Caroline Dumoulin

David Bercovici

Pål Wessel

A continuous plate-tectonic model using geophysical data to estimate plate margin widths, with a seismicity based example

Keywords: Plate-tectonics, plate boundaries, intraplate deformation, toroidal-poloidal partitioning

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A continuous plate-tectonic model using geophysical data to estimate plate margin widths, with a seismicity based example

Introduction

The theories of mantle dynamics and plate tectonics are fundamentally incompatible; the former employs continuum physics while the latter assumes that plates move as rigid bodies and thereby have infinitesimally thin margins. This disparity hampers efforts to couple plate and mantle theories, causing two problems in particular. First, the discontinuity between plates causes plate traction on the mantle to be infinite [START_REF] Hager | A simple global model of plate dynamics and mantle convection[END_REF]. Second, the same discontinuity renders the vorticity (strikeslip shear) and divergence (which represents sources and sinks of mass in the lithosphere) fields to be comprised of unphysical and mathematically intractable singularites [START_REF] Bercovici | A continuous kinematic model of plate-tectonic motions[END_REF]Bercovici, 1995a]. The Earth's plates are for the most part not discontinuous; i.e., intraplate deformation is common and many plate margins have significant widths. Thus, efforts to understand how plates are coupled to or generated from mantle flow should not seek to achieve discontinuous plates; such plates are abstractions based on the simple assumptions of the standard plate model, and are therefore unrealistic as well as unrealizable. However, the actual continuity of plates needs to be assessed, and then incorporated into a global plate model; the resulting model would be more compatible with mantle flow models, and represent a more realistic goal for plate generation theories.

To this end, we here extend the continuous kinematic plate model of [START_REF] Bercovici | A continuous kinematic model of plate-tectonic motions[END_REF], which introduced finite margin widths into a global model of plate motions. In particular, we use analytically continuous (i.e., infinitely differentiable) functions to describe both plate geometry and plate-margin width. In this paper, the mathematical model of the plate geometries is considerably refined to prevent the excessive smoothing of plate boundaries used by [START_REF] Bercovici | A continuous kinematic model of plate-tectonic motions[END_REF]. Moreover, we demonstrate how geophysical data on intraplate deformation can be incorporated into the model to constrain plate margin widths; for purposes of demonstration, we use, as an example data set, the global seismicity distribution. This leads to a marked improvement over the ad hoc constant margin width assumed in the standard plate model, or in the simple model of [START_REF] Bercovici | A continuous kinematic model of plate-tectonic motions[END_REF]. We refer to this seismicity based application of the model as the SEISMAR (seismicity-inferred margin-width) plate model. The SEISMAR model of continuous plate motions introduced here is a basic example of how data on intraplate deformation can be incorporated into a global plate-tectonic model.

In this paper, we will first present the theoretical model for representing plate shapes and, in particular, develop the necessary improvements to facilitate more realistic plate boundaries. We will then discuss how margin width is determined, for example, from global seismicity. Finally, we will examine the implications of the resulting seismicity-based SEISMAR model on global velocity, in particular the associated deformation fields (vorticity and divergence) and kinetic energy partitioning between toroidal and poloidal parts.

Plate shapes and boundaries

The analysis of how to represent the shape of a tectonic plate with an analytically continuous function is discussed in detail in [START_REF] Bercovici | A continuous kinematic model of plate-tectonic motions[END_REF]. We will here review the more salient points of the theory and explain in detail when we have differed from or improved on that earlier work.

The horizontal velocity field of the Earth's surface, given the motion of tectonic plates can be written in a single equation:

¡ £¢ ¥¤ §¦ ©¨ § "! $# % '& )( (1)
where ¦ is the position vector of a point on the surface of the Earth, at longitude # and latitude & , and 0 and are the angular velocity vector and shape function of the 1 32 54 plate, respectively. The angular velocities of the presentday plates is determined from the NUVEL-1A Pacificplate-fixed Euler poles [START_REF] Demets | Effect of recent revisions to the geomagnetic reversal time scale on estimates of current plate motions[END_REF]] added to estimates of the instantaneous Pacific-hotspot pole from [START_REF] Pollitz | Episodic North American and Pacific plate motions[END_REF] joint inversion of North American and Pacific plate motions. The shape function is defined to be 1 within the plate, and 0 outside the plate. For the standard plate model, has a discontinuous transition from 1 to 0, whereas the transition is continuous and smooth (i.e., infinitely differentiable) in the continuous model. The procedure for determining in the continuous model requires several steps as discussed below [see also [START_REF] Bercovici | A continuous kinematic model of plate-tectonic motions[END_REF].

Plate boundary filtering.

To make the model of the tectonic plates analytically continuous, the plate boundaries must be smoothed at least slightly to remove minor discontinuities (e.g., ridgetransform offsets). This is necessary to allow the boundary to be modelled as a mathematically single valued function (see next subsection below). In [START_REF] Bercovici | A continuous kinematic model of plate-tectonic motions[END_REF], smoothing was performed until each plate boundary could be represented as a single-valued polar function in which the polar origin was at some point inside the plate. To that end, a very large filter width (7000km Gaussian full-width; see [START_REF] Bercovici | A continuous kinematic model of plate-tectonic motions[END_REF]) was necessary so that the most non-circular plates (e.g., nearly horse-shoe shaped ones such as the Austro-Indian plate) could be sufficiently rounded out. Invariably, each resulting smoothed plates became a crude first order representation of the original plate, and smaller plates, such as the Cocos plate (with its lopsided hour-glass shape), became almost entirely circularized. However, to employ data on inptraplate deformation (such as seismicity) to estimate margin widths, it is necessary that the zones of high deformation (e.g., dense seismicity) coincide accurately with the plate boundaries. Therefore, to preserve plate boundary structures we use a a much finer filter width (1000km Gaussian full-width); the original and smoothed (filtered) plate shapes are shown in Figure 1. To represent the plate boundaries with single valued functions therefore requires additional manipulation, as discussed below.

