Crustal tilts around the Kalabsha fault, North West of the Nasser Lake, as predicted by least-squares collocation of levelling data

DOI : 10.4197/Mar. 23-1.1

3

Crustal tilts around the Kalabsha fault, North West of the Nasser Lake, as predicted by least-squares collocation of

levelling data

Gamal El-Fiky

Department of Hydrographic Surveying, Faculty of Maritime Studies, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia

E-mail: gamal_elfiky@yahoo.com

Abstract: Least-squares prediction using an empirically deduced local covariance function was performed to investigate the horizontal gradient of vertical movements (tilting movements) and possible position of block boundary around the Kalabsha fault, north west of the Nasser Lake, Egypt. Levelling data covering the Kalabsha area in the interval 1986 - 1992 were used. First approximated a covariance function of the rates of vertical crustal movement with a Gaussian form function is estimated and used it to segregate signal (rate of vertical movement rate) and noise in observed data. Then, the signal at grid points of 400 × 400 m² mesh covering the Kalabsha area was estimated. After that, the estimated signal was differentiated in space to calculate the tilting movements. Finally, the position of possible fault zones or blocks boundaries in the Kalabsha area was delineated from the obtained continuous distribution of tilts of the deformed surface. By the present method, the subsidence in the southern part of the Kalabsha network relative to its northern was well reproduced.

The pattern of tilt movements shows a low zone of tilt movements in the middle of the studied region surrounded by high tilt areas. The rate of tilt movements reaches about 0.35 microradian/yr in the southern high tilt area. Moreover, the delineated position of the fault zone or the block boundary in the area is in a good agreement with the well- known trace of Kalabsha fault.

Keywords: Levelling Data, Nasser Lake, Least-Squares Prediction, Kalabsha Fault, Vertical Movements, Crustal Tilt.

Introduction

The least-squares prediction (LSP) method is a part of the least-squares collocation (LSC) method that has been developed by Moritz (1962) for the reduction of gravity data. Application of this method to crustal deformation has been made by El-Fiky et al. (1997). It could be a very useful tool for studying tilting movements and position of possible fault zones or crustal block boundaries. In the present article, we apply this technique to investigate the tilting movements and position of possible Kalabsha fault using the levelling data in the Kalabsha area.

The algebraic polynomials of specified degree can be used to describe the tilting movements (Vanicek and Christodulides, 1974). Kato and Kasahara (1977), for example, used the spline function in the time- space representation of non-linear movement along levelling routes.

However, such purely mathematical approach could sometimes yield spurious results. We would prefer to use some physical information included in data itself. One of relevant techniques is the LSC (Hein, 1986; El-Fiky et al., 1997 and Tawfik et al., 2011) whereby a tectonic signal in the data is extracted by the statistical analysis of data itself. LSC is able to make parameter estimation, filtering and prediction of the signal by least-squares adjustment. Thus, we prefer to use the LSC technique. Among capabilities of the technique we use nonparametric estimate of the signal of crustal movement to determine the tilting movements from the levelling data, which is called as the least squares prediction (LSP) technique.

The LSP technique is developed to predict the distribution of tilting movements in the Kalabsha area. For this purpose, we first derived an empirical covariance function (ECF) for this area using the levelling data in the interval 1986-1992 and used it to estimate significant signal, in the present case vertical movement rate. Generally, main benchmarks in levelling network of studied region are established about every 2 km in two routs perpendicular to Kalabsha fault. However, the distance between these two levelling routes is quite large. Therefore prediction of the rates of vertical deformation and tilting movements is important for studying the areal distribution of such crustal activities. In the present investigation, emphasis will be laid on the characteristics of the distribution of predicted tilting movements and block boundary in the area.

The Kalabsha area is the most recent active area in the northwestern part of Nasser Lake. It lies on a large western embayment of the Nasser Lake. An earthquake of magnitude 5.3 (Ms) took place on 14 November 1981 in the unpopulated area of Kalabsha, along the Kalabsha fault, 70 km southwest of Aswan City (Fig. 1). This earthquake is considered a very important event, as it is located not far from the Aswan High Dam (Fig. 2). After the occurrence of this earthquake, a long-term study program was performed to study horizontal and vertical crustal deformation by the geodetic techniques (Vyskocil and Tealeb, 1985). This program includes the establishment and measurement of local geodetic networks around potential faults (Fig. 3) as well as establishment and measurement of sub-regional and regional geodetic network to connect the local ones (e.g., Vyskocil et al., 1991). In addition, several levelling networks crossing these faults were also established within the local geodetic networks in order to study the vertical deformations, which are associated with the seismic activity in the area (Fig. 2).

