed using a helicopter. Consequently,
their observation interval is about 3 m on the flight profile, and the observation density is very high. The gravity
gradiometry surveys allowed for high observation density, high resolution, and high sensitivity to the subsurface structures; therefore, these surveys contribute
greatly to the earth science and resource engineering
fields in terms of being useful and powerful tools for theestimation of subsurface structures.

Correspondence: kusu@sci.u-toyama.ac.jp
Graduate School of Science and Engineering for Research (Science),
University of Toyama, 3910 Gofuku, Toyama 930-8555, Japan

Various analysis techniques using gravity gradient tensors have been suggested and discussed (e.g., Zhang
et al. 2000; Beiki 2010; Martinez et al. 2013; Cevallos
2014; Li 2015). These are considered to be so-called inversion techniques. A semi-automatic interpretation
method that can extract subsurface structure characteristics without geological and geophysical data input has
also been developed and applied to field data (e.g.,
Cooper 2012; Ma 2013; Ferreira et al. 2013).
A typical semi-automatic interpretation method is anedge emphasis technique that uses extraction techniques
to find locations (namely, edge) where the potential fieldchanges abruptly due to density variations. The horizontal gravity gradient method and vertical gravity gradientmethod (e.g., Evjen 1936; Elkins 1951; Tsuboi and Kato
1952; Blakely and Simpson 1986) are classic edge emphasis techniques. In recent years, higher and keener extraction techniques have been suggested (e.g., Miller andSingh 1994; Cooper and Cowan 2006; Sertcelik and
Kafadar 2012; Zhang et al. 2014). In addition, attention
has been paid to techniques that evaluate the shape of© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to
the Creative Commons license, and indicate if changes were made.

Kusumoto Progress in Earth and Planetary Science (2017) 4:15

the potential field (e.g., Koenderink and van Doorn
1992; Robert 2001; Zhou et al. 2013; Cevallos 2014).
Among these methodologies, a technique for estimating the dip of geological boundary using the gradienttensor of the potential fields has been developed (e.g.,
Beiki 2013). Beiki and Pedersen (2010) showed that the
maximum eigenvector of the gravity gradient tensor
points to the causative body (Fig. 1a). Since this property
is common in the potential fields, Beiki (2013) applied it
to a magnetic anomaly in the Åsele area (Sweden) andobtained useful information on the dip of the dikeswarms. Kusumoto (2015), considering that the basement consists of an aggregate of high-density prisms(Fig. 1b), applied Beiki’s technique (Beiki and Pedersen
2010; Beiki 2013) to the estimation of fault dips. Thismethod provided results wherein the fault dip estimatedby the gravity gradient tensor harmonized with the dip
observed from seismic surveys (Kusumoto 2015, 2016a).
In addition, the dip of an earthquake source fault of the
Kumamoto Earthquake that occurred in April 2016 estimated from the gravity gradient tensor also corresponded with the dip of the fault model (normal fault of60°), thus explaining the crustal movement observed by
GNSS (Global Navigation Satellite System) (Kusumoto
2016b). The range for which this method is applicable iswide from low dip to high dip (e.g., Beiki 2013; Kusumoto
2015, 2016a, 2016b), although it has some numerical instability to the vertical fault (e.g., Kusumoto 2015).
Although analyses using the gravity gradient tensor
have yielded excellent results in subsurface structure estimations and edge detections, gravity gradiometry surveys have been conducted in only a few areas, limiting
the tensor data available. If we were to carry out theseanalyses in areas where gravity gradiometry surveys havenot been conducted yet, we would have to use the tensor
estimated from existing gravity anomaly data.
The procedure for estimating the gravity gradient tensor
from gravity anomaly data has already been suggested by
Mickus and Hinojosa (2001). This technique estimates the

