Gaussian curvature and the relationship between the shape and the deformation of the Tonga slab

Geophys. J. Int. (1996) 127,311-327 Gaussian curvature and the relationship between the shape and the deformation of the Tonga slab Sarah Nothard,' Dan McKenzie,' John Haines2 and James Jackson'9" Bullard Laboratories, Madingley Road, Cambridge, CB3 OEZ, UK 'Institute of Geological and Nuclear Sciences, PO Box 132, Wellington, New Zealand Accepted 1996 June 24. Received 1996 June 21; in original form 1996 February 19 SUMMARY We investigate a particular potential cause of deformation within the subducting Tonga slab: that associated with material that moves over a template while remaining in contact with it. In such a situation both the location and the style of deformation within the material depend, in a predictable way, on the shape of the template, and in particular on its Gaussian curvature. We look for such an association in the Tonga slab, using earthquake locations to define the slab shape and their focal mechanisms to indicate the style of deformation. Only in one place, at 25"s and 5-6 km depth, does the style of the faulting in the earthquakes demonstrably correspond with that required by the Gaussian curvature if the slab were moving over a template. Although the Gaussian curvature in other parts of the slab, particularly near the 'hook' at its northern end, would also require deformation if the slab were moving over a template, the pattern of earthquake mechanisms were in those places is not clear enough to confirm the association. Although we are limited by our ability to resolve only the coarsest features (b 3 km) of the slab shape, we reach the important conclusion that deformation in response to motion over a template is not the main cause of the intermediate and deep seismicity in the Tonga slab. Most of the earthquakes, and all the biggest ones, occur where, even if the slab were moving over a template, it would not need to deform to do so. Some other explanation is required for these earthquakes. Key words: deformation, earthquake location, subduction, Tonga slab. 1 INTRODUCTION In what way is the shape of a subducting lithospheric slab related to its deformation? In this paper we investigate this question in the Tonga subduction zone, where the Pacific plate is subducting beneath the Australian plate and the Lau Basin at the Tonga Trench, and a dipping zone of seismicity indicates the present position of the Pacific slab to a depth of 67km in the upper mantle (Sykes 1966). Earthquake locations and mechanisms are the only available measures of the shape and deformation of the slab. In this paper we examine whether the focal mechanisms of earthquakes within the slab are related to its shape. Various things could happen to the subducted slab as it moves through the upper mantle. (1) The shape of the slab could be formed in the trench. It could then be transported into the mantle without a significant further change in shape, in which case we would expect no * Corresponding author. relationship between the locations and mechanisms of the earthquakes within the slab and its shape. (2) The slab could shear, buckle, bend, detach or fragment (Davies 198) in some general way. For example, it could bend and unbend about a horizontal axis (Fig. la) as it does to create an outer trench rise (Chapple & Forsyth 1979) or an intermediate-depth double seismic zone (Kawakatsu 1985, 1986a,b; Zhao, Hasegawa & Kanamori 1994). The deformation resulting from the subduction of a spherical shell of oceanic lithosphere along an approximately linear trench, such as Tonga, is actually a 3-D problem (Yamaoka, Fukao & Kumazawa 1986). 2-D bending about an axis is an oversimplification as no consideration is taken of lateral or alongstrike strain. A more general velocity field within the deforming slab is also possible (Nothard 1995), as illustrated in Fig. l(b). (3) The slab could move over a template corresponding to its shape. Such a template might, for example, be maintained by flow in the asthenosphere. In this situation, the shape of the slab remains unchanged relative to an observer on the overriding plate, but the slab may be required to deform (and 1996 RAS 311 on 21 January 218

312 S. Nothard et al. Figure 1. Cartoons to illustrate the different ways in which a slab might deform as it is subducted through the upper mantle. (a) It may bend and unbend about a horizontal axis resulting in the outer trench rise and an intermediate-depth double seismic zone. (b) Distributed deformation may result in a more general velocity field within the slab. (c) It may move over an uneven surface or template corresponding to the shape of the slab, and deform to maintain its shape. in particular to change its surface area) as it moves over the template. In the same way, a carpet (the slab) moving over a lump in the floor (the template) would need to stretch on the top of the lump to fit the shape exactly (Fig. lc). If deformation of this type occurs by faulting, it requires a predictable relation between the location and type of faulting and the shape of the slab or template. The purpose of this paper is to see whether such relations exist in the Tonga subduction zone. We assume that the position and shape of the subducting lithospheric slab is defined by a surface fitted through the earthquake locations within the Wadati-Benioff zone. We then look for any association between the shape of the slab and the deformation revealed by the location and mechanisms of earthquakes within it. 2 GAUSSIAN CURVATURE We investigate the relationship between the seismicity and the shape of the Tonga slab using the Gaussian curvature (K) of the slab surface (Stoker 1969). Within the tangent plane to a surface at a given point there is one direction for which the curvature of the surface is a maximum, and another orthogonal direction for which it is a minimum. These are the principal curvatures of the surface. Their corresponding radii of curvature, R1 and R2, are the two principal radii of curvature of the surface at that point, and the Gaussian curvature (Gauss 1828; Love 1934) at that same point is 1 K=- RIR2 The principal curvatures are: _- 1 - Bll R~ [i + (aj/ax;)2-p2 (3) where x3 = j(xl, x2) defines the shape of the surface. xi and xi are parallel to the two directions of principal curvature, which are rotated through an angle with respect to x1 and x2, and -- af 8.f - af cos - -sin -, ax; 8x1 8x2 aj -- aj 8.f - sin - + cos -, ax; ax1 3x2 azf B11= - COS - 2- a2j sin2. = - ax: ax, axz ax; ax? a2f sin cos + - a2f a2f a2f a2f a2j sin cos + 7cos2= ~ ax: ax, ax, 8x2 ax: BZ2 = - sin2 + 2- (4) (7) The angle through which the coordinate frame is rotated to align the axes with the directions of the principal radii of curvature at each point on the surface is given by A contour plot of Gaussian curvature indicates where structures occur on a surface. If both principal curvatures are non-zero the surface is said to have double curvature (Lisle 1994). For example, in a dome or basin Rl and R2 both have the same sign and so K > (Fig. 2a). For a saddle, K <, as Ri and Rz have opposite signs (Fig. 2c). For a parabolic, cylindrical or planar surface K = because one of the radii of curvature is infinite (Fig. 2b), and the surface is then said to have single curvature. m K=O (b) Figure 2. The Gaussian curvature, K, relates to the structures on the slab surface as follows: (a) K >, (b) K = and (c) K i. 1996 RAS, GJI 127, 311-327 on 21 January 218