Original Plate Boundaries

Smoothed Plate Boundaries

Plate division

We begin with the 13 major tectonic plates of the Earth as listed in [START_REF] Demets | Current plate motions[END_REF][START_REF] Demets | Effect of recent revisions to the geomagnetic reversal time scale on estimates of current plate motions[END_REF]. The Australian and Indian plates are divided along their diffuse margin as indicated by [START_REF] Wiens | A diffuse plate boundary model for Indian ocean tectonics[END_REF] (Figures 1 and2). The actual dividing line is a linear function of longitude

#

and latitude & , i.e., a straight line in a cylindrical equirectangular projection (in which longitude and latitude are essentially treated as Cartesian coordinates); the end points to this line are

# ¢ ¢¡ ¤£ ¦¥ % '& ¢ ¤ ¤£ § and # ¨¢ © ¢ £ ¦¥ % '& ¨¢ £ .
With the finer smoothing employed in this paper, it is not possible for each plate boundary to be automatically represented with a single-valued function. Thus it is necessary to further subdivide several plates along arbitrary pseudoboundaries. These pseudoboundaries are chosen to maximize the circularity of the resulting subplates; for example, a horshoe shaped plate would be divided across its center curve, while an hour-glass shaped plate would be divided along its neck. In some cases, it is necessary to divide particularly large plates into 3 and 4 subplates. To ensure that the pseudoboundaries do not significantly influence the final model, it is necessary that a subplate and its complement (i.e., the adjacent subplate sharing the common pseudoboundary) fit precisely back together. This is discussed in more detail in Section 3.3.

Plate divisions near the equator are made by pseudoboundaries that are simply straight lines in an equirectangular projection (i.e., & is a linear function of

#

). However (as discussed below) the plate boundary function for a given plate is found in a frame of reference for which that plate does not contain a geographic pole and is far from the zero meridian (to avoid discontinuous changes in longitude); i.e., each subplate is moved to a frame such that its effective center of mass is near the equator and the ¡ meridian [see [START_REF] Bercovici | A continuous kinematic model of plate-tectonic motions[END_REF].

However, pseudoboundaries located at high latitudes can undergo significant distortion when being rotated to the equator; thus to preserve the simplicity of these pseudoboundaries on rotation, they are determined by great circles with specified endpoints. Information on all pseudoboundaries is given in

where " and % in this case are defined such that the pseudoboundary is the intersection of the surface of the Earth with the plane B ¢ " DC $ E% GF (where

! C % F % B ( ¢ H ¦ H ! 5 87 9 & 5 87 9 # % 5 87 9 & 9 G@ A2
# % 9 G@ A2 & )( ) that passes through the center of the Earth and the endpoints listed in Table 1. All divided plates are shown individually in Figure 2; the subplates are also labelled to facilitate further discussion.

The boundary function

As stated above and in [START_REF] Bercovici | A continuous kinematic model of plate-tectonic motions[END_REF], each plate or subplate is rotated to a frame of reference where its center is at a longitude of ¡ I and latitude of I .

(The center of the plate is defined as its effective center of mass wherein equal masses are placed at evenly

! $# ¥¢ % '& ¢ ( is ¦ ! $# ¢ % '& ¢ ( ¢ ¨ § ! $# ¢ ¤ # ¢ ( ¨$ ! & ¢ ¤ & ¢ ( ¨£ (4)
The pseudo-angle between the line connecting these two points and the line connecting

! $# ¢ % '& ¢ ( to the first plate boundary data point ! $# ¢ % '& ¢ ( is ¡ ¢ )( 10 ¢2 3 ¢ & ¢ ¤ & ¢ # ¢ ¤ # ¢ ¤£ ¤ ( 10 ¢2 3 ¦¥ & ¢ § ¤ & ¢ # ¢ ¤ # ¢ ©¨( 5)
(The terms 'pseudo-distance' and 'pseudo-angle' are used because these quantities treat & ¢ and # ¢ as if they were Cartesian coordinates; see [START_REF] Bercovici | A continuous kinematic model of plate-tectonic motions[END_REF].) An arbitrary boundary point

! $# ¢ % '& ¢ ( has pseudo-distance ¦ ¢ ¦ ! $# ¢ % '& ¢ ( .
¦ ! ¡ ( ¢ 3 ¨¦ "! $# (6)
and is the number of resampled points. In this way, ¦ ! ¡ ( is expressed as an analytic function.