Page 2 of 10

gravity gradient tensor from spatial distribution of gravity
anomalies by the Fourier transform. Since the database ofgravity anomalies has been prepared, studies using thegravity gradient tensor estimated by Mickus and Hinojosa’s
method will progress in the future. On the other hand, it
is difficult to apply this method directly to gravity anomalies obtained by gravity surveys conducted on a profile
employed frequently in active fault research.
In dense gravity surveys researching fault structures indetail, profiles were set perpendicular to the fault andshort-spaced gravity observations were taken along the
profiles (e.g., Iwano et al. 2001; Inoue et al. 2004). It is
important to find the fault shape, especially its dip, inthese studies because the fault dip affects the area of
disaster occurrence (e.g., Abrahamson and Somerville
1996; Takemura et al. 1998) and is an important parameter in numerical simulations for hazard map creation(e.g., Irikura and Miyake 2011). Consequently, in twodimensional gravity surveys for faults, a fault dip estimated from the eigenvectors of the gravity gradienttensor calculated from the gravity anomaly would be ofadditional value. In addition, since this analysis technique does not require vast calculation times, I expect
it will be an effective new technique for analyzing highresolution data obtained densely, i.e., through dense
gravity surveys for fault research and also airborne
gravity gradiometry surveys.
In this study, I first introduce the technique for the estimation of the gravity gradient tensor from a gravity
anomaly on the profile. After that, I discuss the relationship between fault dips and eigenvectors of the gravity
gradient tensor and apply its result to gravity anomaly
data obtained on the profile crossing the Kurehayama
Fault in Toyama, Japan.

Methods/Experimental
Gravity gradient tensor on the profile

Gravity gradient tensor Γ on the profile is defined as follows (e.g., Beiki and Pedersen 2011)

Fig. 1 Schematic illustration of the maximum eigenvectors for two-dimensional (2D) structures such as dykes and faults. a Basic model. In thisfigure, v1 is the maximum eigenvector of the gravity gradient tensor and points to the causative body. The angle α between the surface andthe maximum eigenvector is the dip of the causative body. b Fault model. A basement consists of an aggregate of high-density prisms, andthe angle, α, indicates the fault dip

Kusumoto Progress in Earth and Planetary Science (2017) 4:15

g xx g xz
Γ¼
g zx g zz

Page 3 of 10ð1Þ

Here, gxx, gxz, gzx, and gzz are each component of thetensor and are defined as the first derivative of gravity
vector components gx and gz for each direction. In
addition, gravity vectors gx and gz are given by the first
derivative of gravity potential, W, namely, gx = ∂W/∂x
and gz = ∂W/∂z. As the gravity potential satisfies the Laplace equation, ∂2W/∂x2 + ∂2W/∂z2 = gxx + gzz = 0, we findthe relationship gzz = −gxx. Also, the relationship isknown to be gxz = gzx because the gravity gradient tensor
is a symmetric tensor (e.g., Torge 1989).
Relationship between subsurface structure and gravity
anomaly

In the two-dimensional analyses, a structure in one direction is assumed to be infinite. Although this assumption is not realistic, it is a good approximation in fault
structure analyses and gives us some practical analysis
techniques. In calculations of the gravity gradient tensor
from the gravity anomaly, we need gravity anomaly
values at different heights. Consequently, I will show
the relationship between two-dimensional subsurface
structures and gravity anomalies in this subsection
before estimating the gravity gradient tensor from thegravity anomaly.
As the simplest subsurface model, I set a twodimensional double layer model consisting of a sedimentary layer and a basement (Fig. 2). Horizontal
positions are given by x, and vertical positions are
given by z. Depth is zero (z = 0) on the surface, and z
increases with depth. As shown in Fig. 2, an averageboundary depth between the sedimentary layer andbasement is defined as z = D (>0). When the boundary surface at point Q(x’) deviates by h(x’) from theaverage boundary depth (Fig. 2), gravity anomaly gz(x) at
the point P(x) on the surface caused by this deviation is
given by the following equation (e.g., Blakely 1996).
g z ðxÞ ¼ 2γΔρZ ∞ Z Dþhðx0 Þ

z0
ðx−x′ Þ2 þ z02−∞ D

dx′ dz′ð2Þwhere γ is the gravitational constant and Δρ is the density contrast between the sedimentary layer and basement. The integration on z in Eq. (2) is solved as:
Z Dþhðx0 Þ
D