Deformation of the Tonga slab 313 However, in this paper it is the change in Gaussian curvature over the surface that is more important than the Gaussian curvature itself, as this indicates the nature of the deformation necessary if the surface is to change from one shape to another. If the surface is moving over a template, and if the Gaussian curvature (K) is constant everywhere along the path, then the surface area can be locally conserved and the surface can move along the template without extension or compression [Love (1934), based on a theorem of Gauss (1828)l. Thus, if then the surface (in this case the subducting slab) can act as a flexible but inextensible sheet. For example, the simple folds that can be obtained from a sheet of paper (K=O) without stretching all have one radius of curvature which is infinite, so 6K = and area is conserved, but the sheet of paper (K = ) cannot be formed into a surface with double curvature, such as a dome and or saddle (K # ) without some stretching or contraction because 6K #. Two planar or single-curvature surfaces, such as two sheets of paper, can slide over each other and remain in contact without a change in surface area. By contrast, two surfaces with varying double curvature, such as egg-boxes (with domes, basins and saddles), are unlikely to be able to slide over each other and remain in contact without deforming by extension or compression (unless a particular path is chosen for which 6K = ). In this paper we apply these simple ideas to the Tonga slab. If the slab is moving over a template in the mantle then we would expect an association between that shape and the deformation of the slab (revealed by earthquakes). If the slab is following a path for which 6K =O then no associated deformation is necessary; any deformation that does occur must have some other origin. Conversely, if the path is such that 6K #, the associated deformation should be of a type that accommodates the necessary change in surface area of the slab as it moves. Bevis (1986) used Gaussian curvature to examine whether a hemispherical, inextensible sheet representing an oceanic plate could deform into a dipping, subducting slab without a change in its surface area, i.e. whether 6K = throughout the transformation. For a plate at the Earth's surface, R, and R, both have the same sign, and hence K is positive. For the subducting slab R1 is the radius of curvature of the arc or trench at the Earth's surface and R, that of the slab. Bevis ( 1986) noted that most subduction zones have a concavoconvex appearance (inclined seismic zones are generally convex in cross-section, and arcs concave landwards), which implies a negative Gaussian curvature. He therefore concluded that 6K # during the transformation. By contrast, in a study of the shape of the Nazca plate Cahill & Lsacks (1992) noticed that on a local scale the Gaussian curvature of the slab remains positive. When the trench shape is concave seawards, the subducting slab has a convex-up geometry, and where the trench shape is convex seawards, the subducting slab is usually concave-up. That is, the descending plate appears to respond to the changing sense of the lateral curvature of the plate boundary by reversing the sense of its curvature in the mantle. In this way the Gaussian curvature can remain positive, as in the original spherical oceanic plate, but nowhere do the values of Gaussian curvature (9) calculated by Cahill & Isacks (1992) within the subducting slab have the same magnitude as those of the Earth's spherical cap, so that once again 6K #. Thus both Bevis ( 1986) and Cahill & Isacks (1992) concluded that inclined seismic zones have shapes that are not attainable by a flexible but inextensible spherical cap, i.e. that the lithospheric slab does not conserve its surface area as it enters the trench-in contrast to the view of, for example, Yamaoka et al. (1986). In this paper we are concerned with the slab's deformation after it has passed the trench and entered the mantle. The number of earthquakes, the accuracy of their locations and the length of time covered by the earthquake record are not sufficient to examine whether the shape of the subduction zone is changing with time relative to an appropriate external reference frame such as the overriding plate. If, however, the shape of the dipping zone does not change relative to the overriding plate, the rate of change of the Gaussian curvature at a moving point on the slab can be deduced if it is assumed that the subduction process follows steady streamlines. For example, that direction could be the plate velocity calculated at the Earth's surface using the pole of rotation between the Pacific and Australian plates from DeMets et al. (199), which is then rotated onto the plane of the slab to represent the direction of the subducted Pacific plate material in the inclined seismic zone relative to the overriding plate. 3 OBTAINING A SURFACE TO REPRESENT THE SLAB SHAPE For over 2 km, from 35"s to 15"S, the Tonga-Kermadec subduction zone forms an approximately linear feature. The only deviations are where the Louisville Ridge (Fig. 3) collides with the trench at about 25"s (Christensen & Lay 1988) and at its northern termination where the arc forms a sharp westward bend or 'hook'. This hook is also identifiable in the deep seismicity and shows that the whole northern part of the slab is contorted into a complex shape. At this northern end of the subduction zone, between and 3 km depth, the shape of the Tonga slab defined by the seismicity strikes approximately north-south following the overall strike of the trench (Isacks & Barazangi 1977; Fischer, Creager 8z Jordan 1991). At greater depths (3 to 5 km) the slab then flattens to form a bench-like feature that bends sharply westwards to form the hook below 5 km depth (Chiu, Isacks & Cardwell 1991). We define the shape of the slab surface by first determining its perpendicular distance from a reference plane through the seismicity in the structurally simple region of the slab (outlined by the dashed box, Fig. 3). In the larger region (solid box, Fig. 3), a planar approximation is no longer valid, and instead the reference surface is described by an 8th-order polynomial function, In both cases, bi-cubic splines then define the perturbations of the slab surface about the reference surface. Once the shape of the slab surface has been determined by matching earthquake locations, earthquake focal mechanisms from the Harvard CMT catalogue (January 1977 to February 1994) are rotated and projected along a local normal to the surface (Fig. 4). The association between the shape of the surface (its Gaussian curvature), and the earthquake focal mechanisms is then examined. 1996 RAS, GJI 127, 311-327 on 21 January 218