Plate shape and plate margin functions

Given the plate boundary function ¦ ! ¡ ( , the plate shape function of any given plate is

! $# ¢ % '& ¢ ( ¢ &% $ ( 10 ¢2 (' ¢ ¦ ! ¡ ( ¤ ¦ ) 10 ! ¡ ( £ 32 (7)
where ) 0

is the plate margin half-width (in the pseudocoordinates employed here) along the line connecting the points

! & ¢ % # ¢ ( and ! & ¢ % # ¢ ( ; ) 0
is not the margin halfwidth normal to the boundary. The plate margin function ) 0

is the crucial addition to the plate tectonic model since it incorporates information about margin width and intraplate deformation. How this function is determined is demonstrated in the following section. ) 0 is determined via geophysical data on intraplate deformation. A variety of data sets could be used to estimate ) 0 , including, for example, global seismicity, stress distributions [START_REF] Zoback | Lithospheric stress patterns: A global view[END_REF], fault distributions, and possibly a variant of gravity and bathymetry [START_REF] Smith | Bathymetric prediction from dense satellite altimetry and sparse shipboard bathymetry[END_REF][START_REF] Sandwell | Marine gravity anomaly from Geosat and ERS-1 satellite altimetry[END_REF]]. While no one data set is clearly preferable, in this paper we employ seismicity for the sake of demonstration, for its utility and accessibility, and because it is the classic delineator of tectonic boundaries. Since the plate margin function is only meant to indicate the presence of (and not amount or force of) deformation, the seismicity data is not weighted, e.g., by earthquake moment or magnitude which contain information about the amount of displacement along with crustal/lithosphere strength, etc. The use of geologically recent seismicity data of course gives a very current estimate of plate margin structure and thus is relevant for models (e.g., mantle flow models) which need to employ instantaneous motion. We discuss below how the data are chosen, used and manipulated in order to yield the plate margin function ) 0

. We refer to the resulting seismicity based application as the SEISMAR (seismicity-inferred margin-width) plate model.

The seismicity data

We use seismicity data for the period from 1928 to 1990 in the June 1992 version of the Global Hypocenter Data Base. While earthquake locations are less reliable prior to 1960, we choose to have a large data set to facilitate resolution of plate margins (i.e., to minimize gaps in seismicity along plate boundaries). Certain constraints, however, are placed on the chosen earthquakes. First, we allow only earthquakes whose focal depth is less than or equal to 29 km; this is done because indeterminant focal depths are, by default, asigned a value of 30 km [F. Duennebier, pers. comm, 1996]. Earthquakes deeper than 30km are eliminated as they are less representative of plate boundary deformation and interactions (and are liable to be influenced by deep earthquakes which are well removed from plate boundaries). We also choose only earthquakes with magnitude greater than 4 (given that small earthquakes have poorly determined locations). In the end, we obtain the locations of 24,721 earthquakes (Figure 3). However, when necessary we delete intraplate earthquakes that are clearly unrelated to plate deformation (e.g., earthquakes on Hawaii which are primarily from landslides and volcanic activity).

Determination of margin widths

For a given plate we consider only the subset of the seismicity data which includes 1) earthquakes on the plate itself, and 2) earthquakes outside the plate but within ¢ ¢¡ of the plate boundary (and not including earthquakes associated with other plate boundaries). We then calculate the position of each earthquake in terms of the pseudo-distance ¦ and angle ¡ .

We next divide the given plate into wedges of equal angular extent ¡ and bin the seismicity data on each wedge (Figure 4). Thus the number of earthquakes ¤£ ¦¥ at pseudo-distance ¦ and a given ¡ is in fact the number of earthquakes at ¦ between ¡ ¤ ¡ and ¡ $ ¡ (Figure 4). If §£ ¦¥ versus ¦ at the given ¡ had a nor- mal (Gaussian) distribution centered at ¦ ¢ ¦ , then the margin half-width would be the Gaussian half-width (i.e., ¨ times the standard deviation) of the distribution.

However, given the scatter and occurence of outliers in the actual data, we employ robust estimation techniques [START_REF] Rousseeuw | Robust Regression and Outlier Detection[END_REF] and define the margin width as the minimum width of the window (in the units of pseudo-distance) which contains 50% of the earthquakes (again, in the wedge centered on ¡ ). (This step is consid- erably facilitated if one first sorts the earthquake location in each wedge in order of increasing ¦ .) By finding this width for each ¡ we build the raw margin function ) 0 ! ¡ (

(deemed raw as it still requires some additional processing, as discussed below).

The chosen wedge size ¡ differs between plates be- cause, for example, the ¡ for a small plate is larger than that for a big plate since the smaller plate is likely to have fewer earthquakes along its boundary. In general we use between 60 and 70 wedges for the bigger plates, and 20 to 30 for the smaller ones. Moreover, we can only calculate a margin width in a wedge if it has more than 4 earthquakes. If there are fewer than 4 earthquakes in the wedge we assume the margin width is the average of the widths of the two surrounding wedges.

To determine the confidence in the inferred margin width, we also record the total number of earthquakes on each wedge © ! ¡ ( . We then smooth the function ) 0 ! ¡ ( by a Gaussian filter that is weighted by © ! ¡ ( ; in this way values of ) 0 that have higher confidence are more strongly weighted. (The Gaussian filter is arbitrarily chosen to have a half-width that is ¨ ¡ ). Therefore, our smoothed margin width at the 1 2 54

value of ¡ is ) 0 ! ¡ ( ¢ ) 0 ! ¡ ( © ! ¡ ( ! 3 # 3 # "! $# # % & © ! ¡ ( ! 3 # 3 # "! '# # (8)
However, the pseudoboundaries (where some plates are divided) must be left intact so the parts of a plate will fit back together again along the boundary as tightly as possible. The calculation of the margin widths for the pseudoboundaries is discussed below.

