AN ABSOLUTE PLATE MOTION MODEL BASED ON HOTSPOT REFERENCE FRAME

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1 CHINESE JOURNAL OF GEOPHYSICS Vol.60, No.5, 2017, pp: DOI: /cjg AN ABSOLUTE PLATE MOTION MODEL BASED ON HOTSPOT REFERENCE FRAME ZHANG Qiong 1, WANG Shi-Min 2, ZHAO Yong-Hong 1 1 Department of Geophysics, Peking University, Beijing , China 2 Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing , China Abstract Based on the new generation model of relative plate motions, the MOVEL model, an absolute plate motion model is built by least squares inversion of the observed hotspot trend data. A systematic test for the consistency between the MOVEL model and all the available hotspot data adopted by previous studies of absolute plate motion models shows there are outliers in the data set. Accordingly, two new methods to reject outliers, such as, step-wise rejection and global search rejection, are proposed. Combined with the chi-square test for the total residual and the normal test for the residual frequency distribution, the best model is selected step by step. A T87 model is eventually obtained. This model provides reasonable fitting to 87 globally distributed hotspot trends, but the predicted plate velocities are systematically lower than the observed rates of volcanic migration along hotspot tracks with the deviation up to 4 cm a 1. Such a deviation may be explained as a consequence of systematic errors possibly associated with the observed hotspot rate data, or due to relative motions between hotspots caused by mantle return flow. For either case, however, this study s results demonstrate that the hotspot trend data are capable of defining a global hotspot reference frame independently and effectively, which could be conveniently applied to studies on absolute plate motion, mantle convection and true polar wander. Key words Absolute plate motion; Hotspots; Least squares inversion; Outliers rejection 1 INTRODUCTION Plate tectonics are the most important tectonics in the earth s lithosphere. The theory of plate tectonics, which was born in the 1960s, states that the lithosphere of the earth is made up of a series of rigid plates of different sizes, which float on top of the asthenosphere that is hot and more fluid. The plates have relative movements between each other (Morgan, 1968). The lithosphere s tearing and collision or dislocation caused by plate s relative movement formed many magnificent geological movements (Fu et al., 1992, 2007). The plates movement not only dominates the tectonic movement pattern of the whole lithosphere, but also reflects the movement of the underlying mantle. Plate movement plays a very important role in the study of deep earth dynamics. The plate movement includes the relative plate motion (RPM) and the absolute plate motion (APM) relative to the deep mantle. According to the theory of plate tectonics, the lithosphere deformation is mainly concentrated in the plate boundary area, and the inner plate is mostly rigid. Thus, we can quantify the motion of the plate on the Earth s surface using the theory of rigid body fixed-point motion. i.e. Euler s theorem, and then establish a model of the plate s motion. Several generations of relative plate motion models have been successfully established using data such as transform fault, seafloor spreading rate and seismic slip direction (DeMets et al., 2010, 2011). The APM model describes the movement of a rigid plate relative to the deep mantle, so we need a reference to represent the deep mantle, i.e. we need to define the APM reference frame. The APM reference frames defined in the previous studies mainly include the hotspot reference frame, the NNR (no-net-rotation) reference system, and the average mantle reference system based on shear wave splitting data and so on. In addition, the paleo-latitude change derived from paleomagnetic and paleoclimatic data (Bai, 2011) and the geographical distribution of the ridge and trench also have a certain constraint on the absolute motion of the plate. zhangqiong0503@pku.edu.cn *Corresponding author: smwang@ucas.ac.cn