Fig. 1. Location of studied area.

Egypt

Cairo

Nasser Lake

Kalabsha Area

In this study we used precise levelling measurements across the Kalabsha fault during the period from 1986 to 1992 (Fig. 2), and applied the least-squares prediction technique, to show that it is effective for studying tilting movements and possible position of the fault zone or block boundary in this high seismic area. Brief discussion on the tectonic implication of the result is also discussed.

Fig. 2. The Kalabsha area lies along the Kalabsha fault on a large western embayment of the Nasser Lake. The E-W and the N-S fault systems in the northwest Nasser Lake as well as the epicenters distribution of the earthquakes, which occurred around the northern Nasser Lake for the period from 1982 to 2000 (ARSC, 2001), are also shown on the figure.

Least-Squares Collocation Technique

Least-squares collocation is a mathematical tool developed by Moritz (1962). It evolved from statistical methods to interpolate gravity anomalies, but has been developed to have a much wider application both inside and outside of physical geodesy (e.g., El-Fiky and Kato, 1999a;

1999b). The basic equation of least-squares collocation is given as (Moritz, 1972):

32.3E 32.5E 32.7E 32.9E 33.1E

23.2N 23.4N 23.6N 23.8N 24.0N

Nasser Lake

High Dam 1982 - 2000

Kalabsha Fault Seiyal Fault

Gazalla FaultGabel El-Baraqa Abu Dirwa Fault Kur El-Ramla Fault

Kalabsha

Area

l = AX + t + n (1) Where l is the measurements vector, A is a known coefficient matrix, X is the vector of unknowns or parameters, t is the signal vector at observation points, and n is the vector of the noise which represents the measuring error. Thus, the vector of measurements l consists of a systematic part, AX, and two random parts, t and n. The noise vector n and the signal vector t have an average value of zero and the covariance matrices C_n and C_t can describe their statistical behavior, respectively.

Fig. 3. Configuration of the local Kalabsha geodetic network. The thin dashed line indicates the Kalabsha levelling network. The thick dashed line in meddle part represents trace of the well-known part of Kalabsha fault. A part of the regional levelling line is shown in the figure.

Suppose that the systematic or modeling error is removed in advance by the network adjustments (AX = 0). Then we consider only the non-parametric part, so that equation (1) may be written in the following form:

32.47 32.48 32.49 32.5 32.51 32.52

23.51 23.52 23.53 23.54 23.55 23.56 23.57

7/1 8/2

8/1

1/4

o o o

o o

o

o o o o o o 3/1 o

3/2 2/2

2/1 1/3

4/1

4/2

5/2 5/1 6/1

5/3 1/2

1/1 7/2

6/2 KHL

To High

Dam

Reg iona

l leve lling lin

e.

l = t + n (2) This equation simply means that the spatially distribution of geodetic data l is given by the summation of tectonic signal t and noise n.

Noise is assumed to be limited only to the levelling benchmark or its adjacent local regions, while tectonic signal can have wider spatial correlations (El-Fiky and Kato, 1999a). We further assume that the data field (rate of vertical movements) is isotropic and homogeneous, so that the covariances of data are only a function of site distance (El-Fiky et al., 1997). We then remove average rate of vertical movements of each levelling benchmark to make non-biased data set and calculate variance- covariance:

Variance: C_l(0) = (

i 1=

∑

^{N li2 )/N , (3)}

Covariance: C_l(d_p) = (

i j<

∑

^{Np lilj )/Np , (4)}

where N is the total number of data points and N_p is the number of data points within a specific distance interval d_p. The distances between two data points d are divided into finite discrete intervals P, in which N_p (1 ≤ p ≤ P) data pairs whose distance drops in the interval (p -1)δ< d_p≤ (p)δ, (p = 1, 2, 3, ...P). Here, variances C_l(0) are estimated at each observational site whereas covariances C_l(d) are estimated for all of site pairs within the assigned distance interval.