"
#
2
2
1
ðx−x′ Þ þ ðD þ hðx′ ÞÞ
dz ¼ log2
ðx−x′ Þ2 þ z02
ðx−x′ Þ2 þ D2
z0

′

ð3Þ
here, if h(x’) is much smaller than D, namely, h(x’) << D,(D + h)2 is {D[1 + (h/D)]}2 ≈ D2(1 + 2 h/D) = D2 + 2Dh,
Eq. (3) would be rewritten as follows:"#2
1
ðx−x′ Þ þ D2 þ 2Dhðx′ Þ
dz ≈ log2
ðx−x′ Þ2 þ z02
ðx−x′ Þ2 þ D2"#
1
2Dhðx′ Þ¼ log 1 þ
2
ðx−x′ Þ2 þ D2

Z Dþhðx0 Þ

z0

D

′

ð4Þ
In general, if −1 < ξ ≤ 1 in log(1 + ξ), we have the following approximation (e.g., Gradshteyn and Ryzhik 2007)
1
1
1
logð1 þ ξ Þ ¼ ξ− ξ 2 þ ξ 3 − ξ 4 þ ⋯
2
3
4
∞
p
X
pþ1 ξ
ð−1Þ
¼
p
p¼1

ð5Þ

The second term, 2Dh/[(x − x’)2 + D2], in Eq. (4) issmall because D > > h. We can use Eq. (5) to derive a
linear approximate equation of Eq. (4). By neglecting
higher terms of ξ, Eq. (3) or (4) is rewritten as follows:Z Dþhðx0 ÞDz0ðx−x′ Þ2 þ z02

dz′ ≈Dhðx′ Þðx−x′ Þ2 þ D2

ð6ÞConsequently, we obtained the following equation.
g z ðxÞ≈2γDΔρ

Fig. 2 Model of subsurface structure. A double-layer model consistingof a sedimentary layer and a basement is assumed here. D is theaverage depth of the stratum boundary, and h(x’) is the deviationof the boundary from the average. Here, the deviation is assumed
to be very small, i.e., h(x’) << DZ ∞hð x′ Þ2−∞ ðx−x′ Þ þ D2

dx′

ð7Þ

Here, I introduce a new function, ϕ, defined by:φðxÞ ¼

1x2 þ D2and Eq. (7) is rewritten as follows:ð8ÞKusumoto Progress in Earth and Planetary Science (2017) 4:15

g z ðxÞ ¼ 2γDΔρ

Z ∞ 
−∞

  
φ x−x′ h x′ dx′

Page 4 of 10ð9ÞThis form is convoluted, and we obtain Eq. (10) byapplying the Fourier transformation to Eq. (9)Gz ¼ 2γDΔρΦH

ð10Þ

where, Gz, Φ, and H are Fourier transforms of gz(x),ϕ(x), and h(x), respectively. As is well known, theFourier transform of Eq. (8) is (e.g., Blakely 1996;
Gradshteyn and Ryzhik 2007)
Φ¼

π −Djk jeDð11ÞHere, |k| = ikz = |kx| (e.g., Blakely 1996) and kx is thewave number in the x direction. Here, I employed theFourier transform, F, of a function f(x) defined as follows(e.g., Blakely 1996):
F¼

Z ∞
−∞

f ðxÞe

−ikx

dx

ð12Þ

By Eq. (11), Eq. (10) is rewritten as:Gz ¼ 2πγΔρHe−Djk jð13ÞThis is the relationship between gravity anomaly on
the profile and two-dimensional subsurface structure.
Relationship between gravity anomaly and gravity
gradient tensor

As shown in the previous section, the gravity gradienttensor is given by the second derivative of the gravity
potential. The relationship between gravity anomaly gz
and gravity potential W isZ