2 Zhang Q et al.: An Absolute Plate Motion Model Based on Hotspot Reference Frame 457 The lithospheric no net rotation is an approximation to that the total amount of lithosphere movement is zero. Only when the viscosity of the asthenosphere is transversely uniform and the moment associated with the boundary of the plate is symmetrically applied, can the non-rotation be equivalent to zero net torque (Argus et al., 1991, 2011). Actually, it is ubiquitous that the viscosity of the asthenosphere is not transversely uniform and the moment is asymmetrically applied on both sides of the trench due to plate subduction, which forms a serious challenge to the rationality of using the NNR reference system to represent the deep mantle. The wave velocity anisotropy reflected by the seismic shear wave splitting is related to the shear motion of plate relative to the lower mantle, so the reference frame established by the shear wave splitting direction is more likely to represent the asthenosphere or upper mantle, not the deep mantle in the mean sense. The hotspot reference frame is the reference frame which can represent the deep mantle with a most clear physical meaning, and the research of APM model based on the hotspot reference frame is also sufficient. The hotspots origined from the mantle are indicated by a series of linearly arranged volcanoes that are numbered in one direction. Morgan (1971, 1972a, 1972b) has argued that the mantle plume rising from the core-mantle boundary forms a hotspot at the bottom of the lithosphere, and the plate movement above the mantle hotspots forms a surface volcanic chain. He made the assumption that hotspots are fixed to each other relatively. According to this assumption, the trend of the hotspot chain and the migration rate of the hotspot directly indicate the direction and rate of the plate s absolute motion, which can be easily used to constrain the APM model. With the progress of observation technology, the division of the plate is increasingly finer, and the amount of observation data is increasing and the precision is improving. The research on the APM model is expanding, and the APM models experienced five generations so far as shown in Table 1. On the basis of the RPM model, we can combine the hotspots data to establish the corresponding APM model. It can be seen from Table 1 that the APM model on the basis of the RPM model from the previous generations has been studied comparatively well while the study of the APM model based on the latest generation of MORVEL model is blank. Taking into account that the MORVEL model is the result of much work by DeMets, Gordon and other scholars in the past 20 years, it represents a comprehensive update of the NUVEL1A model. The establishment of an APM model consistent with the MORVEL model has important scientific significance. Table 1 Evolutionary history for plate motion models First generation Second generation Third generation Fourth generation Fifth generation LP68 CH72 RM1 P071 RM2 NUVEL1 (A) MORVEL RPM (Le Pichon (Chase, (Minster (Chase, (Minster (DeMets et al., (DeMets et al., 1968) 1972) et al., 1974) 1978) et al., 1978) 1990, 1994) et al., 2010) R S Data T GPS Sum Plate number R, S age (Ma) <10 <5 3 3 <3 HS2-NUVEL1 (Grippand Gordon, 1990); T22A AM1 P073 AM1-2 APM (Hotspot (Wang and Wang, 2001); (Minster (Chase, (Minster reference frame) HS3-NUVEL1A (Gripp et al., 1974) 1978) et al., 1978) and Gordon, 2002); MM07 (Morgan and Morgan, 2007) Note: R represents the spreading rate; S represents the earthquake slip direction; T represents the transform fault azimuth; APM represents the absolute plate motion model.