Plot of the covariances with respect to distance would be a curve that naturally diminishes with distance (Fig. 4). A mathematical function could be fitted to the covariances and subsequently used to compute each element of covariances matrix. One of the simple mathematical functions to represent such plot would be a Gaussian function in the following form,

C_l(d) = C_t(0) exp (-k²d²) , (5) which is adopted here as the empirical covariance function (ECF). Where C_t(0) and k are constants. Two parameters C_t(0) and k are estimated from a covariance plot of the data. C_t(0) is the expected variance at the observation station and C_n(0) = C_l(0) - C_t(0) is considered as the noise component at the site. k is an indication of how far the correlation

distance reached, which has the dimension of inverse distance. Fig. (3) shows the obtained variance-covariance plot and ECF. Then, parameters C_t(0), and k were fitted to data to define ECF.

Once such ECF is obtained, we can estimate signal S at any arbitrary location in the studied area by the following equation.

S = C_stC_L^-1l, (6)

where the matrix C_st is the covariance matrix between the data points and other grid points where signals are to be estimated. It is composed of elements c_st (1 ≤ t ≤ N, 1 ≤ S ≤ m, where m is the number of grid points whose signals are to be estimated); and c_st is given by c_st = C_t (0) exp (- k²d_st²), where d_st is distance between the data site and the predicted point.

The above formula was used to reconstruct vertical velocity (signal) at grid points of about 400 m × 400 m mesh covering the study region.

Fig. 4. The empirical covariance function of a Gaussian form C(d) = C_t(0) exp(-k²d²) of the rates of vertical crustal movements. Estimated numerical values of the parameters C_t(0), C_n(0), and k are shown in the figure.

Mathematical Formula of Tilting Movements

In the present study we try to transfer the levelling data into tilting movements and finally into block boundary. The concept of analysis is based on the general assumption that a physically, geometrically and elastically defined thin plate is deformed. This deformation, V, determined from vertical rates is used as input-parameters for calculating

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 0.00

0.04 0.08 0.12 0.16 0.20 0.24

d (km) C (0) = 0.19 mm/yr_s ²

C (0) = 0.01 mm/yrr ² k = 0.7511 km^-1 C (0)

C (d) (mm/yr) l2

l

Ct(0) = 0.19 (mm/yr)² Cn(0) = 0.01 (mm/yr)² k = 0.7511 km^-1

the strain tensor within the plate (e.g., Timoshenke and Woinowsky, 1959).

We used the LSP method for interpolating the components of the vertical displacements. The components of the vertical strain V_x, V_y, V_xy, V_xx, V_yy are then the first and second spatial derivatives of the interpolated vertical displacements.

If the covariance function for the rate vertical displacement V is obtained, then from equation (6) we have

V = C_VvC_vv^-1v, (7)

with V being the vector of signal to be estimated; C_Vv is the cross- covariances matrix between the signal V at the grid points and each element v_i of the signal at the observation points v (where i=1,2 …N).

C_vv is the variance matrix of the signal v; and v is the vector of known vertical displacements.

Covariance function is a function of distance d, because v is assumed to be homogeneous and isotropic as it is mentioned before. If we adopt the Gaussian function as the covariance function i.e., C_v(d) = C_vs exp (-k²d²), then we have

C_Vvi = C_vt exp (-k²d_i²) , i = 1, 2, …N (8) Where C_Vvi is the element of vector C_Vv = (C_Vv1 C_Vv2 …C_Vvi ,… C_VvN)^T, and d_i = [(x-x_i)² + (y-y_i) ²]^1/2 is the distance d_i between signal at a grid point z and observation point z_i, (x, y) are the plane coordinates of z and (x_i, y_i) are those of z_i.