W ¼ − g z dz

ð14Þ

From Eq. (13), the equation giving the gravity anomaly
at point P’(x) of an arbitrary height z from the surface(Fig. 2) is obtained in the Fourier domain as follows:
Gz ¼ 2πγΔρHe−ðDþzÞjk jð15Þ

By integrating this equation to z and substituting z = 0,
we obtain the gravity potential at the surface. If the Fourier transform of the gravitational potential is represented by U, from these calculations, the U would begiven by Gz as follows:
U¼

1
Gz
jk j

ð16Þ

As the x direction component of gravity anomaly isgiven by the first derivative in the x direction of thegravity potential W, the gx in the Fourier domain, Gx,would be given by a differential formula in the Fourier
domain (e.g., Blakely 1996) as follows:

Gx ¼ ik x U

ð17Þ

From Eq. (16), we obtained
Gx ¼

ik x
Gz
jk j

ð18Þ

gxx in the Fourier domain is given by
Gxx ¼

−k 2x
Gz
jk j

ð19Þ

We can obtain gxx by applying the inverse Fouriertransform to Gxx, and gzz would be obtained from the relationship of gzz = −gxx. The other component gzx (=gxz)
would be given by:
Gzx ¼ Gxz ¼ ik x Gz

ð20Þ

where Gzx and Gxz are the Fourier transform of gzx
and gxz.
Here, although I showed a technique to calculate thegravity gradient tensor in the Fourier domain, there isanother technique to calculate the tensor by a simple
finite-difference method (e.g., Blakely 1996) of gravity
vectors gx and gz in the space domain.
Relationship between subsurface structures andeigenvectors

As indicated by Beiki and Pedersen (2010), the maximum eigenvector of the gravity gradient tensor points
to the causative body of the gravity anomaly (Fig. 1a).
They also pointed out that the minimum eigenvector of
the tensor indicates the strike direction of structures
such as dikes in three-dimensional analyses. Since there
are two perpendicular eigenvectors of the gravity gradient tensor in the two-dimensional analyses, it is expected that the minimum eigenvector of the tensor will
point to the low-density causative body or medium if the
maximum eigenvector of the tensor points out highdensity causative bodies such as a dike in a low-density
layer such as a sedimentary layer.
To clear this inference, I calculated the gravity gradient tensor on the profile caused by the model shown in
Fig. 3 and investigated the dips of the maximum andminimum eigenvectors of the tensor. The model shownin Fig. 3 has a width and height of 0.25 and 2.0 km,
respectively.
Each component of the gravity gradient tensor causedby two-dimensional structures such as the dike shown in
Fig. 3 is given by Telford et al. (1990). The relationshipbetween eigenvectors and structural boundaries will be
discussed widely in this study; I therefore employed calculation formulas given by Talwani et al. (1959). Talwani
et al. (1959) show well-known calculation formulas giving gx and gz for two-dimensional arbitrary structures

Kusumoto Progress in Earth and Planetary Science (2017) 4:15

Page 5 of 10

Fig. 3 Model of subsurface structures. Here, rectangular causative
body of width and height of 0.25 and 2.0 km, respectively, isassumed. Densities of a medium and a causative body are ρ2and ρ1, respectively

closed by a polygon. In this study, I obtained gzx (=gxz)
and gxx components by the numerical differentiation ofgz and gx, and the gzz component was given by gzz = −gxx.
A simple finite-difference method (e.g., Blakely 1996) was
employed for these numerical differentiations. In addition,
the dip of each eigenvector (α) was calculated by vz
α ¼ arctan
vx

ð21Þ

where vx and vz are x and z components of eacheigenvector.