314 S. Nothard et al. 17' 175' 18' 185' -1' -15' -2' -25' -3' -35' 17' 175" 18' 185' 19' Figure 3. Bathymetry and seismicity of the Tonga-Kermadec subduction zone. The bathymetry contours are at 2 km intervals and earthquake locations are from the ISC catalogue (1964 to 199). Only earthquakes reported by more than 5 stations are plotted. At the northern termination of the zone a sharp westward bend or 'hook is evident in both the trench and the deep seismicity. The dashed line outlines the area in which the slab's shape was approximated to a plane. The solid lines delimit the area of slab represented by the polynomial and spline surfaces. Figure4. Cartoons to show how a typical earthquake mechanism will appear in the various types of projection used in this paper. The cross indicates the location of the P-axis on the focal sphere. Black quadrants are compressional first motions. (a) Map view showing the strike of the trench (N-S) with the slab dipping in the direction indicated by the arrow. The mechanism is a lower-hemisphere projection. The P-axis of the mechanism indicates down-dip shortening of the slab as proposed by the model of Isacks & Molnar (1971). (b) The mechanism is plotted on a vertical cross-section through the slab and is a back-hemisphere projection. The P-axis is now more obviously oriented in a down-dip direction and the arrows indicate the direction of shortening within the slab. (c) The focal mechanism is a back-hemisphere projection onto the average or best-fit plane through the seismicity. The view is along a normal to the plane. The mechanism looks like a thrust fault with its P-axis in a downdip direction, parallel to the average plane, resulting in shortening of the slab in a down-dip direction. on 21 January 218 1996 RAS, GJI 127, 311-327

Deformation of the Tonga slab 315 3.1 The simple case: a planar approximation We first defined an average plane through the seismicity of the Tonga subduction zone within the region outlined by the dashed box in Fig. 3. Hypocentre locations from the following sources were chosen to maximize the number of well-located earthquakes: (1) those reported in the ISC catalogue with the number of recording stations 2 1 for the years 1964 to 199 (reliable teleseismic hypocentres are typically those with the greatest number of P-wave arrival-time reports used to locate the earthquake); (2) Harvard CMT solutions for January 1991 to February 1994; and (3) earthquakes from the CMT catalogue for 1977 to 199 that did not appear in the ISC group above. We are only concerned with earthquakes within the subducting slab (intra-plate). Earthquakes shallower than 1 km were not used because uncertainty in their depths makes it unclear whether they are intra- or inter-plate in origin. The best-fit plane is the result of minimizing the sum of the squares H for all earthquakes (i): N H = 2 [a.(xi - E)]' - I(a.a - I), (1) I where xi is the position vector of each earthquake, Z is the average position, a is the unit vector normal to the plane and L(a a - 1) is a Lagrange multiplier or normalization factor. The earthquake locations were projected along the normal to the reference plane. In order to reduce the number of data points and also to smooth the surface, the values of distance from the plane were averaged over 25 x 25 km squares on the planar surface. A continuous, non-planar, irregular surface was then fitted through the grid points using a program taken from Swain (1976) and the minimum curvature method of Briggs (1974). This procedure produces a spline surface with continuous second derivatives. The number of earthquakes and their locations make it possible to study only the largescale features of the non-planar surface, so a Gaussian filter with a half-width of 3 km was then used to smooth the surface so that only the longer-wavelength features remain. The large-scale features (> 3 km) of the non-planar surface are not dependent upon the catalogue of earthquake locations used. We ran tests using different catalogues and subsets of the catalogues and found that only the short-wavelength details (< 3 km) are affected. The minimum curvature routine fits a surface such that curvature occurs predominantly where there are data, and so the maximum values of K occur where the earthquakes are. Figs 5(a) and 6 illustrate the view perpendicular to the average plane through the seismicity (along the z-axis) with the x-axis along strike and the y-axis down dip (Fig. 4). The smooth non-planar surface (Fig. 5a) has 'lows' top right and bottom left, and 'highs' top left and bottom right (right is the southern end of the arc, and left the northern end), with a zone of transition in between. In this projection, high regions can be thought of as hills above the average plane, and low regions as basins. The bend in the strike of the trench at 25"s (Fig. 3) and the steepening of the inclined seismic zone from a dip of -45" in the south to -56" farther north contribute to the distribution of the 'lows' and 'highs' on the average plane shown in Fig. 5(a). This shape could also be achieved by a slab that has torn along its down-dip length and forms two separate planes or flaps of subducted material. Barazangi et al. (1973), Bevis & Isacks (1984) and Yamaoka et al. ( 1986) all propose, however, that the Tonga-Kermadec seismic zone is continuous; tearing is inferred to be necessary only at the northern end where hinge faulting has been proposed (Isacks, Sykes & Oliver 1969). The lateral contortion of the downgoing slab at depth may be partly a consequence of the inward bending of the inextensible spherical lithosphere as described by Frank (1968) and Strobach (1973). For this study, however, it does not matter whether the surface is continuous or torn. If the slab is torn and if the trend of the tear is the same as the direction of the plate velocity, no deformation has to result from the tear because the material will be able to move such that 6K =. Another possibility is that the shape of the slab we obtained in Fig. 5(a) is influenced by a 'double seismic zone', where the earthquakes occur in two parallel but separate zones, often with distinctly different focal mechanisms. Kawakatsu ( 1985) describes a double seismic zone in Tonga from -17.5"s to -24.53 between depths of 6 and 2 km. McGuire & Wiens ( 1993; personal communication) extended this work and have identified an incomplete intermediate-depth double seismic zone that runs from 17"s to 28"s at 7-3 km depth. The gaps in their proposed double seismic zone occur most notably around the Louisville Ridge. If the two sides of this double seismic zone are not equally represented in the seismicity, because, for example, the upper layer is seismically active on one side of the Louisville Ridge and the bottom layer on the other side, then the bend in the fitted surface (Fig. 5a) from low to high along strike may result from fitting only the different strain regimes of the slab and not its actual shape or location. However, the suspected double seismic zone at this location (1-3 km depth) is not based on many earthquakes. This is clear from the plot of rotated CMT solutions in Fig. 5(c). The slab to the north of the Louisville Ridge (at -6 km along strike) shows large downdip 'thrust' mechanisms (in the projection normal to the slab) and only one small down-dip extension earthquake, whereas to the south of the intersection with the Louisville Ridge the slab is relatively aseismic above 3 km depth. As we will show later, even if the slab does contain a double seismic zone at these depths it is not important for this work, as we have found that the areas where an association between the shape of the slab and the seismicity can be identified are near the bottom of the slab. The Louisville Ridge is a linear aseismic ridge (Larson & Chase 1972), several thousand kilometres in length, defined by a line of shallow (4 km) bathymetry (Fig. 3; Christensen & Lay 1988). It intersects the trench obliquely, apparently affecting the bathymetry and trend of the trench, and corresponds to a region with a low level of shallow seismicity compared with elsewhere along the arc (Kelleher & McCann 1977). The influence of the subducting ridge on the shape of the slab must also be considered. If the ridge is subducted along the trend suggested by Larson & Chase (1972) (an extrapolation of its current trend), it would have been subducted obliquely and its intersection with the arc would sweep rapidly southwards. It is then possible that the Pliocene-Recent opening of the Lau Basin may be related to the subduction of the Louisville Ridge (Isacks & Barazangi 1977), and perhaps to the flattening of the inclined seismic zone below 3 km at the northern end of the Tonga arc. However, Giardini & Woodhouse (1986) have proposed instead that a zone of low seismicity at depth 1996 RAS, GJI 127, 311-327 on 21 January 218