3 458 Chinese J. Geophys. Vol.60, No.5 Under the premise that the plate relative motion model is completely specified, the APM model (established from RPM) can be established by inversion to determine only three parameters (such as the Pacific plate angular velocity vector). Fortunately, the hotspot data has far more than three parameters, so it s possible to test the consistency of hotspot and compatibility between hotspot data using APM models. Based on previous research on the APM model based on the NUVEL1A model, scholars found that there was a system incompatibility between hotspot trend data and hotspot rate data (Wang and Liu, 2006). Whether this incompatibility is formed by the systematic error of hotspot rate data (Morgan and Morgan, 2007), or by the relative movement between hotspots (Wang and Wang, 2001; Wang and Liu, 2006) it is still an unresolved question. According MORVEL model, we built an APM model based on hotspot reference frame with hotspots data as much as possible. After a statistical test of the consistency between all the hotspots of the APM model and the MORVEL model, we found that there are outliers in the dataset. Correspondingly, two new methods (step-wise rejection method and global search rejection method) are proposed to extract the outliers, and then a new APM model (T87 model) is obtained. The model can reasonably fit the overwhelming majority of globally distributed hotspot data, while the rate predicted by T87 model is systematically smaller than the observed hotspot migration rate. The following sections detail the process of creating the best fit model and a discussion of T87 model results. 2 OUTLIER DATA ELIMINATION METHOD Under the premise that the relative motion between designated plates completely obeys the MORVEL model, the absolute motion of all plates relative to the hotspot reference system depends only on three independent parameters. In this paper, we select three components of the Eulerian vector (ie, the angular velocity vector) of the Pacific plate as the independent model parameter. Under the constraint of observed hotspot data, the iterative weighted least squares method can be used to invert for the Eulerian vector of the Pacific plate relative to the hotspot reference system. We adopt the same algorithm and formula in the inversion, which is used for establishing the NUVEL1 model (DeMets et al., 1990) and the MORVEL Model (DeMets et al., 2010). We have obtained globally distributed 94 hotspot directional (or trend) data and 42 hotspot rate data by sorting all the hotspot data used to constrain the APM model (merging and repeating the data). Because the previous research has found that there is a system incompatibility between the hotspot directional (or trend) data and the hotspot rate data, this paper only uses the hotspot trend data to build the APM, and then tests the compatibility of the hotspot directional data and the hotspot rate data by comparing the differences between the predicted plate rate and the observed hotspot rate. The inversion calculation shows that although the APM model can be obtained under the constraints of all 94 hotspot trend data, the misfit for a small part of the data is very large, indicating that these data may be outliers. Since the optimal objective function is the sum of the squared residuals for all the fitting data under the least squares criterion. Although the numbers of outliers are small, the effect on the optimal fitting result is large, so we must reject the possible outliers. In the study of previous plate motion models, there was no systematic rejection problem of outliers except seldom individual studies. Wang and Wang (2001) used empirical methods to remove individual data. In this paper, two new methods, namely, step-wise method and global search rejection method are proposed. Combined with statistical methods such as chi-square test of total residual and normal distribution test of residual distribution, we try to search for the best fit APM model. Step-wise method searches for the data which cause the largest error. For example, for 94 hotspot trend data, we do an inversion once to find data with the largest fit error and remove it. Then the remaining dataset has a number of 93, and then do inversion again with the new dataset to find data with the largest fit error. This process can be repeated until the remaining data is less than three which is not enough to constrain the model effectively. The step-wise method requires less computational effort, and can remove the outliers efficiently, but it may

4 Zhang Q et al.: An Absolute Plate Motion Model Based on Hotspot Reference Frame 459 result in improper rejection, that is, the objective function value (total residual) reduction is not necessarily the largest after the elimination of the maximum residual data. To futher verify that we reject the right outlier data, we propose another method named global search rejection method. This method still removes one data at a time, but treats each data in the current dataset as a potential rejetion object (not necessarily the data with largest weighted residuals). It globally searches the data which makes the total residual reduction the largest by comparing with the value of the objective function after removing data one by one. The flow chart of step-wise method and global search rejection method are shown in Fig. 1a and 1b, respectively. (a) (b) Fig. 1 Flow chart of outlier rejection (a) Stepwise rejection method; (b) Global search rejection method. Totalresiduals Datasetnumberofmodels Fig. 2 Variations of optimal model parameters and data fitting residuals in the stepwise rejection and global search rejection processes (a), (b) and (c) are curves for the latitude, longitude and angular velocity, respectively, of the Euler vector for the Pacific plate, and (d) for total residual.

5 460 Chinese J. Geophys. Vol.60, No.5 Figure 2 compares the optimal model parameters and the fitting residual curves obtained by using the step-wise method and global search rejection method. The results show that the two methods have the same results when the number of outlier data is less than ten, and that the true outlier data has been completely removed after ten data are rejected. Thus, the two outlier rejection methods will give the same result in this paper. The data rejection process should be stopped after the outliers have been completely rejected. We use the chi-square method to test whether the dataset includes outliers. For the weighted least squares inversion, the intrinsic implicit assumption is that the weighted residuals are subject to the normal distribution and that the total residuals of the N independent data are fitted by three independent model parameters Fig. 3 Variations of the total residual associated with the optimal model and the 95% confidence interval of chi-square distribution as functions of degree of freedom of data fitting will obey the chi-square distribution with freedom of N-3. So, the total residual value corresponding to the optimal model after completely removing the outliers should fall within a reasonable chi-square distribution confidence interval. Fig. 3 shows that the total residuals of the models with freedom (corresponding to the total data number in the range of 78 to 88) are within the 95% confidence interval of the chi-square distribution, so the optimal APM model should be selected from these 11 models whose total number of data ranges from 78 to 88. To further select the optimal model, we examine the residual frequency distribution and compare it with the standard normal distribution curve. It is found that the residual frequency distributions of the model whose Fig. 4 Frequency distrubution of modeling residuals compared with the standard normal distribution curve for 4 optimal models fitting data