Results and discussion
Density structures and eigenvectors

Figure 4a shows distributions of the maximum (red) andminimum (blue) eigenvectors of the gravity gradient tensor caused by the model structure (Fig. 3) whose density
contrast (Δρ = ρ1 − ρ2) is 200 kg/m3. Figure 4b shows distributions of the maximum (red) and minimum (blue)
eigenvectors of the tensor caused by the model structure(Fig. 3) whose density contrast (Δρ) is −200 kg/m3. In eachfigure, the lengths of all the eigenvectors are the same.
From Fig. 4a, it is found that the maximum eigenvectorof the gravity gradient tensor points to a high-density
causative body if the body is embedded in the low-density
medium. In this case, the minimum eigenvector of thetensor points to the low-density medium around thehigh-density body. On the other hand, the minimum

Fig. 4 Eigenvectors of the gravity gradient tensor caused by thesubsurface model shown in Fig. 3. The maximum eigenvector andminimum eigenvector are shown by red and blue, respectively. a
Eigenvectors on the high-density causative body (light green). Whenthe high-density causative body was given in the low-density medium,the maximum eigenvector of the gravity gradient tensor points to thecausative body and the minimum eigenvector points to the lowdensity medium. b Eigenvectors on the low-density causative body(light yellow). When the low-density causative body was given in
the high-density medium, the minimum eigenvector of the tensor
points to the causative body and the maximum eigenvector points
to the high-density medium

eigenvector of the gravity gradient tensor points to a
low-density causative body if the body is embedded in
the high-density medium. In this case, the maximum
eigenvector of the tensor points to the high-density
medium around the low-density body. From theseresults, in the two-dimensional analyses, it was shownthat the maximum eigenvector points to a highdensity causative body and the minimum eigenvectors
points to a low-density causative body.
In Fig. 4, there are vectors pointing to the area z < 0.
This indicates that α is negative. Structures exist underground, and the negative α is not realistic. Consequently,
I will add π to α if α is negative.
Fault types and eigenvectors

In calderas and/or sedimentary basins, high-density andlow-density materials are in contact with each other via
normal faults and/or reverse faults. In gravity anomalies
and gravity gradient tensors, differences in fault type areKusumoto Progress in Earth and Planetary Science (2017) 4:15

defined as differences in density structure. As it was
shown that the behavior of each eigenvector is dependent
on the density structure in the previous subsection, I investigated the relationship between eigenvectors and fault
type by the simplified sedimentary basin models.
Figure 5a is a simplified sedimentary basin model inwhich the sedimentary layer is in contact with the basement by normal faults, and Fig. 5b is a simplified sedimentary basin model in which the sedimentary layer isin contact with the basement by reverse faults. Density
contrast between sedimentary layer and basement is assumed to be −200 kg/m3.
In Fig. 6, I showed distributions of the maximum (red)and minimum (blue) eigenvectors of the gravity gradienttensor caused by these models. In each figure, thelengths of all the eigenvectors are the same, because weare interested in the fault dip and only angle informationis necessary for this study.
From Fig. 6a, it is found that the dip of the maximum
eigenvector of the gravity gradient tensor closely follows
the dip of the normal fault. When the basement distributes near the surface, the maximum eigenvector points
in the vertical direction to the high-density basement.
The effect of the high-density basement is weak in thesedimentary layer area, while the effect of the lowdensity sedimentary layer is strong; therefore, the minimum eigenvector points in the vertical direction to the
low-density sediment and the maximum eigenvector
points in the horizontal direction.
When the boundary is a reverse fault, from Fig. 6b, it
is found that the dip of the minimum eigenvector of thegravity gradient tensor indicates the dip of the fault well.

