Mapping the admittance and correlation functions on the Martian surface using a spherical wavelet filter

Transcription

1 1 Mapping the admittance and correlation functions on the Martian surface using a spherical wavelet filter M. Kido JST Corp. Domestic Res. Fellow at JAMSTEC, Japan D. A. Yuen Dept. of Geology and Geophysics and Minnesota Supercomputer Inst., Univ. of Minnesota, USA Short title: MAPPING ADMITTANCE AND CORRELATION ON MARS

2 2 Abstract. The admittance function, Z, the ratio between the gravity and the topography, is a basic tool for estimating the lithospheric dynamical state on planets. Usually Z is determined in the wavelength domain over a selected region. Recently developed localization techniques have allowed us to focus on the local admittance of a region of interest. However, its reliability also depends on the correlation between the gravity and topography fields. In this study we have employed a spherical wavelet filter and developed a new technique for calculating simultaneously the local admittance function and the correlation over a spherical surface. We applied this wavelet formalism for analyzing the data from the Mars Global Surveyor mission. Our results reveal strong lateral variation of the admittance values in different scales and the reliability of the admittance result over different regions, which is corroborated by the map of the correlation for the corresponding scales.

3 3 Introduction Planetary bodies, such as Venus, Mars and the Moon rely greatly upon analysis of the gravity and the topography for providing valuable information concerning their dynamical behavior of the mantle and state of the lithosphere. An important constraint concerning the elastic thickness of the lithosphere is the admittance function Z (e.g., McKenzie and Fairhead, 1997), between the Fourier transform of the free-air gravity anomaly and the surface topography. This quantity is traditionally cast in the wavelength domain and is taken over 1-D profiles (e.g., McNutt and Parker, 1978). In contrast to the Fourier method, which is a global approach, wavelet transforms represent a recently novel mathematical technique (e.g., Mallat, 1998), which allow one to examine multi-dimensional fields locally and at different length scales. Recently Kido et al. (2002) have devised a 2-D isotropic wavelet-like transform for a spherical surface. With wavelet transformed gravity and topography, localized admittance function and correlation between the two fields can also be constructed as a series of maps with different scales. In this work we will employ this new tool for constructing multiscale maps describing the admittance function and correlation over the Martian surface. McKenzie et al. (2002) have recently shown by using admittance function over selected regions with Fourier analysis that there exist sharp lateral variations in the effective elastic thickness on the Martian lithosphere. Furthermore, McGovern et al. (2000) made localized admittance study on some interesting tectonic regions on Mars. Our principal

4 4 aim is to reveal with our spherical transform filter that lateral variation in admittance on over the sphere as a 2-D map and discuss its robustness with localized correlation analysis. Spherical wavelet filter In general, wavelet transform is a convolution between a mother wavelet function and an original data field. Here, we used a isotropic (axis-symmetric) mother wavelet F lw,σ(ξ), which is based on the zeroth order Bessel function J 0 with a spherical correction and Gaussian window, presented in Kido et al. (2002); F lw,σ(ξ) = ( ) 2 l 2 lw ξ w exp 2σ [ ξ J 0 (l w ξ) sin ξ exp ( ] σ 2), (1) where l w is analogous to a continuous wavenumber corresponding to the spherical harmonic degree. σ is proportional to a window size and have been set to be 3.0 in this study. There is a trade-off between the spatial and wavelength localization (Mallat, 1998). A larger σ puts more weighting on wavelength localization rather than spatial localization, and vice versa. ξ is an angular distance from the center of the function F. When the center of the function F is located at a geographical point (φ 0, θ 0 ), F lw,σ(ξ) can be calculated anywhere on the sphere and is represented as F lw,σ,φ 0,θ 0 (φ, θ). Here, for a given field G(φ, θ), the resultant waveletized field G lw,σ at a point (φ 0, θ 0 ) is a convolution with the mother wavelet over the spherical surface Ω; G lw,σ(φ 0, θ 0 ) = 1 G(φ, θ)f lw,σ,φ 4π 0,θ 0 (φ, θ)dω. (2) Ω