316 S. Nothard et al. (23"S, 179.5"W, Fig.3) corresponds to the position of the subducted Louisville Ridge. If they are correct, the bottom of the slab must have sheared southwards relative to the trench by some 5 km (Giardini & Woodhouse 1986). The strip of low seismicity may then coincide with the trace of the subducted ridge and with the change in shape of the surface (Fig. 5a) along strike, from low to high at the top, and from high to low near the base. 3.1.I Accuracy of the earthquake locations The earthquake locations used to determine the average plane and the slab surface are from the ISC and the CMT catalogues spanning the time period from 1964 to 1994, and no relocations were performed to improve the definition of the average plane through the seismicity as we believe this would not have significantly improved the fit to the reference plane. For example, the change in the locations of Tonga earthquakes that were relocated in a test using a hypocentroid decomposition algorithm (Jordan & Sverdrup 1981) were generally very much smaller (- 5-1 km) than the misfit (27 km, see below) between the average plane and the earthquakes (Nothard 1995). Moreover, the irregular slab surface was fitted to the average, binned, hypocentre locations, and the averaging will have reduced the influence of errors from mislocations. Therefore, performing relocations for all the earthquakes used in this study would not have significantly improved the fit of the earthquakes to the irregular surface. 3.1.2 Accuracy of the surface The Wadati-Benioff zone is located within the subducting oceanic lithosphere and is assumed to represent the overall shape of the subducting slab. The shape of the irregular surface shown in Fig. 5(a) follows approximately the middle of the seismogenic part of the Wadati-Benioff zone. North (22.28) South (27.83) 1 2 3 D m s z ' 4 "s 5 h X 3 Y 6 2 3 4 5 6 7 8 9 1 Distance along Section (km) Figure 5. (a) Contour plot of the slab surface. Contours are at 5 km intervals above or below the average plane. Regions above the slab are indicated by solid contours and regions below by dashed contours. Dots are the CMT and ISC earthquakes that were used to define the surface. (b) An oblique section through the surface and the seismicity along the solid line in Fig. 5(a) to illustrate the fit of the smoothed (black line) and unsmoothed (grey line) surfaces to the seismicity projected along a normal onto the section. Earthquakes within 5 km either side of the line are projected onto the section. Open circles are earthquakes projected from the right of the line and solid circles earthquakes projected from the left. (c) Harvard CMT solutions (1977-1994) projected on to the average plane through the seismicity. These are back-hemisphere projections and the black quadrants are compressional. The area of the fault plane solution is proportional to the square root of the earthquake seismic moment. (d) The CMT solutions in (c), but all plotted at the same size. on 21 January 218 1996 RAS, GJI 127, 311-327

N A b E! on 21 January 218 Up-dip Distance (km) Up-dip Distance (km) w 3 ( O O N w Ul rn o S 8 8 8 8 8 v @ @ 8 - - m e Q 6 8 d @@ 'N - - ' -.o 'P - - -Ul 3 'Q, :8 --I - - -A ru P. :....... cn Q, 4 a, *- 't Q, Ul P w N 4 Q, P.