Page 6 of 10

Fig. 6 Eigenvectors of the gravity gradient tensor caused by thesimplified sedimentary basin models shown in Fig. 5. The maximum
eigenvector and minimum eigenvector are indicated by red andblue, respectively. a Eigenvectors on the sedimentary basin formed
by normal faults. When the sedimentary layer is in contact with thebasement by normal fault, the dip of the maximum eigenvector
follows the dip of the normal fault. b Eigenvectors on the sedimentarybasin are formed by reverse faults. When the sedimentary layer is incontact with the basement by reverse fault, the dip of the minimum
eigenvector follows the dip of the reverse fault

The maximum eigenvector on the basement points inthe vertical direction to the high-density basement, andthe minimum eigenvector points in the horizontal direction. Since the low-density sediment distributes near thesurface in the sedimentary layer area, the minimum
eigenvector points vertically.
From these results, it was concluded that if the structural boundary is a normal fault, its dip can be estimatedfrom the dip of the maximum eigenvector of the gravity
gradient tensor, and if the boundary is a reverse fault, its
dip can be estimated from the dip of the minimum
eigenvector of the tensor. In addition, in the area awayfrom the boundary, it was found that the maximum
eigenvector on the basement and the minimum eigenvector on the sediment point in the vertical direction,and the maximum eigenvector on the sediment and theminimum eigenvector on the basement point in thehorizontal direction, regardless of whether the boundary
is a normal fault or reverse fault.

Subsurface structures and eigenvectors

Fig. 5 Simplified sedimentary basin model. Light yellow and whiteareas indicate sedimentary layer and basement, respectively. a
Sedimentary basin model where sedimentary layer is in contact
with the basement by normal faults of 45° dip. b Sedimentarybasin model where the sedimentary layer is in contact with thebasement by reverse faults of 45° dip

By simple numerical simulations, it was found that the
maximum eigenvector of the gravity gradient tensor
points to a high-density causative body and that theminimum eigenvector points to a low-density causativebody. In addition, it was found that the dip of the maximum eigenvector of the tensor closely follows the dip ofthe normal fault and that the dip of the minimum eigenvector closely follows the dip of the reverse fault.

Kusumoto Progress in Earth and Planetary Science (2017) 4:15

As mentioned above, Beiki and Pedersen (2010) havealready pointed out that the maximum eigenvector of
the gravity gradient tensor points to the high-density
causative body. The result in Fig. 4a confirms that theirresults are true for the two-dimensional analyses as well.
When the basement distributes near the surface, themaximum eigenvector points in the vertical direction.
This property also shows that Beiki and Pedersen (2010)
are correct, and the idea of the basement as an aggregateof high-density prisms (Fig. 1b), suggested by Kusumoto(2015, 2016b), would not be incorrect.
As to why the dip of normal fault was given by the dipof the maximum eigenvector of the gravity gradient tensor, I considered that the lower part of the boundary
structure (fault) exists inside the low-density area morethan its upper part. Therefore, because the gravity gradient tensor is most sensitive to the subsurface structures near the surface, the structure shown in Fig. 5a
was considered a high-density body that intruded into
the low-density layer, and the dip of the normal fault
was given by the dip of the maximum eigenvector. I believe Kusumoto (2015, 2016a, 2016b) was able to obtain
results that coincided with seismic surveys since he estimated the fault dip in normal fault regions by the maximum eigenvector of the tensor.
On the other hand, when the maximum eigenvector
points to high-density causative bodies embedded in
low-density medium or low-density causative bodies embedded in a high-density medium, the minimum eigenvector points to the low-density mediums or to thecausative bodies. Beiki and Pedersen (2010) have not explicitly referred to analyses of low-density causative bodies using eigenvectors. Since it is necessary to analyze
anomalies caused by low-density bodies in the field, it
seems that the result, in which the minimum eigenvector points to the low-density bodies, would play animportant role in subsurface structure estimation, although this is the result of two-dimensional analysis.
In addition, it was found that the dip of the minimum
eigenvector of the gravity gradient tensor gave the dip of
the reverse fault. As to the reason why the dip of reverse
fault was given by the minimum eigenvector of the gravity gradient tensor, I considered that the lower part ofthe boundary structure (fault) exists inside the highdensity area more than its upper part. Namely, because
this structure was considered a low-density body that intruded into the high-density layer, the dip of the reverse
fault was given by the dip of the minimum eigenvectorof the gravity gradient tensor.
As is understood from the results and discussions obtained in this study, selecting a suitable eigenvector for
estimating the fault dip is important. If the study area is
not too wide and prior geological information is available, the eigenvector that should be employed forPage 7 of 10