Substituting equation (8) into (7), we get the vertical displacement in terms of the covariance function

V =

i 1=

∑

^{N Cvt exp (-k2di2)Cvv-1vi , (9)}

Then by space differentiation of equation (9), we can easily get the following equations

dV/dx = V

x

=

i 1=

∑

^N

^-2C

^vt

^k

²

^(x-x

ⁱ

^{) exp (-k}

²

^d

ⁱ²

^{) C}

^vv-1

^v

ⁱ

^,

dV/dy = V

^y

=

i 1=

∑

^N

^-2C

^vt

^k

²

^(y-y

ⁱ

^{) exp (-k}

²

^d

ⁱ²

^{) C}

^vv-1

^v

ⁱ

^,

(10)

From these equations the total tilt, T and the azimuth of tilting θ can be obtained as follows:

T= (V_x² + V_y²)^1/2 (11) θ = tan^-1 (V_x / V_y), (12) Finally, we would take advantage of this new technique to delineate position of the fault zones or in other word crustal blocks boundaries from rates of vertical movements in the studied region based on the hypothesis that tilting axes of land blocks usually coincide with block boundaries. This hypothesis means that maximum gradient change of tilting is coincided with possible fault zones in the studied area. Here we try to use the obtained continuous distribution of tilt of the deformed surface to delineate the position of the Kalabsha fault.

Levelling Data of Kalabsha Area

The Kalabsha levelling network was established in 1985 in a direction approximately perpendicular to the extension of the Kalabsha fault. The first measurements for this network were performed in January 1986 and repeated in January 1988, October-November 1989, and January 1992. The configuration of this levelling network is superimposed on the Kalabsha geodetic network in Fig. (3). The precise automatic Kern GK2A, invar rods with a metallic cover of 3.0 m length and heavy footplates were used to carry out the measurements. The Kalabsha levelling measurements were designated as first-order in double run, i.e., the measurements of different heights were run twice, forward and backward measurements.

El-Fiky et al. (2002) has already studied the spatial and temporal variation of the above levelling data in Kalabsha area using a dynamic adjustment, in which not only corrections to the assumed height but also rate of vertical movement linear with time at the benchmarks were assumed to be unknowns. Their results for the interval 1986-1992 are

reproduced in Fig. (5). Hence, we here use the LSP method (equation 6) to segregate the signal (rate of vertical movements) and noise in the estimated rate vertical movements by El-Fiky et al. (2002) for the interval 1986-1992. Then, the tilt movements and the position of Kalabsha fault will be investigated for the same interval using the above developed LSP technique.

Fig. 5.A contour map of the rate of vertical crustal movement relative to point 1/3 in the Kalabsha area based on linear interpolation for the interval 1986-1992. The unit is mm/yr. The figure is reproduced from El-Fiky et al., (2002).

Results and Discussions

In order to apply the LSP, we consider the estimated rates of vertical crustal movement in the interval 1986-1992 as the original observation (l in equation 1), in which the random part caused by vertical crustal movement as the signal (t) and the systematic part (AX) caused by systematic errors. The systematic part is assumed to be eliminated by

32.48E 32.49E 32.50E 32.51E 23.51N

23.52N 23.53N 23.54N 23.55N

23.56N 2/2

1/3

5/3 5/1 1/2

5/2

1/1992 - 1/1986

network adjustment (El-Fiky et al., 2002). This allows us to conduct the least-square prediction (equation 2) and estimate parameters, C_t(0), C_n(0), and k. Using these estimated parameters, it is possible to predict the signal (the rate of vertical crustal movement) at any arbitrary point by equation (6).

The parameters of the covariance function (Gaussian function) employed in the present study are the variance of signal C_t(0) = 0.19 (mm/yr)², k = 0.7511 km^-1, and variance of noise C_n(0) = 0.10 (mm/yr)², respectively.

After dividing the Kalabsha area into cells of 400 m × 400 m, we estimated the rate of vertical crustal movements at all grid points from the observed signal at the benchmarks for interval 1986-1992.

Fig. 6.Distribution of the predicted rates of the vertical crustal movements in the Kalabsha area estimated by LSP method for the interval 1986 - 1992. The unit is mm/yr. The open circles indicate the levelling benchmarks used in this study.