318 S. Nothard et al. North South 1 2 3 4 5 6 7 8 7 p 6 Y Y al 5.- 3 x 4 3 3 E # B 2 1 4--v-T- 1 2 3 4 5 6 7 Along-strlke Distance (km) 3 25.n 2 U A 3 15 North South 1 45 5 55 6 65 7 Along-strike Distance (km) Figure 6. (a) Contour plot of the Gaussian curvature (K) of the slab surface. Solid contours are positive (resulting from a 'hill' or 'basin') and dashed are negative (resulting from a 'saddle'). The arrow represents the direction of Pacific plate motion rotated into this coordinate system to indicate the approximate probable direction of flow of the slab material. The contour interval is 2 x km-2. (b) A close-up of region A, shown in (a). Earthquake mechanisms are backhemisphere projections onto the average plane through the seismicity as in Fig. 5(d) and are all plotted equal in size. We can now compare the fit of the earthquake locations to the best-fit plane, the unsmoothed surface and the smoothed surface. We define a standard deviation, B, such that where ei is the height of the surface above the average plane and xi is the distance of the ith earthquake from the average plane, calculated for all n earthquakes. The surface is defined by m model parameters. Surface type (1) best-fitting plane (2) unsmoothed surface (3) smoothed surface Standard deviation (r (km) 27.15 9.49 19.98 The unsmoothed surface has the lowest value of misfit (B), which is effectively the same as the standard deviations of the values for the individual earthquakes about the local 'binned' average value. Both surfaces are better approximations to the shape of the slab than the average plane. The amplitude of the smoothed surface may appear to underestimate the magnitude of the height of the earthquakes above the average plane. This is the result of a trade-off between fitting the data exactly and smoothing the surface to obtain only those longer-wavelength features that can be considered real. In globally minimizing the curvature, while locally fitting the data, the unsmoothed surface has undulations that are not totally supported by the seismicity, whereas the smoothed surface may underestimate the curvature. We regard the smoothed surface as a more realistic approximation to the actual shape of the slab than the unsmoothed surface. This is a subjective judgement, with a trade-off between the presumed thickness of the Wadati-Benioff zone and its presumed shape: the surface becomes more contorted as the thickness of the zone is decreased. The solution to the problem of whether the spatial distribution of earthquakes is best described by a thin contorted zone or by a thicker uncontorted one (Isacks & Barazangi 1977; Bevis & Isacks 1984) would be aided by more (or better-located) earthquakes. The number of earthquakes available and the accuracy of their locations are not sufficient to infer small contortions or tears within the surface or to warrant using a highly contorted surface. Hence we prefer to use the smoothed surface. In fact, the surface has probably been over-smoothed, as it is important to be sure that the features in the contour plot of Gaussian curvature (Fig. 6a) are required by the data, rather than simply allowed by it. The two principal curvatures can now be determined for each point on the smoothed surface. The value of K = 1/RlR2 is contoured in Fig. 6. 3.1.3 Earthquake mechanisms Focal mechanisms of earthquakes from the Harvard CMT catalogue (nearly all of them with M, > 1'' Nm or M, > 5.3) are projected onto the best-fit plane through the seismicity in Figs 5(c) and (d). The individual elements of the earthquake moment tensor are rotated into the same coordinate frame as the average plane, that is, with the x-axis in an along-strike direction, and the y-axis in the dip direction. The best doublecouple solution is viewed as a back-hemisphere projection onto the plane and Fig. 5(c) is equivalent to a map view looking down the normal to the slab. (It is irrelevant to this study whether the best double-couple or the complete moment tensor solution is used; it is the relationship between the style of deformation and the shape that is of interest.) Isacks & Molnar (1969) proposed a correlation between the depth of a subducted slab and the distribution of strain axes in focal 1996 RAS, GJI 127, 311-327 on 21 January 218

Deformation of the Tonga slab 319 mechanisms. They suggested that a slab reaching the 67 km discontinuity, such as Tonga (Giardini 1984), is shortening down its entire length as it encounters resistance at the density contrast between the upper and lower mantle. Hence the focal mechanisms projected onto the surface of the plane should show predominantly down-dip compression (looking like thrust faults in this projection, Fig. 4). Fig. 5(c) shows that this is true for the largest of the earthquakes. However, earthquakes with smaller seismic moments show a wider variety of mechanisms in this projection (Fig. 5d): down-dip compression ( thrust.faults); along-strike extension ( normal faults); and strike-slip motion. We now suggest an explanation for some of these mechanisms. 3.1.4 Gaussian curvature of the surface If the Gaussian curvature (K) is constant along the path between two points then the slab can move over the uneven surface between them without a change in its surface area. Fig. 6(a) shows a.contour plot of K, viewed normal to the slab surface. In regions of constant K, the slab may still deform and the surface can still be non-planar (see Fig. 5a), but the material does not need to be stretched or compressed within the plane of the surface as it moves. The regions of constant K coincide with the largest intermediate and deep earthquakes [compare Figs 5(c) and 6(a)], the majority of which show down-dip compression ( thrust ) mechanisms. We therefore conclude that these large earthquakes are unlikely to result from the deformation of the slab as it moves over an irregular surface or template. The regions of positive K in Fig. 6(a) correspond to hills or basins on the slab surface. The region marked A in Fig. 6(a) is shown in close-up in Fig. 6(b). If the slab material were to move over a template of this shape in the direction of the plate velocity (the white arrow, relative to the overriding plate), the change in Gaussian curvature at A would require a relative increase in surface area in that region. This is a possible explanation for the change from compressional ( thrust ) to extensional ( normal ) mechanisms at the centre of the region of positive K. Although the extensional mechanisms in region A have small seismic moments (Fig. 5c), they do coincide spatially with the positive region on the plot of K (Fig. 6b). The extensional mechanisms have P axes that are approximately normal to the slab surface but the T axes change in orientation around the dome-like structure. Thus one interpretation of this deformation pattern is that the slab material is stretching to accommodate the change in shape, in the same way as a carpet moving over a lump in the floor stretches on top of the lump to remain in contact with it (Fig. lc). Schneider & Sacks (1987) observed a similar relationship between shape and seismicity in the contorted region near the base of the subducting Nazca plate below 18 km depth. They noticed that the orientations of the T axes of composite focal mechanisms tend to point towards the contortion and also become aligned in an along-strike direction where the plate flattens out at depth (similar to the orientation of the mechanisms in Fig. 6b). In other areas of the slab that exhibit a non-zero Gaussian curvature (Fig. 6a) there are insufficient earthquakes to evaluate any association between orientation of the mechanism and shape of the slab. 3.2 Dealing with the hook : a polynomial surface Having explored the Gaussian curvature on the structurally simple region of the Tonga subduction zone, we now look at the more complex region outlined by the solid box in Fig. 3. The shape of the slab in this region is defined using the hypocentre locations of more than 18 earthquakes published in the ISC catalogue (1964 to 199). In order to minimize the number of poorly located earthquakes that would result in larger uncertainties in the definition of the slab, we used only those earthquakes whose P-wave arrivals had been reported by more than 5 stations (see e.g. Cahill & Isacks 1992). Once again, earthquake locations are averaged into bins and the resultant surface is smoothed across neighbouring bins to preserve only the long-wavelength features of the surface. Earthquakes that occurred shallower than 1 km depth are not included in the surface fitting procedure. Previously, we defined the shape of the slab using a minimum curvature surface about an average or best-fit plane through the seismicity. At the northern end of the Tonga subduction zone the geometry of the slab is more complex. The slab bends sharply to the west below 5 km depth and the curvature of the hook is sufficiently large that mapping the surface onto the same plane as the rest of the slab would cause singularities and repetition of surface values for any one (x,y) plane coordinate. Also, it is meaningless to project the focal mechanisms of the earthquakes within this region onto the average plane when the geometry of the slab is almost perpendicular to it. We need to define the slab geometry by a continuous surface onto which the mechanisms can be projected along the local normal of the surface. We choose a coordinate system and reference frame for the surface such that one of the coordinates is constant for a given depth. Initially, the shape of the slab is described using a polynomial function in (x, y), where x and y are the horizontal coordinates of the earthquake in a Mercator projection (which are conformally related to the points on the Earth s surface). The x- and y-coordinates are mapped conformally to coordinates ([, q), where [ is the coordinate in the direction of the horizontal projection of the down-dip lines and q is the orthogonal along-strike coordinate. The coordinate system is described in Fig. 7. The depth of the surface is a function of the horizontal down-dip coordinate ([) and is independent of the along-strike coordinate (q). Conversely, the horizontal down-dip coordinate (c) is a function of the depth, and is constrained by the depth of each earthquake. The surface is defined up to the trench, unlike in the previous section where the plane was only considered below 1 km depth. The trench is the curve [(x, y) =, which constrains one edge of the surface and is independent of the earthquakes. Points along the trench are weighted more highly than the earthquake locations in the fitting procedure for the initial conformal mapping in order to force the surface to match the line of the trench accurately. The polynomial surface is then used as the average surface (equivalent to the plane used previously) about which the shape of the smaller-wavelength features of the slab is described by a bi-cubic spline function. The details of the derivation of the surface are discussed in Nothard (1995) and in Appendix A. Both the standard error (CT) of the misfit between earthquake locations and the polynomial surface and the appearance of the horizontal projection of the surface were used to determine 1996 RAS, GJI 127, 311-327 on 21 January 218