estimating the fault dip correctly would be selectedbased on the information. If the study area was a fault
area where normal faults were mainly distributed, themaximum eigenvector of the gravity gradient tensor
would be employed for estimating the fault dip. If thestudy area was a fault area where reverse faults weremainly distributed, the minimum eigenvector would be
employed.
In the three-dimensional study for high-density causative bodies, it is pointed out that the minimum eigenvector is parallel to the strike direction of the structure(Beiki and Pedersen 2010; Beiki 2013). However, in thetwo-dimensional analyses, the strike direction of thestructure is perpendicular to x- and z-axes and does notappear in the analyses. As it is difficult to directly compare the properties of the minimum eigenvector obtained in different dimensions, in the future, it would benecessary to discuss detailed properties of the minimum
eigenvector.

Application to field data

As an application of the techniques, I estimated the dipof the Kurehayama Fault located in Toyama, Japan. The
Kurehayama Fault is a reverse fault located at the centerof the Toyama basin, and it strikes in the NNE-SSW direction (Fig. 7). The length of the fault is about 22 km,
and the fault dip is about 45° (e.g., The Headquarters for
Earthquake Research Promotion 2008; Toyama City
2013). The Toyama City has carried out seismic surveys
and dense gravity surveys crossing this fault (Toyama
City 2013). Toyama City (2013) set three profiles crossing the Kurehayama Fault, and the dense gravity surveys of 50 m spaced measurements have been conductedon these profiles, although spacing of several hundred
meters has been usually employed for these surveys.
Here, I used gravity anomaly data on the profile located at the shoreline. Figure 8 shows the Bouguer
anomaly in which the Bouguer density of 2260 kg/m3
was assumed (Toyama City 2013). The indication“Kurehayama Fault” shown in this figure indicates arough fault location.
I applied the techniques to the Bouguer anomaly andobtained the gravity gradient tensor shown in Fig. 9.
Figure 10 shows distributions of the maximum eigenvector (red) and the minimum eigenvector (blue) of
the gravity gradient tensor. Since the Kurehayama
Fault is a reverse fault, I focus on the dip of theminimum eigenvector. From Fig. 10, it is found thatthe dip (α) of the Kurehayama Fault was about 138°.
Since the angle α is measured clockwise from the surface (x-axis), it seems that the obtained dip indicatesthe dip of the reverse fault of 42°. This fault dip isconsistent with conventional data.

Kusumoto Progress in Earth and Planetary Science (2017) 4:15

Page 8 of 10

Fig. 9 Gravity gradient tensor (gxx, gxz (=gzx), gzz) on the profile.
These are estimated from the Bouguer anomalies on the profile
shown in Fig. 8. The component of gxx and gxz is calculated bya finite-difference method of gravity vectors gx and gz in the spacedomain. The “Kurehayama Fault” shown in this figure indicates a roughfault location by Toyama City (Toyama City 2013). The unit of the gravity
gradient tensor is given in E (Eötvös), and 1 E = 0.1 mGal/km

estimating the depth of the estimated dip or the dip in
the arbitrary depth.
Fig. 7 Location map of the study area. Kurehayama Fault is areverse fault located in the center of the Toyama Basin, Toyama
Prefecture, Japan. Its location has been estimated by topographic,
geological, and geophysical data. The red line and brown lines denotethe estimated location of the Kurehayama Fault, Toyama City (Toyama
City 2013), and The Headquarters for Earthquake Research Promotion(The Headquarters for Earthquake Research Promotion 2008),
respectively. Blue line a - b indicates the dense gravity survey profile,which has gravity observation points at about 50 m intervals

Conclusions
In this study, I showed techniques for estimating thegravity gradient tensor from gravity anomalies on the
profile and for estimating the fault dip by eigenvector of