32.48E 32.49E 32.50E 32.51E 23.51N

23.52N 23.53N 23.54N 23.55N 23.56N

2/2

1/3

5/3

5/1 1/2

5/2

Fig. (6) shows the predicted vertical crustal movement rate during this interval by LSP. On the other hand, as it is already mentioned above, Fig. (5) indicates the distribution of the rate of vertical crustal movements for the same interval (El-Fiky et al., 2002). It should be noted that the contour map of the vertical rates in Fig. (5) was drawn by linear interpolation where no errors were considered in the data. Comparing Fig. (5) with Fig. (6), it is readily seen that the contour map by linear interpolation (Fig. 5) and by the LSP method in this study (Fig. 6) are approximately the same as a whole, but some differences exist. In the previous study (El-Fiky et al., 2002), the adjusted rates of vertical movements were considered as true values and the contour lines were most influenced by the closest known values. On the other hand, the LSP uses empirical covariance function, deduced from all the adjusted values, and predict the values at unknown points by removing the observation errors through the filtering effects of parameter C_n(0) in the covariance function. Therefore, the irregularities in the known data, which may be due to the observation errors, are removed to some extent. Furthermore, it may be seen that most of the subsidence of short wave length located at in southern part of the studied region is filtered out after applying this technique. Remaining patterns in this figure may well represent areal vertical deformations due to tectonic origin. Fig. (6) clearly represents the subsidence in the southern part of the levelling network relative to its northern part. The subsidence rate of this area is about -1.4 mm/yr. This subsidence could be explained in relation to water level in the lake during the epochs of measurements.

On the other hand, the estimated vertical rates at grid points (Fig. 6) are differentiated spatially using equation (10) to estimate the crustal tilts in the present data period. Fig. (7) shows the obtained tilting movements in the interval 1986-1992. The tilt vectors in this figure shows a low tilt zone in the middle of the studied region surrounded by high areas of tilting movements. At the southern edge of this low zone the trace of Kalabsha fault is run, as will be discussed later. The amount of tilt movements reaches 0.35 microradian/yr at the southern part of the studied area. This high tilts rate might be due the high seismic activity in the studied area and/or the fluctuation in the water level of the Nasser Lake in the present interval (El-Fiky et al., 2002). The high tilt movements in the southern part of the studied region (Fig. 7) seem to be correlated with the high subsidence there (Fig. 6).

Fig. 7.Direction and magnitude of tilting movements in the Kalabsha area estimated by the developed LSP technique for the interval 1986 – 1992. The unit used is microradian/yr. The open circles indicate the levelling benchmarks used in this study.

Finally, the obtained continuous distribution of tilt of the deformed surface in the Kalabsha area was used to delineate position of fault zones or blocks boundaries, based on the hypothesis that tilting axes (lines of maximum gradient change) of land blocks usually coincide with block boundaries. Fig. (8) shows the position of the fault zones obtained from tilts data in the Kalabsha area. As shown, the fault zone, which represents the Kalabsha fault in this area, has been clearly delineated. This fault zone runs in the middle of Kalabsha area and divides the area into the northern and the southern regions (Fig. 8). It coincided with the well- known trace of Kalabsha fault (Fig. 2). The Kalabsha fault was identified as the source of the 14 November 1981 earthquake (Kebeasy et al.,

32.48E 32.49E 32.50E

23.52N 23.53N 23.54N 23.55N

23.56N 2/2

1/3

5/3

5/1 1/2

5/2

0.35 microradian/yr

1982). The Kalabsha fault is well expressed on aerial photographs. Its total length is about 300 km from the Kalabsha embayment in the east to the west crossing the Sinn El-Kaddab plateau (Issawi, 1969, 1978).

Geological and seismological evidence indicate that the Kalabsha fault is a right lateral strike slip fault (Issawi, 1969 and Kebeasy and Gharib, 1991). On the other hand, seismic activity in the Kalabsha area is continuing at a low level since the half of 1980s till now, and mainly occurred near the epicenter of the main earthquake of 1981 along the Kalabsha fault near Gebel Marawa (ARSC, 2001). This seismic activity is believed to have originated tectonically and only triggered by the lake (e.g., Kebeasy et al., 1982).

Fig. 8. The estimated position of fault zones or crustal block boundaries in the Kalabsha area estimated from the tilting movements in Fig. (5). The open circles indicate the levelling benchmarks used in this study.

The present study has shown that the LSP with an empirical covariance function is a useful tool to delineate the picture of the tilts movements and the position of possible fault zones where vertical crustal deformations are available. However, a further study with a dense

32.48E 32.49E 32.50E 23.51N

23.52N 23.53N 23.54N 23.55N

23.56N 2/2

1/3

5/3

5/1 1/2

5/2

MP MP

MP

MP

MP

levelling data and a longer interval is necessary to confirm the obtained results in this study.