As with the boundary function ¦ ! ¡ ( , the margin function can be put into an analytic form with discrete Fourier transforms, though since

) 0

is defined only on wedges and not each boundary point, it is defined (usually resampled to radix 2 points) at fewer values of ¡ .

Pseudoboundary margin widths

The creation of pseudoboundaries has no (or negligible) effect on the final model of plate kinematics as long as the effects of these boundaries' margins essentially cancel each other once all the subplates are pieced back together again. The shape functions of a subplate and its complement (i.e., the adjacent suplate sharing the pseudoboundary) should add to unity on the pseudoboundary. As a simple example, the 1-D hyperbolic-tangent step function (centered at C ¢ )C and of margin width )

) and it's mirror image add to unity:

! $ ( 10 ¢2 ' ! C ¤ C ) ( $ ! ¤ ( 10 ¢2 (' ! C ¤ C ) ( ¢ (9)
But these functions only add to unity as long as they each have the same margin width. Moreover, since complementary subplates have the same Euler pole, there is no differential motion across their pseudoboundary and thus the pseudoboundary has little or no effect on strain-rate, vorticity and divergence fields. However, the actual description of margin widths on pseudoboundaries is not as precise or as simple as implied by (9) given that the boundaries are arbitrary curves on a spherical surface, and the margin widths are measured from completely different points on a subplate and its complement.

To make a subplate and its complement fit as tightly as possible along a pseudoboundary the definition of the margin must be precise. The margin width must have a consistent mathematical definition that is frame invariant (in particular, independent of the frame of reference of the subplate in which the margin width is measured). Thus, it is best to have an exact mathematical description of the margin in the original spherical coordinates # and & . For precision we delineate the margin region of the plate with a top-hat or box car function centered along the pseudoboundary, representing, say, a distribution of fictional earthquakes (see Figure 4); i.e., this function is 1 inside and 0 outside the margin region. Since a pseudoboundary curve is given by the single valued function & ' ! ! $# ( (see equations ( 2) and ( 3)), the margin region is therefore defined by the function ) © if the boundary approaches a North-South tangent; see also [START_REF] Bercovici | A continuous kinematic model of plate-tectonic motions[END_REF]). With this definition, the margin width measured from the centers of two adjacent subplates will be derived from the same formula.

£¡ ! $# % '& )( ¢ ¤ $ 9 ¦¥ 2 ¥ ¤ ¢ & ¤ & ' ! ! $# ( ) ! $# ( £ ¨¨ § (
With the margin region for a given plate's pseudoboundary defined in the (

#

, & ) coordinate system, we need to measure the width of the region in the rotated polar coordinate system ( ¦ , ¡ ). For a given constant-¡ line (that in- tersects the pseudoboundary) the values of ¢¡ are calculated for increasing ) 0 may require many incremental steps in ¦ to properly resolve the margin region, sometimes as many as 50,000 steps.) This is done for a sufficient number of

¡ along the pseudoboundary (typically about 20 values).

In this way we obtain a sampling of the function ) 0 ! ¡ ( along the pseudoboundary. This sampling is appended to the filtered sampling of ) 0 ! ¡ ( along the true boundaries (with seismicity data) to yield a complete margin function ) 0 ! ¡ ( for ¡ ¢¡ . Incorporation of the total margin function for each plate into the associated plate shape function (see equation ( 7)) comprises the essential part of the SEISMAR plate model. Figure 5 shows an example of the seismically inferred margin width for a plate with considerable variability in its margin structure, i.e., the Eurasian plate. Where the concentration of earthquakes is dense the margin width is relatively narrow and well defined. However, where earthquakes are poorly concentrated, the margin width is appropriately broad (e.g., the north-east margin of the Eurasian plate).

Results of the SEISMAR Model

Divergence and Vorticity Fields

The Earth's plate motions can be divided linearly into poloidal and toroidal parts [START_REF] Hager | Subduction zone dip angles and flow driven by plate motion[END_REF][START_REF] Hager | Kinematic models of large-scale flow in the Earth's mantle[END_REF]. Poloidal flow represents divergent motion (ridges and subduction zones) and thus reflects upwelling and downwelling in the mantle. Toroidal motion represents purely horizontal rotational motion as exists in the spin of plates and strike-slip shear. Poloidal motion is typical of convective flow. However, the issue of how and why the purely horizontal and dissipative toroidal motion is generated in the plate-mantle system remains a fundamental, yet largely unresolved, aspect of our understanding of how plate tectonics is linked to mantle convection [see [START_REF] Bercovici | A simple model of plate generation from mantle flow[END_REF]Bercovici, , 1995aBercovici, ,b, 1996[START_REF] Bercovici | Generation of plate tectonics from lithospheremantle flow and void volatile self-lubrication[END_REF]Zhong & Gurnis, 1995a,b, 1996].