The estimated fault dip would be the dip near thesurface because the method employs the gravity gradient tensor, which is sensitive to subsurface structuresnear the surface. Since it is important to know quantitatively which depth the estimated fault dip is, in the
future, it would be necessary to develop a techniqueFig. 8 Bouguer anomalies on the profiles (after Toyama City 2013).
The Bouguer density of 2260 kg/m3 is assumed. The “Kurehayama
Fault” shown in this figure indicates a rough fault location by Toyama
City (Toyama City 2013). The unit of the gravity anomaly is given in
milligal, and mGal = 10−5 m/s2

Fig. 10 Eigenvectors of the gravity gradient tensor on the profile
shown in Fig. 7. The maximum eigenvector and minimum
eigenvector are indicated by red and blue, respectively. The dipsof eigenvectors are given clockwise from x-axis to z-axis. Since it
is known that the Kurehayama Fault is a reverse fault, we focus on theminimum eigenvector of the tensor. The “Kurehayama Fault” shown inthis figure indicates a tentative fault location in Toyama City (Toyama
City 2013). The average dip of the minimum eigenvector in the
Kurehayama Fault zone shown by a rectangle with dashed lines
is about 138°, and this angle indicates that the Kurehayama
Fault would be a reverse fault of 42°. In addition, the maximum
eigenvectors on the right side of the “Kurehayama Fault” shownin this figure point to the vertical direction, and the minimum
eigenvectors in the left side of the “Kurehayama Fault” point tothe vertical direction

Kusumoto Progress in Earth and Planetary Science (2017) 4:15

the observed or calculated gravity gradient tensor on the
profile. I also investigated its properties by numericalsimulations.
From numerical simulations, it was found that the
maximum eigenvector of the tensor points to a highdensity causative body and that the dip of the maximum
eigenvector closely follows the dip of the normal fault.
In addition, if the basement distributes near the surface,
the maximum eigenvector points to the vertical direction. They have been pointed out already in previous
studies, and the results shown in here confirmed that
their results are true for the two-dimensional analyses aswell. On the other hand, it was found that the minimum
eigenvector of the tensor points to a low-density causative body and that the dip of the minimum eigenvector
closely follows the dip of the reverse fault. Since eigenvector analyses of the anomalies caused by the lowdensity causative body have not been discussed explicitly
in previous studies, these results would play an important role in estimations of subsurface structures in the
future. From these results, it was found that the eigenvector of the gravity gradient tensor for estimating fault
dips is determined by fault type, and we would estimate
the fault dip correctly if we were to employ suitable eigenvectors based on prior information.
As an application of suggestions, I estimated the dip ofthe Kurehayama Fault located in Toyama, Japan, and obtained the fault dip of about 42° as the dip of the minimum eigenvector of the gravity gradient tensor because
the fault is the reverse fault. This dip harmonized withconventional geological information.
Since the analysis technique shown in this study does
not require complex calculations and vast calculation
times, it will be an effective technique for analyzing
high-resolution data obtained densely by not only dense
gravity surveys for fault research but also airborne gravity or gravity gradiometry surveys.
Acknowledgements
The author is most grateful to the two anonymous reviewers for theirconstructive reviews and comments on the manuscript. In addition, theauthor is most grateful to Yuichi Hayakawa for his editorial advices andcooperation. The manuscript was improved by these reviewers’ comments
and suggestions. This work was supported partially by JSPS (Japan Society
for the Promotion of Science) KAKENHI Grant Numbers 15K14274, 16H05651,
17K01325. The author is grateful to JSPS.
Funding
This work was supported partially by JSPS KAKENHI Grant Numbers
15K14274, 16H05651, and 17K01325.
Author’s contributions
SK planned this study and conducted all the calculations and discussion. He
also drafted this manuscript.
Authors’ information
SK is an associate professor at the University of Toyama.

Page 9 of 10

Competing interests
The author declares no competing interests.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in publishedmaps and institutional affiliations.
Received: 30 November 2016 Accepted: 24 May 2017

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