32 S. Nothard et al. Horizontal Projection -.. trench lines of constant 5 down-dip lines with q constant 6 r;c Z) Figure 7. Cartoons to illustrate the coordinate system used to define the larger region of the slab surface. The picture on the left shows the horizontal projection of the along-strike (q) and down-dip (c) coordinates and on the right is a cross-section through the slab showing the depth dependence of the [ coordinate. the order of polynomial that generated the 'best-fitting' surface. The 8th-order polynomial was chosen as the 'best-fitting' one and the horizontal projection of this surface is shown by the grid in Fig. 8(a). The standard error (a) falls from 147 km for a zero-order surface (equivalent to a plane) to 61 km for a 1storder surface, and then to 43 km for an 8th-order surface. The standard error is still 43 km for a 14th-order surface. The coefficients of the lower-order terms are much more significant than the higher-order ones and the fit is not improved by using a higher-order polynomial: 5th-order polynomials were the highest order we considered. The grid is fairly regular, indicating that the 8th-order polynomial surface is quite planar and is not able to match the sharp westward curve of the deep seismicity, as there are insufficient degrees of freedom to fit the sharp curvature of the hook. After the initial surface was fitted to the earthquakes and to the trench we then discarded 23 of the total of 1832 events (- 1 per cent) from the fitting procedure because they were located far away (- 1 km) from the surrounding earthquakes. These anomalous earthquakes are discussed in the following section. We then used the polynomial surface as the average surface (equivalent to the reference plane used earlier) about which a spline surface (also described earlier) was fitted through the seismicity to obtain the adjusted surface. The horizontal projection of the adjusted surface defined by both the spline and polynomial functions is shown in Fig. 8(b). The along-strike and down-dip lines are no longer orthogonal because the down-dip coordinate [l(x, y)] is altered, while keeping the along-strike coordinate [~(x, y)] constant, with the result that the down-dip lines along which q is constant are the same in Fig. 8( b) as in Fig. 8(a), whereas the along-strike lines, along which depth is constant, are changed (see Appendix A). At shallow depths the adjusted surface in Fig.8(b) has a near-constant dip and a simple shape along strike, which is consistent with the simple geometry identified by Isacks & on 21 January 218 Barazangi (1977). The curvature of the 'hook' at the northern end of the slab has now been matched by the shape of the grid, as has the bench-like region around 16"S, 177"W where the lines of constant depth become further apart as the gradient of the slab decreases. In the central part of the region (- 2"s) the slab has a steeper dip near the bottom than further to the <fl south (-25"S), where the dip of the Wadati-Benioff zone appears to be constant for all depths (-5"). The steep dip trench shown by the net in Fig. 8(b) at -2"s below 4 km still does not completely match the complex shape defined by the seismicity in this region (as we discuss below). 3.2.1 Cross-sections through the adjusted surface depth ( z ) Fig. 9 shows cross-sections through the Wadati-Benioff zone. Each section is numbered according to the down-dip line along which it is drawn: line 1 is the northernmost down-dip line on Fig. 8( b). The sections are therefore not necessarily perpendicular to the trench in a Mercator projection. The earthquakes within the grid boxes bordering the section line are included in the cross-section and therefore the total width of seismicity included in each section is two boxes, or approximately 5 km. The solid line is a section through the adjusted surface and the dashed line is through the original polynomial surface. The main geometrical features of the Tonga slab can be identified in these cross-sections. For example, section 6 crosses the bench-like feature at the northern end of the Tonga arc where the dip of the slab decreases between about 3 and 4 km depth (Fig. 8b). The constant-depth lines for the deeper seismicity are bent sharply to the west making the 'hook', but the trend of the shallower seismicity continues to strike approximately north-south. The adjusted surface (solid line) is clearly a better fit to the seismicity than the polynomial reference surface (dashed line). Cross-section 14 is farther south and cuts through the most complex part of the deep Wadati-Benioff zone. The seismicity forms an 'S' shape below 4 km depth as the section steepens and appears to overturn. The adjusted surface was unable to match an overturning or vertically dipping slab because of the general problem of repeating points in the coordinate system (see Appendix A). Section 18 shows the slab beneath the Tonga islands and crosses through the location of the double seismic zone at - 1-3 km depth identified by Kawakatsu (1985). The adjusted surface is not well located at shallow depths because the seismicity splits into two groups (as is seen near 1 km depth). This section contains four of the earthquakes (open circles, Fig. 9) that were discarded from the data set at shallow depths (1-15 km). They occurred consistently deeper than and trenchwards of the main trace of the slab and probably form the lower half of a double seismic zone. If they had been included in the fitting procedure the adjusted surface would have left the trench at a very steep angle and then risen again or flattened to the horizontal in order to fit them. From section 22 southwards the dip of the Wadati-Benioff zone becomes gradually shallower and demonstrates little structure. These southern sections cross the region approximated by a plane earlier. Other earthquakes that were discarded in the fitting procedure are identifiable as open circles in the cross-sections shown in Fig. 9. Six of them occurred at approximately 1 km depth to the west of the main trace of the slab (e.g. section 26, 1996 RAS, GJI 127, 311-327