5 5 We used the GMM2B model (Lemoine et al., 2002) for the Martian gravity field, which is given by a set of spherical harmonics coefficients up to degree and order 80 (Fig. 1a). The Martian topographic field is taken from GTM090AA model (Smith et al., 2000), given up to degree 90 (Fig. 1b). For both data sets, hydrostatic components due to planetary rotation has been omitted. The waveletized gravity and topographic fields for l w = 8 (424km), 16 (212km), 32 (106km), and 64 (53km), transformed by Eq. (2), are shown in Fig. 1c and 1d, respectively. Fig. 1. Fig. 1 Admittance and spatial correlation The admittance is a transfer function which represents a ratio of amplitude between gravity anomaly and topography for a selected wavelength component. Here we define the spatially localized admittance as a least squares fit of the slope in G lw,σ(φ, θ) versus T lw,σ(φ, θ) distribution, which is plotted only in the Gaussian window centered at a point of spatial localization. The window is represented by a local weighting over the entire sphere in the least squares fit; ( ) 2 lw ξ W lw,σ(ξ) = exp, (3) 2σs where s has been set to be 2.0, which results that the weighting window is wider than wavelet transform window by a factor of 2.0. Finally, the spatially localized admittance

6 6 Z lw,σ at a point (φ 0, θ 0 ) is expressed as a 2-D map; W lw,σ(ξ) T (φ, θ) G(φ, θ)dω Ω Z lw,σ(φ 0, θ 0 ) = W lw,σ(ξ) T. (4) 2 (φ, θ)dω Ω The spatial correlation between G lw,σ(φ, θ) and T lw,σ(φ, θ) is also an important quantity for the interpretation of the admittance, because the admittance theory has an implicit assumption that two fields are highly correlated. Piromallo et al., (2001) presents the waveletized correlation in Cartesian basis functions for comparison between seismic tomographic images in different depths. Here, we define the spatially localized correlation C lw,σ between the waveletized quantities G lw,σ(φ, θ) and T lw,σ(φ, θ) around a point (φ 0, θ 0 ) in the spatial domain with the weighting function of W lw,σ(ξ) in the same way as in Eq. (4); C lw,σ(φ 0, θ 0 ) = W lw,σ(ξ) T (φ, θ) G(φ, θ)dω Ω W lw,σ(ξ) T 2 (φ, θ)dω W lw,σ(ξ) G 2 (φ, θ)dω Ω Ω. (5) Thus the calculated Z lw,σ(φ, θ) and C lw,σ(φ, θ) based on GMM2B and GTM090AA models are respectively shown in Fig. 1e and 1f, for l w = 8, 16, 32, and 64. Since the admittance analysis requires high correlation in the two fields, poorly correlated regions (given in Fig. 1f) on the admittance map (Fig. 1e) are masked by superpositing black color in Fig. 1g. Although C lw,σ(φ, θ) is conceptually close to the coherence analysis used in isostatic response (e.g., Forsyth, 1985, Simons et al., 2000) we used this quantity for evaluating the reliability of the admittance field. From now on, we will discuss about the admittance map based on Fig. 1g.

7 7 Results and discussion We now discuss on this way of viewing the Martian gravity and topographic fields from a wavelet standpoint in Figs. 1c and 1d. We have considered four different length scales, corresponding to spherical harmonic degrees of 8, 16, 32 and 64 respectively. We emphasize that these perspectives of the gravity and topography are novel in geophysics because we have used here for the first time the spherical wavelet filter on the Martian surface. Particular interest is the prominent signature of the Tharsis region at all of the scales examined, when we compare Figs. 1c and 1d with Figs. 1a and 1b, describing the original gravity and topographic fields. One can also discern Valles Morineris and Elysium up to l w =64. However, artificial oscillations with small amplitude in the waveletized gravity fields is appeared in many areas for l w =64, which has already been observed in the original gravity map in Fig. 1a. Admittance analysis with correlation becomes more important by excluding such areas. The reader is kindly requested to consult our web page kido/marsgrl/ for more high-resolution figures. We need to go to the admittance map for a better understanding of the lateral differences of the Martian lithosphere, because of the predictive power of admittance function (Ricard et al., 1984). In Fig. 1e and 1f, we show the 2-D maps of the localized admittance and correlation. Localized admittance study on Venus using a windowed function consisting of spherical harmonics has been developed by Simons et al., (1997), and this method has been applied to Mars by McGovern et al., (2000). The major