6 Zhang Q et al.: An Absolute Plate Motion Model Based on Hotspot Reference Frame 461 total number of data is 87 is closest to the standard normal distribution in the 11 candidate models, so we choose this model as the optimal APM model, and named it as T87 model. Fig. 4 shows the contrast of residual frequency distribution with standard normal distribution curve of the T88-T85 models. 3 RESULT AND DISCUSSION The Eulerian vector of Pacific plate obtained from T87 model is E, S, /Ma, the confidence can be completely determined by the covariance matrix. In the cartesian coordinate system, the covariance matrix of the Eulerian vector of the Pacific plate is a b c b d e c e f = rad 2 Ma 2. (1) The a, d, f are the variance of the Euler vector of pacific plates in (0 N, 0 E), (0 N, 90 E) and 90 N direction, respectively. The b, c, e are the covariance between the Cartesian coordinates. The Euler vectors of global plates abosolute motion could be obtained by adding the result of RPM model (MORVEL model) with the Euler vectors of Pacific plate which we get from APM model (T87 model). The results are given in Table 2. Table 2 Euler vectors of absolute plate motion predicted by the T87 model Euler vector Euler vector Euler vector Plate Plate Plate Longitude Latitude Rate Longitude Latitude Rate Longitude Latitude Rate name name name ( E) ( N) ( /Ma) ( E) ( N) ( /Ma) ( E) ( N) ( /Ma) Pa In Sa Am Jf Sc An Lw Sm Ar Mq Sr Au Na Su Ca Nb Sw Co Nz Yz Cp Ps Eu Ri We compare the predicted absolute plate motion rate and direction by T87 model with the observed data shown in Fig. 5. The average fitting residuals of 94 hotspots data calculated from the T87 model are 16.9, while that of 87 hotspot data except 7 outlier data are Four hotspots in the seven outliers came from Antarctic plate (the Balleny hotspot, Scott s hotspot, the bouvet hotspot and the Heard hotspot), other three are from Eurasian plate (Hainan hotspot), Somalia plate (Comores hotspot) and Nubia plate (Great Meteor hotspot), seperately. Fig. 5b shows that 81% of the total data is located below the diagonal straight line, indicating that there is a systematic deviation between the absolute motion rate of the plate predicted by the T87 model and the observed hotspot rate. This result is consistent with the results of the APM model based on the NUVEL1A model, indicating that there is a system incompatibility between the hotspot trend data and the hotspot rate data, and the incompatibility can not be improved by updating the NUVEL1A model to the MORVEL model. The reason for this incompatibility may has very important geodynamic significance and it is worthy of further study. Figure 6 depicts the plate velocity vector predicted by the T87 model, as well as the observation data at 41 hotspots. Comparing the predicted rate with observed hotspots trend, Fig. 6 shows that at most hotspots, the difference between the predicted plate velocity and the hotspot velocity is roughly opposite to the observed hotspot velocity, and the maximum value reaches 4 cm a 1.