Conclusion

The least-squares prediction technique has been applied to investigate tilting movements and position of the fault zone in Kalabsha area using the levelling data for the period from 1986 to 1992. In order to apply this technique, we first derived an empirical covariance function (ECF) for this area using the levelling data in the interval 1986-1992 and used it to segregate signal (rate of vertical movement rate) and noise in observed data. Then, the signal at grid points of about 400 m × 400 m mesh covering the Kalabsha area has been estimated. After that, the estimated vertical rates at grid points have been differentiated spatially to estimate the crustal tilts in the present data period. Finally, the obtained continuous distribution of tilt of the deformed surface in the Kalabsha area have been used to delineate position fault zones or blocks boundaries, based on the hypothesis that tilting axes of land blocks usually coincide with block boundaries. Based on this study we can draw the following conclusion.

The pattern of the rates of vertical movements estimated by LSP shows a subsidence in the southern part of the Kalabsha network relative to its northern part.

The pattern of tilts movements shows a low zone of tilts movements in the middle of studied region surrounding by high areas of tilting movements. The maximum amount of tilt movements is about 0.35 microradian/yr in the southern high tilt area. The high tilt movements in the southern part of the studied region show a good correlation with the high rate of subsidence there.

The position of the fault zone in the Kalabsha area has been clearly delineated. It coincided with the well-known trace of Kalabsha fault, which is well expressed on aerial photographs.

Acknowledgments

The author is very grateful to the president of the National Research Institute of Astronomy and Geophysics (NRIAG), Helwan, Egypt, Cairo for providing the levelling data used in this study.

References

ARSC (Aswan Regional Seismological Center) (2001) Internal report, National Res. Inst. of Astro. and Geophys., Helwan, Egypt, 250 p (in Arabic).

El-Fiky, G.S. and Kato, T. (1999a) Continuous distribution of the horizontal strain in the Tohoku district, Japan, predicted by least-squares collocation, J. of Geodynamics, 27: 213-236.

El-Fiky, G.S. and Kato, T. (1999b) Interplate coupling in the Tohoku district, Japan, deduced from geodetic data inversion, J. Geophysical Research, 104: 20361-20377.

El-Fiky, G.S., Kato, T. and Fujii, Y. (1997) Distribution of the vertical crustal movement rates in the Tohoku district, Japan, predicted by least-squares collocation, J. Geodesy, 71: 432- 442.

El-Fiky, G.S., Mousa, A. and Tealeb, A. (2002) Temporal change of vertical deformation around the Kalabsha fault, northwest of Nasser Lake, deduced from dynamic adjustment of leveling data, Scientific Bulletin, Faculty of Engineering, Ain Shams University, 37 (3):

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Hein, G.W. (1986) A model comparison in vertical crustal motion estimation using levelling data, NOAA Technical Report No. 117 NGS 35, Rockville, Md.

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Kebeasy, R. M. and Gharib, A. A. (1991) Active faults and water loading are important factors in triggering earthquake activity around Aswan Lake, J. Geodynamics, 14: 73-85.

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Moritz, H. (1972) Advanced least squares methods, Report of Dep. of Geod. Science, No. 175, Ohio State Univ., Columbus, U.S.A.

Tawfik, H.S., El-Fiky, G.S. and Salama, I. (2011) Crustal strains field in the vicinity of the Northern Part of Nasser Lake, Egypt, as deduced from GPS measurements and its tectonic implications, Civil Eng., Research Magazine (CERM), Faculty of Eng., Al-Azhar University, Cairo, Egypt, 33: 537-561.

Timoshenko, S.P. and Woinowsky, S. (1959) Theory of plates and shells, McGraw-Hill, New York.

Vaniček, P. and Christodulides, D. (1974) A method for the evaluation of vertical crustal movements from scattered geodetic relevelling, Cand. J. Earth Sci., 11: 605-610.

Vyskočil, P. and Tealeb, A. (1985) Report on activities for monitoring recent crustal movements at Aswan in the period 1983-1985, Bull. of National Res. Inst. of Astronomy and Geophysics, Helwan, Egypt.

Vyskočil, P., Zeman, A., Tealeb, A., Mahoud, S.M. and El-Fiky, G.S. (1991) Vertical movements around the Kalabsha fault, northwest of Aswan Lake, from precise levelling, J. Geodynamics 14: 249-262.

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