In this section we show how the SEISMAR model resolves the poloidal and toroidal parts of the Earth's present-day plate motions. In particular, we will examine the horizontal divergence and vertical vorticity of the plate motions which are the most detailed manifestations of the poloidal and toroidal fields. In the standard plate model divergence and vorticity are essentially impossible to resolve as they are singularities (due to the assumption of zero margin width). However, in the continuous plate model, and the SEISMAR model in this paper, these quantities are quite resolvable and contain considerable detail about deformation of the Earth's surface. Although this model also has certain drawbacks and artifacts it gives us the first quantitative estimate of the Earth's surface divergence and vorticity fields.

Theoretical background

The two-dimensional velocity on the surface of a sphere can be expressed with a Helmholtz representation:

¡ £¢ ¤£ ¦¥ $ £ ¨! § © ( ( 13
)
where £ is the horizontal gradient, ¥ is the poloidal potential, © is the toroidal stream function, and § ¨is the unit vector in the radial direction. Horizontal divergence is

¢ £ ¡ £¢ £ ¨¥ (14) while radial or vertical vorticity is §¢ § £ ¨¡ ¢ ¤ £ ¨© £ (15)
Divergence and vorticity are thus also representative of the poloidal and toroidal fields, respectively. However, since the divergence and vorticity involve gradients of ¥ and © , they enhance the finer small-scale features and thus permit a more detailed description of the poloidal and toroidal fields.

To calculate the divergence and vorticity fields directly we would simply find the appropriate horizontal gradients of the surface velocity field. This is possible in the continuous plate model, but not so in the standard plate model. In the continuous plate model we would obtain (from substituting (1) into ( 14) and ( 15)):

¢ ¤ ! " " & $ # 5 87 9 & " " # (16) $ ¢ $ # " " & ¤ ! 5 87 9 & " " # (17)
where ;these take the form of

¢ ! 5 87 9 # $ % 9 G@ A2 # ( 5 87 9 & $ & 9 1@ 2 & (18) # ¢ ! 5 87 9 # $ % 9 G@ A2 # ( 9 1@ A2 & ¤ & 5 87 9 & (19) ! ¢ % 5 87 9 # ¤ 9 1@ A2 # (
" " ¢¡ ¢ ¤£ ) 0 9 ¦¥ !5 ' ¨¢ ¦ ¤ ¦ ) 0 £ ¨ § ¤ © ¨ § ¦ ¤ ¦ ¡ ¢ ¡ ¤ ¡ § ¤ § £ ¡ ¦ ¤ ! ¦ ¤ ¦ ( 2 )
¡ § For the standard plate model the shape function is essentially constructed from discontinuous step functions and therefore derivatives of do not exist (i.e., they are singularities). One can however determine the divergence and vorticity from the velocity field without taking gradients of ¡ . This method involves vector-spherical- harmonic transforms [see O 'Connell et al., 1991] and yields the spherical harmonic transform of and , not the physical quantities themselves. Using this method one finds that the spherical harmonic transforms of and $ are

¡ ¢ ¤ "! ¦# £ $ ¡ &% ¡ (22) ¡ ¢ ¤ '! ¦# £ )( § 0$ ¡ % 21 ¡ (23)
where is the solid angle, and $ ¡ 3%

is the complex con- jugate of the normalized spherical harmonic $ ¡ of de- gree 4 and order

¡

. In this way divergence and vorticity can be calcluated from the expansions

¢ 5 ¡ 3 ¡ $ ¡ (24) §¢ 5 ¡ 3 ¡ $ ¡ £ (25)
where 6 is the maximum 4 used.

Field representations

The divergence and vorticity fields for the standard discontinuous plate model (using ( 22)-( 25) with 6 ¢ ¥ ; O 'Connell, pers. comm., 1992) are shown in Figure 6. Even though the vector spherical harmonic transforms from which these are derived do not directly involve singularities, the divergence and vorticity fields for this model are intrinsically singular. This causes the excessive ringing or Gibbs phenomenon extant in both fields. Moreover, the magnitudes of and depend on 6 , i.e., where the spherical harmonic series is truncated. Some of the broader true features are discernible in the divergence and vorticity fields, such as the East Pacific Rise, and the Southeast Indian Ridge. However, the Gibbs effect obscures any finer or more complex features, such as those around the Philippine plate. Finally, it is worth noting for the sake of later comparison that the divergence field has the largest amplitude feature which occurs the along East Pacific Rise. The maximum vorticity is about 80% of the maximum divergence and occurs on the southwest side of the Philippine plate (though for the Earth's present day plate motions based on vector spherical harmonic transforms of the velocity field for the standard discontinuous plate tectonic model (with zero margin widths). the structure of the entire Philippine plate is obscured by ringing).

The divergence and vorticity fields for the continuous, SEISMAR plate model (using ( 16) and ( 17)) are shown in Figure 7. Since these fields are calculated directly, i.e., without spectral transforms, they exhibit no ringing. While the SEISMAR model does yield some artifacts (discussed below), it also gives a detailed picture of and , clearly showing the effects of even the smallest plates. The magnitudes of the divergence and the vorticity fields are not artifacts of the model resolution but arise from the Euler pole and seismicity data. Thus the SEISMAR model gives the first quantitative estimate of the strength of these fields.

Fine details of the divergence and vorticity field can be resolved with the SEISMAR model that cannot be dis-tinquished with the standard model. For example, the small positive divergence embedded in a convergent zone near the south-eastern boundary of the Philippine plate is indicative of the Ayu trough [START_REF] Weissel | Evidence for Eocene oceanic crust in the Celebes Basin, in The Tectonic and Geologic Evolution of Southeast Asian Seas and Islands[END_REF]; this feature would not be permitted in the standard model since it would be cancelled out by the negative divergence of the Yap and Marianas Trenches.