7 462 Chinese J. Geophys. Vol.60, No.5 Fig. 5 Comparison of the T87 modeling predictions with the observed hotspot trends (a) and rates (b) Fig. 6 Comparison between the T87 predicted plate velocity and the observed hotspot velocity at hotspots with rate data available There are two explanations for the results in Fig. 5b and Fig. 6. The first explanation is that there is a systematic error in the hotspot rate data and the maximum error is 4 cm a 1. Morgan and Morgan (2007) argue that hotspot rate data depends on volcanic age dating, whereas previous volcanic age data may have a large error and tend to give a low age, which lead to a larger hotspot volcanic migration rate. If this interpretation is true, the fact that the T87 model rationally fits the 87 globally distributed hotspot data with three independent parameters will undoubtedly provide strong support for the hypothesis that hotspot is fixed. However, if the first interpretation does not hold, that is, the hotspot rate data does not have a large and systematic error, then the results in Fig. 5b and Fig. 6 reflect the presence of a few centimeters movement per year between hotspots. The fact that the absolute plate motion rate is systematically smaller than the observed hotspot migration rate may be caused by the return flow of the mantle (Wang and Wang, 2001; Wang and Liu, 2006). If it is further assumed that the hotspot originates from the bottom of the mantle, the ratio of the pridicted plate rate and observed hotspot migration rate can be used to estimate the viscosity ratio of the lower mantle and upper mantle (Wang and Wang, 2001; Bercovici, 2003; Boschia, 2008). Using the method proposed by Wang and Wang (2001), the ratio of plate-hotspot velocity of T87 model is 0.832, and the ratio of lower mantle and upper mantle viscosity is about 16. This result is approximately in agreement with those of

8 Zhang Q et al.: An Absolute Plate Motion Model Based on Hotspot Reference Frame 463 post glacial rebound, the mantle viscosity structure and also consistent with the numerical simulation results of mantle convection. Steinberger and O Connell (1998) and Steinberger (2000) have found that when the magnitude of lower mantle viscosity is 1 2 orders higher than the upper mantle, the hotspots usually move at a rate of one order lower than plate, up to 3 cm a 1, and the moving direction of the hotspot is roughly opposite to that of the absolute motion of the plate. 4 CONCLUSION Based on the latest generation of relative plate motion model (MORVEL model), this paper uses the least squares method to invert the observed hotspot trend data, and get the best fit APM model. We propose two new methods namely step-wise method and global-search method to eliminate the outliers in hotspot dataset, and combine the chi-square test of the total residual with the normal distribution test of the residual frequency to filter the opotimal model, and then a new APM model named T87 is obtained. This model can reasonably fit the globally distributed 87 hotspot trend data; by comparison with the 41 hotspot rate data the plate motion rate is systematically smaller than the hotspot migration rate and the maximum deviation is 4 cm a 1. If such a deviation is caused by the systematic error of the observed hotspot rate, it indicates that the error of the hotspot rate data is large and therefore the hotspot data should not be used to constrain the plate motion. On the contrary, if the observed hotspot rate is reliable, it suggests that there exists relative motion between hotspots, and the hotspots movement reflected by the T87 model is consistent with the characteristics of the mantle return flow. To distinguish which one is more in line with the actual situation between the two above explanations, it needs more indepth study in the future. However, regardless of which one is ture, the fact that T87 model reasonably validates the 87 globally distributed hotspot data with three independent parameters strongly suggests that the hotspot trend data can define the global hotspot reference frame independently and effectively. The hotspot reference frame of this definition can be easily applied to the study of plate absolute motion, mantle convection and true pole wander. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China ( , , ) and the Chinese Academy of Sciences Hundred Talents Program project. We thank the reviewers for helpful suggestions to this paper. We also thank Qiongyao Liu, Rebecca Lee and Adrienne Knok for contribution to this paper. References Argus D F, Gordon R G No-net-rotation model of current plate velocities incorporating plate motion model NUVEL-1. Geophysical Research Letters, 18(11): Argus D F, Gordon R G, DeMets C Geologically current motion of 56 plates relative to the no-net-rotation reference frame. Geochemistry, Geophysics, Geosystems, 12(11): Q Bai X H The method of using paleomagnetic data to mensurably define the plate motion [Master Thesis] (in Chinese). Fuxin: Liaoning Technical University. Bercovici D The generation of plate tectonics from mantle convection. Earth and Planetary Science Letters, 205(3-4): Boschi L, Becker T W, Steinberger B On the statistical significance of correlations between synthetic mantle plumes and tomographic models. Physics of the Earth and Planetary Interiors, 167(3-4): Chase C G The N plate problem of plate tectonics. Geophysical Journal International, 29(2): Chase C G Plate kinematics: The Americas, east Africa, and the rest of the world. Earth and Planetary Science Letters, 37(3):

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