\c4 36 Figure 8. (a) Horizontal projection of the 8th-order polynomial surface. The grid lines oriented along strike are lines of constant depth on the surface. The down-dip lines are orthogonal to them. The grid lines are fairly regularly spaced implying that the surface is still quite planar and it is unable to match the shape of the hook. (b) Horizontal projection of the adjusted surface. The adjusted surface is formed by continuous spline functions describing the perturbations about the polynomial surface required by the seismicity. The along-strike lines (in the q direction) are constant-depth contours on the surface. The numbers along the trench correspond to the numbers of the down-dip lines. 25. on 21 January 218-1 5' -2" - 175' 18" 185' 19' w - - --I- (a) 8th Order 15'!O' -1 5' -2' 175' D 18' - - - - - 185' 19' I- - (b) Adjusted -15' 2' -25O!5' -25' 175' 18' 185' 19' 175' 18' 185' 19'

322 S. Nothard et al. 7 6 5 4 3 2 1 7 6 5 4 3 2 1 1 2 3 4 5,- Section 6 I ' I ' I 6 ' I, I,, I,7 16 6 i am Section 1 I 1 I I ' I I, I, I, 17 7 6 5 4 3 2 1 7 6 5 4 3 2 1 1 2 3 Section 18 6 7 7 6 5 4 3 2 1 7 6 5 4 3 2 1 1 2 3-4 ], ),,,,,,,,,. Section 22-5 - 6 7 Figure 9. Cross-sections through the adjusted grid (Fig. 8b). Axes are both in kilometres. The section number refers to the dip-line along which it is drawn. Filled circles mark the location of earthquakes used in fitting the surface. Open circles are ISC earthquakes that were not used but still have the number of reporting stations 2 5; this group includes the 23 discarded events. Fig. 9) and probably within the overriding plate. These earthquakes may be mislocated, although some were located using 22 stations. Six other discarded events occurred in a deep group more than 1 km from the main trace of the slab (not shown in Fig. 9). These events may have occurred in a continuous slab that is lying flat on top of the 67 km discontinuity in this region (van der Hilst 1995). In general the shape of the adjusted surface is a good approximation to the shape of the seismicity seen in the crosssections. However, because of uncertainties in the depths of 1996 RAS, GJI 127, 311-327 on 21 January 218

Deformation of the Tonga slab 323 shallow earthquakes and the fact that the surface is unable to match double seismic zones, we only consider features below about 2 km, where the fit is generally good, except for the S-shaped region below 4 km depth at - 19"s (section 14, Fig. 9). 3.2.2 Principal directions of curvature For each point on the adjusted surface the orientations and magnitudes of the two principal curvatures can be found, and their horizontal projections are shown in Fig. 1. The thick lines represent positive curvature and the thin lines negative curvature. The length of each line is proportional to the magnitude of the curvature: the greater the curvature, or the smaller the radius of curvature, the longer the line. The coordinate system chosen for the conformal mapping of the slab means that the curvature can be calculated right up to the trench itself. The principal curvatures along strike are small at the trench (the curvature is that of the Earth) but are high in the down-dip direction as the slab bends beneath the overriding plate. The base of the slab forms undulations along strike with the curvature changing from positive to negative to positive again. The along-strike curvature increases with depth, especially in the region of negative curvature near the hook (Fig. 1). 3.2.3 Earthquake mechanisms The 637 Harvard CMT solutions of earthquakes within the region of Fig. 1 (between January 1977 and February 1994) are projected along the local normal onto the adjusted surface calculated above. Fig. 1 l(a) shows the rotated mechanisms plotted in the (5,~) coordinate system with the area of the focal sphere proportional to the square root of the seismic moment of the earthquake. The southern part of this figure can be directly compared with Fig. 5(c), which shows the earthquake mechanisms rotated onto the average plane through the seismicity. The majority of the largest events, both in the southern and northern parts of Fig. ll(a), are dip-slip mechanisms looking like 'normal' or 'thrust' faults in this projection (Fig. lla). The smaller earthquakes show a wider variety of mechanisms. 175" 18" 185" 19" -1 5" -1 5" -2' -2" -25" -25" 175" 18" 185' 19" Figure 1. Horizontal projection of the directions of principal curvature. Thin lines are negative curvature and the thick lines are positive curvature. The length of each line is proportional to the magnitude of the curvature. 1996 RAS, GJI 127, 311-327 on 21 January 218

South (-28"s) 1 2 3 4 5 6 7 on 21 January 218 North (-13"s) 1 2 e t Y 3 W Along-Strike Direction w N lh c, Q 4 5 6 7 a m,m Q 2 "5 W + W I N -4 i Figure 11. (a) Harvard CMT solutions rotated and projected onto the slab surface, showing also the Gaussian curvature contoured at intervals of 2 x krk2. The area of each mechanism is proportional to the square root of its seismic moment. The region outlined by the solid box is shown in (b), and that outlined by the dashes in (d). (b) The extensional mechanisms in region R are discussed in the text. Mechanisms are equal-sized back-hemisphere projections of the CMT solutions. (c) Section along dip-line 29 showing the group of deep earthquakes (R) that result in extension of the surface of the slab in (b). The arrows indicate the direction of shortening across the slab. (d) Close-up of the region outlined by the dashed box in (a); focal mechanisms vary and no obvious association between Gaussian curvature and deformation style is evident.