In the SEISMAR model the vorticity field has the largest maximum near the northern end of the East Pacific Rise, not near the Philippine plate as implied with the standard plate model. However, two distinct local maxima in the vorticity field occur on the northern and southern corners of the Philippine plate. The maximum positive divergence also occurs on the East Pacific rise, as with the standard model. However the maximum absolute divergence is in fact negative, in contrast to the Of particular interest is the effect of broad plate margins. One can distinctly see that the North-Eastern boundary of the Eurasian plate, which hypothetically cuts across a diffuse seismicity zone in Siberia, is hardly manifested at all in the divergence and vorticity fields; this is entirely due to the breadth of the margin as estimated from the widely and sparsely distributed seismicity data. Moreover, the distribution of seismicity in the Basin and Range province causes the vorticity field over the San Andreas region to be well defined but relatively broad. One can also discern, in both the divergence and vorticity fields, the effect of the diffuse boundary separating the Australian and Indian plates.

Finally, the SEISMAR model clearly shows shades of nonzero vorticity permeating some plate interiors (e.g., the Pacific, Philippine and especially the Cocos plates). This effect is the direct consequence of plate spin which is proportional to plate area and not deformation on the margins (see the first term in ( 17)).

Artifacts of the SEISMAR model

Although the SEISMAR model creates a relatively clear and quantifiable representation of the Earth's divergence and vorticity fields, it also creates some minor artifacts that are worthy of discussion. Foremost is that the pseudoboundaries show a very small but discernible signature in the divergence and vorticity fields. In particular one can observe slight seams in most of the larger plates (e.g, the southwest-northeast running seam in the Pacific plate). (These seams are also emphasized in the figures due to the false illumination projected from the southwest). In the plate interiors these seams are not strong enough to register outside the narrow zero-centered gray range (i.e,. less than 5% of the maximum) of the color scale in Figures 6 and7; however, they do manifest themselves more distinctly where they intersect a true boundary. Although these artifacts are second-order at worst, they demonstrate that there is room for improvement in the continuous plate model, or in the application of seismicity in the SEISMAR model. While considerable care was taken to accurately resolve the pseudoboundaries (so that sections of a divided plate would fit cleanly together again) there are still some problems. First, pseudoboundaries (both the boundary distance and margin width) are not sampled by two adjacent plates from the same coordinate system; thus the boundary shape and width are not measured at the same points or from the same angles by two complementary plates. This scheme for sampling the pseudoboundaries might be made more exact in future studies. Alternatively, it may be possible to essentially apply a high-pass filter to the divergence and vorticity field; i.e., permit only values with absolute values greater than some minimum and thus effectively exclude most of the effects of the pseudoboundaries. This latter method is arbitrary at best and thus not entirely satisfactory.

Another artifact of the SEISMAR model is evident in the spikes and ringing in the divergence and vorticity fields along (i.e., parallel to) the margins. This undoubtedly comes about because of the often highly oscillatory nature of the margin width function ) 0 ! ¡ ( for many plates. This in itself likely reflects the sparse coverage of seismicity data on certain plate boundaries, and/or diffuse boundaries. Even with smoothing weighted by confidence (i.e., number of earthquakes, e.g., from (8)), the resolution of ) 0

in ¡ is still relatively coarse in or- der to permit more than 4 earthquakes per wedge of the plate; this can cause significant variation from one alpha to the next which becomes manifest as spikes and some secondary ringing of and along the boundaries. Again, these artifacts might be reduced by more aggressive filtering, or by using more data on plate margin widths (e.g., other data sets with additional resolution of plate boundaries).

Power spectra and kinetic energy partitioning

The kinetic energies of toroidal and poloidal motions are mostly dependent on the bulk size and shape of the larger plates, and not extensively on the margin width. Thus one should expect that the toroidal and poloidal kinetic energies, their power spectra and their ratio should differ little from one plate model to the next. We therefore examine the effect (or lack of effect) the SEISMAR model has on the various characteristics of the toroidal and poloidal kinetic energies (as well as the power spectra of the di-vergence and vorticity field).

The spherical harmonic kinetic energy spectrum is defined as the kinetic energy for each spherical harmonic degree 4 ; the poloidal and toroidal energy spectra (per unit mass and divided by ¨) are, respectively,

¡ £¢ ¥¤ ! 4 ( ¢ 4 ! 4 $ ( ¡ 3 ¡ 0 ¡ (26) ¡ £¢ §¦ ! 4 ( ¢ 4 ! 4 $ ( ¡ 3 ¡ 0 ¡ £ (27)
Figure 8 (top two frames) shows these spectra and their ratios for both the standard discontinuous plate model and the SEISMAR model. One can see that the SEIS-MAR model has little effect on the amplitude and shape of this spectrum, as expected. The ratio of the energies ¡ £¢ §¦ ! 4 ( ¡ £¢ ¤ ! 4 ( is also shown in the same figure and this emphasizes some minor differences between the two models. The ratio for both models is greater than unity only in the 4 ¢ mode, which represents the amount of net lithospheric spin. The ratio is noticeably larger at 4 ¢ for the SEISMAR model than the standard model. Moreover, the local peak in the ratio at 4 ¢ ©¢© in the standard model is not present in the SEISMAR model. These differences may reflect the use of finite margins in the SEISMAR model because, for example, with more diffuse margins the SEISMAR model gives relative more weight to spin kinetic energy in the toroidal field (which depends on plate area not on margin structure).

For the standard plate model the total poloidal and toroidal kinetic energies, and their ratio are ¡ £¢ ¤ ¢ ¥ (rad/Gyr) ¨, ¡ £¢ ¦ ¢ (rad/Gyr) ¨and ¡ £¢ ¦ ¡ £¢ ¥¤ ¢ £ §¡ . For the SEISMAR model ¡ £¢ ¤ ¢ ¢ ¡ (rad/Gyr) ¨, ¡ £¢ ¦ ¢ (rad/Gyr) ¨and ¡ £¢ ¦ ¡ £¢ ¥¤ ¢ £ §¡ ¢¡ . The SEISMAR poloidal energy is 6% and the toroidal energy 3% lower than those for the standard model, while the energy ratio is higher for the SEISMAR model. While these differences between the models are subtle, they are systematic and thus likely reflect the reduction in energy associated with deformation on margins in the SEISMAR model, and thus a higher proportion of spin energy in the SEISMAR toroidal field.

A perhaps more revealing power spectra involves the squared amplitude of divergence and vorticity at each spherical harmonic degree, i.e., (dark solid) and toroidal (dashed) kinetic energies, and the ratio of toroidal to poloidal energy (gray solid) line for both the discontinuous and SEISMAR plate models (top two frames). Spherical harmonic power spectra for horizontal divergence (solid lines) and radial vorticity (dashed lines) for both the standard and SEISMAR plate models (bottom frame).

¨ ¢ ¡ 3 ¡ 0 ¡ (28) ¨ ¢ ¡ 3 ¡ 0 ¡ £ (
which employ essentially the same information as in the kinetic energy spectra, yet emphasize the small wavelength features. These spectra are also shown in Figure 8 (bottom frame) for both the standard and SEISMAR plate models. Both models show significant peaks in divergence at 4 ¢ , 4 ¢ ¥ and , and 4 ¢ ¥ ; however, only the peak 4 ¢ is reflected in the vorticity spec- tra. Nevertheless, the vorticity spectra appears to track the divergence spectra but only for low degree, i.e. up to approximately 4 ¢ ; for higher degree the spectra seem largely uncorrelated. This may reflect the influence of plate geometry at low degree on both spectra, while at higher degree we see that the structure of divergent/convergent boundaries is largely independent of the structure of strike-slip boundaries. Finally, one can also see the power drop off for the SEISMAR model, indicating that the spherical harmonic series for the SEISMAR model is convergent; no such convergence is indicated for the standard model, as is to be expected for apparent singularities.

Conclusions

In this paper we have presented a continuous kinematic model of present-day plate motions allowing for 1) a realistic representation of plate shapes; and 2) incorporation of geophysical data on intraplate deformation in order to estimate plate margin widths. A variety of data sets contain information about intraplate deformation, such as seismicity and stress distributions. To demonstrate how such data is used in the continuous plate model, we employ global seismicity to constrain plate margin width. The mathematical treatment of how to 1) define analytically continuous plate shape functions, and 2) use geophysical data (seismicity in particular) to construct plate margin width functions are presented in this paper; although the treatment as a whole is relatively convoluted, the individual steps are basic and straightforward. The resulting seismicity-based example, called the SEISMAR plate model (for seismicity inferred margin widths), yields a detailed and quantifiable representation of the horizontal divergence and radial vorticity fields (which are precise measures of poloidal and toroidal motion). Employing finite margin widths allows these fields to be analytically calculable (as opposed to being singularities in the standard plate model) and permits the representation of diffuse margins. The kinetic energy spectra and partitioning of the SEISMAR model are not greatly different from the standard plate model, as expected and desired; most of the minor differences arise because, with finite margins (and thus non-infinite divergence and vorticity), a more accurate weighting can be given to the contribution of plate spin to the toroidal energy. Although the SEISMAR model represents one example of combining geophysical data with a continuous plate model, it provides the first effort to incorporate intraplate deformation data into a global plate tectonic model and is therefore a significant adjustment to the theory of plate tectonics.

Figure 1 :

 1 Figure 1: Original plates boundaries (top) and the same plate boundaries smoothed with a 1000km Gaussian fullwidth filter (bottom), as employed in the continuous plate model.

Figure 3 :

 3 Figure 3: Global seismicity distribution used to determine plate margin widths.

Figure 4 :

 4 Figure 4: Sketch illustrating the determination of margin width along the true boundary and pseudoboundary of a single subplate, in this case the northern Pacific plate (Pacific-1; see Figure 2). On the true boundary, the oblique margin width at angle ¡ is determined by bin- ning the seismicity within a wedge between ¡ ¤ ¡ and ¡ $ ¡ (upper left); the margin is then the narrow- est region which contains 50% of the earthquakes in that wedge (the shaded region, which is exaggerated for the sake of clarity). The margin width of the pseudoboundary (lower right) is found by measuring, along a line of constant ¡ , the width of a box-car function ¢¡ centered on the boundary (long shaded region). The figure, however, is over-simplified and should only be considered illustrative; in particular, ¡ is defined in the original coordinate frame ! $# % '& )( , while the measurements of margin width are done in the rotated frame ! $# £¢ % '& ¢ ( . See text for

  margin half-width measured in the North- South direction. For simplicity, we prescribe the margin to have a constant half-width normal to the boundary (see Figure4) and thus )