Deformation of the Tonga slab 325 5 h E 25 55 Q E X 2 R 2 6 North Along-Strike Direction South I - 5 km \ West Distance (km) East 7 6 5 4 4 s 5 (D 3- n 6 w 7 5 55 6 North Along-Strike Direction South 3 3 h E 4 Y 5 Q " X 2 8 5 a 6 Figure 12. (Continued.) 3.2.4 Gaussian curvature 4 5 6 Fig. ll(a) also shows a contour plot of the Gaussian curvature of the slab. The largest earthquakes have nearly pure dip-slip mechanisms and, as we showed earlier, do not occur within regions of high Gaussian curvature. At the southern end of Fig. 11 (a) the largest earthquakes show down-dip shortening ('thrust' mechanisms), as they did in Fig. 5(c). However, at the northern end there are shallow mechanisms exhibiting both down-dip shortening and down-dip extension in a double seismic zone (Kawakatsu 1986a) at about 2km depth. Towards the bottom of the slab the earthquake mechanisms in the northern region in particular show a strike-slip component, possibly arising from a shear of the bottom of the slab with respect to the trench, as postulated by Giardini & Woodhouse (1986). The earthquakes with smaller seismic moments are of more interest as they show a wider variety of mechanisms and might relate to deformation of the slab as it moves over a template defined by the seismicity. The main region where there seems to be an association between the seismicity and the shape of the slab is shown by the box outlined by a solid line in Fig. ll(a). This region is enlarged in Fig. 11 (b) and is approximately the same area as Fig. 6(b). It is encouraging that Figs 6( b) and 11( b) both show an area of positive Gaussian curvature associated with extensional mechanisms, even though the two surfaces from which the Gaussian curvatures were calculated were obtained by different methods. The curvature of the slab in both cases may be underestimated. If the slab follows a template as it moves, then in this region of positive Gaussian curvature it must stretch. Such an interpretation is consistent with the mechanisms of the smaller earthquakes in this particular region (Fig. lla), as we concluded earlier. An alternative interpretation of these focal mechanisms was offered by Lundgren & Giardini (1994). To illustrate this we show the earthquakes in region R of Fig. ll(b) as a crosssection in Fig. 11 (c). This cross-section is along down-dip line 29 (Fig. 8b) and includes mechanisms from one grid box on either side of the dip line. The mechanisms have been projected onto the plane of the cross-section along a local normal (Fig.4). The group R earthquakes look like 'thrust' mechanisms in this orientation, and the majority show shortening in a direction (indicated by the arrows) that would cause thinning of the slab and extension along strike (see also Fig. 1 lb). The P axes of the extensional mechanisms in Fig. 11 (b) are approximately normal to the slab surface. The majority of the earthquakes in the group to the east of group R (Fig. llc) have mechanisms with a down-dip P axis showing down-dip shortening of the slab along subvertical fault planes identified by Giardini & Woodhouse (1984) and Nothard (1995). Lundgren & Giardini (1994) concluded that as the slab deflects towards the horizontal it no longer acts as a stress guide (because then the P axes would remain in the plane of the slab) but shortens in a nearly normal direction (indicated by the arrows in Fig. llc) as a result of the gravitational forces associated with large volumes of material in non-hydrostatic conditions. Alternatively, as we point out above, the association of these events with a region of positive curvature suggests that they could also be related to the shape of the subducting slab itself. The other region in Fig. ll(a) where small earthquakes with a variety of mechanisms are associated with a change in Gaussian curvature is shown in detail in Fig. ll(d). However, there are insufficient numbers of such earthquakes to see a clear pattern from which to draw any firm conclusions. 1996 RAS, GJI 127, 311-327 on 21 January 218

326 S. Nothard et al. 4 CONCLUSIONS There are various reasons why a lithospheric slab might deform as it is subducted into the mantle. We have searched for one particular potential cause of slab deformation, which is the deformation required if a sheet of material is to move over an irregularly shaped template or surface while remaining in contact with it. In the case of a subducting slab, that template may be provided by, for instance, the pattern of flow in the asthenosphere. Deformation arising from this cause should have a predictable relationship, both spatially and in orientation, with the shape of the template or surface. We have searched for such a relation in the Tonga slab by using the earthquake locations to define the shape of the slab, and both their locations and focal mechanisms to indicate the position and orientation of the deformation within it. One important conclusion is that most of the intermediate and deep earthquakes in the Tonga slab are not obviously associated with the shape of the slab in the manner predicted above: they occur in regions where the slab could move over a template defined by its shape without requiring any internal deformation. Although this conclusion is limited by our ability to resolve details in the slab's shape (we were only able to resolve the coarsest features; and cannot exclude shape changes with wavelengths less than - 3 km), it suggests that a different explanation must be sought for most of the intermediate and deep earthquakes, including all the largest ones. However, we found one area, at about 25.5"s and 5-6 km depth, where there is a plausible association between the shape of the slab and the change in earthquake focal mechanisms, such that this deformation could be related to the contortion of the slab at the bottom of the deep seismic zone. The earthquakes concerned are relatively small, but show a consistent spatial pattern of mechanisms in accord with the slab moving over a template of its own shape-though this may not be the only explanation. There are other regions of the Tonga slab, notably in the 'hook' region to the north where it forms a sharp bend, where the shape of the slab would require internal deformation if it were moving over a template. In these regions the smaller earthquakes have a variety of focal mechanisms, but without the clear pattern seen at 25.5"s. We conclude that, with the exception of the deep part of the seismic zone at 25.5"S, most of the earthquakes in the Tonga slab are not obviously the result of the slab's moving over a template: some other explanation for them must be sought. ACKNOWLEDGMENTS SN and JJ thank the Institute of Geological and Nuclear Sciences and the British Council for support in New Zealand. This work was partially funded by NERC (Grant GR9/175). 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