¦

  (using in (10) the

#

  and & uniquely corresponding to the given ¦ and ¡ ). The margin half- width along this constant ¡ line is thus

  Since the plate margins for the continuous plate model are of finite width, the various derivatives of are calculable. Moreover, since and are scalars they are invariants and thus we need only determine them for each individual plate or subplate in the plate's specific plate-centered reference seismically inferred margin width for the Eurasian plate, superimposed on the local seismicity distribution. The plot shows the shape function for the entire Eurasian plate; however, only values between ¢ £ and £ © are given a single gray shade while all other values are white. frame divergence and vorticity involve derivatives of each plate's shape function with respect to # ¢ and & ¢

  derivatives with respect to ¡ . Although these derivatives are susceptible to Gibbs ef- fects, such effects can be reduced with modest Gaussian filtering in the Fourier domain [see[START_REF] Bercovici | A continuous kinematic model of plate-tectonic motions[END_REF]. We then simply use the value of or the plates are then added together to yield the total divergence and vorticity fields.

  Surfaces of horizontal divergence and radial vorticity (with a world map at the bottom for orientation)

  Same as Figure 6 but with horizontal divergence and radial vorticity calculated directly from the velocity field of the SEISMAR plate model. standard model.

Figure 8 :

 8 Figure 8: Spherical harmonic spectra for the poloidal

Table 1

 1

	(i.e., the plate it divides,

Table 1

 1

	Plate	: Pseudoboundary Information (deg) ¡ (deg) ¨(deg) ¡ ¨(deg) Type of Boundary
	African	43.7231 12.1501 -12.1895 -2.6453	straight line
	Antarctic	183.3320 -65.3357 338.0790 -60.4307	great circle
	Australian	16.0360 -49.3737 119.6860 -10.0926	straight line
		181.5650 -34.3662 170.8820 -23.3943	straight line
	Cocos	254.1260 18.1296 255.3420 19.2305	straight line
	Eurasian	122.1750 22.2206 94.9668 26.3249	straight line
	Nazca	275.0440 1.6506 279.5100 -1.0551	straight line
	N. American 162.3830 54.5400 152.0150 59.3193	straight line
		-62.4904 18.9033 -45.1415 24.3281	straight line
		-34.6069 53.8276 129.1240 50.1203	great circle
	Pacific	188.2280 -14.7373 231.3450 44.8559	straight line
	S. American 336.1190 -56.6925 344.8290 -46.6037	straight line
	AFR-1

AFR-2 ANT-1 ANT-2 IND AUS-1 AUS-2 AUS-3 COC-1 COC-2 EUR-1 EUR-2 NAM-1 NAM-4 NAM-2 NAM-3 NAZ-1 NAZ-2 PAC-1 PAC-2 SAM-1 SAM-2

	Figure 2: Plates divided by pseudoboundaries (with labelled subplates).
	! $# £¢ spaced distances along the boundary.) The coordinates ¢ of this frame of reference are denoted as % '& ( . For a ! $# ¤¢ given plate the pseudo-distance from an arbitrary point within the plate % '&

¢

( to any other point on the globe

  In order to make the plate shape ana-

	lytically continuous, valued function of ¡ ; this function ¦ must be expressed as a single-¦ ! ¡ ( is called the plate-boundary function. Achieving a single-valued ¦ ! ¡ ( requires a prudent choice of the polar origin ! $# ¢ % '& ¢ ! $# ¢ ¢ ( . Here, the optimum % '& ( yields the smallest maximum value of H ¦ ¡ H since this would be zero for the ideal circular boundary while a multivalued ¦ ! ¡ ( would have singularities in its slope. The optimum values of # ¢ ¢ and & are found numerically for all 25 plates and/or sub-plates. Since most plates and subplates are, to first order, elliptically shaped, most of the ! $# ¢ % '& ¢ ( are near ! ! $# ¢ ¡ % ( . Table 2 shows the values of % '&

¢

( for each plate or sub- plate (see Figure

2

for subplate labels).

The plate-boundary function ¦ ! ¡ ( is therefore con- structed from all the points on the boundary; it is then evenly resampled in ¡ with radix-2 number of points and discrete Fourier transformed to obtain

Table 2 :

 2 Polar Coordinate Origin (

	¢	¢
	14 ¡	) for Each

The SEISMAR model: a seis- micity based example

Here we illustrate how the margin width function

  29)

		1 1	2	3 4	6 8 10	20	30 40	60 80 100
		10 1 10 2					Discontinuous Plate Model	1.00
		10 0						0.75
	energy kinetic	10 -3 10 -2 10 -2 10 -1 10 0 10 1 10 2					SEISMAR Plate Model	0.25 0.50 0.75 1.00	P T /KE KE
		10 -1						0.50
		10 -2 10 -2
		10 -3						0.25
		800
		600
	300 l ω 400 2
	&	200
	60 80 100 l 2 D	Discontinuous Plate Model SEISMAR Plate Model
		40
		10 0			10 1 10 1			10 2
				spherical harmonic degree