8 8 advantage of our wavelet approach is that all the computation procedures are carried out in the spatial domain, using a grid data structure. This allow us to handle an analysis for a selected region to be interested in with a high-resolution grid data on the Earth, and also be a powerful method for Lunar analysis, where constructing spherical harmonic coefficients is difficult due to the lack of data over a large region (Wieczorek and Phillips, 1998). Another advantage is that we can estimate the spatial reliability of the admittance map by introducing the localized correlation quantity. For this purpose we show the supercomposition of the admittance and the correlation in Fig. 1g transformed by a one-dimensional alpha-map in the correlation color map. Lithospheric strength can be estimated by a threshold wavelength of admittance drop (Ricard et al, 1984 and see also Fig. 1 in McKenzie et al., 2002). Looking at Fig. 1g, we can discern the threshold in the Tharsis region lies between l w = 16 and 32, which is interpreted as relatively rigid lithosphere. On the other hand, in the Valles Morineris admittance is still small even for short wavelengths, which implies a weak lithosphere in this region. Admittance in Tharsis region does not become zero even for l w =8, which results in degree 1 distribution of admittance for l w =8. This may caused by east-west degree 1 structure in the underlying Martian mantle circulation (Matyska et al, 1998; Zhong and Zuber 2001). On the other hand, we can see that no information can be drawn concerning the north-south dichotomy of Mars at all scales, which is clear in topographic field in Fig. 1b. This can be explained by the lack of coupling of the variations in the crustal thickness of Mars on the underlying convective circulation at the long wavelengths and the effective thickness of the elastic lithosphere

9 9 at shorter wavelengths. We can observe negative admittance for l w =16 in Utopia, Isidis, and Argyre region, which are considered as craters caused by large impacts. However Hellas, the largest crater in Mars, has no strong signal in the admittance. A possible explanation for this is due to a different infill material from those in other craters. We show the efficacy of the wavelet approach over classical spectral techniques in Fig. 1g according to the color-map scheme the black regions represent unreliability, whereas the white, red and blue regions can be considered to be reliable. From Fig. 1g we can see that there are regions of reliability even at the shortest scales. In the classical fully spectral approach, global correlation in such short scales would be too small to be used for analysis. McKenzie et al., (2002) has performed regional admittance study using a highly reliable raw orbital data. Our localized admittance value is basically compatible with their results. Even considering a possible error in the spherical harmonic gravity data we used here, mapping the admittance over the sphere yields significant information concerning the lateral variation of the Martian lithosphere, which would help us understand the underlying dynamical cause of the Tharsis bulge. We are now in a position to apply this spherical wavelet technique to the Earth, Venus and the Moon. Acknowledgments. We thank discussion with Alain Vincent. This research has been supported by the geophysics program of the US National Science Foundation.

10 10 References Forsyth, D. W., Subsurface loading and estimates of the flexural rigidity of continental lithosphere, J. Geophys. Res., 90, , Kido, M., D. A. Yuen, and A. P. Vincent, Continuous wavelet-like filter for a spherical surface and bandpass filter in spherical harmonics. submitted to Phys. Earth Planet. Inter., Lemoine, F. G., D. E. Smith, D. D. Rowlands, M. T. Zuber, G. A. Neumann, D. S. Chinn, and D. E. Pavlis, An improved solution of the gravity field of Mars (GMM-2B) from Mars Global Surveyor, J. Geophys. Res., 106, , Mallat, S., A Wavelet Tour of Signal Processing, 505 pp., Academic Press, Boston, McNutt, M. K., and R. L. Parker, Isostasy in Australia and the evolution of the compensation mechanism, Science, 199, , Matyska, C., D. A. Yuen, D. Breuer, and T. Spohn, Symmetries of volcanic distributions on Mars and Earth and their mantle plume dynamics, J. Geophys. Res., 103, , McGovern, P. J., S. C. Solomon, D. E. Smith, M. T. Zuber, G. A. Neumann, J. W. Head, Localized gravity/topography admittance on Mars, Lunar Planet. Sci., 31, abstract 1792.pdf, McKenzie, D., and D. Fairhead, Estimates of the effective elastic thickness of the continental lithosphere from Bouguer and free air gravity anomalies, J. Geophys. Res., 102, , 1997.

11 11 McKenzie, D., D. A. Barnett, D.-N. Yuan, The relationship between Martian gravity and topography, Earth, Planet. Sci. Lett., in press, Piromallo, C., A. P. Vincent, D. A. Yuen, and A. Morelli, Dynamics of the transition zone under Europe inferred from wavelet cross-spectra of seismic tomography, Phys. Earth Planet. Inter., 125, , Ricard, Y., L. Fleitout, and C. Froidevaux, Geoid heights and lithospheric stresses for a dynamic Earth, Annal. Geophysicae, 2, , Simons, M., S. C. Solomon, B. H. Hager, Localization of gravity and topography: constraints on the tectonics and mantle dynamics of Venus, Geophys. J. Int., 131, 24 44, Simons, F. J., M. T. Zuber, and J. Korenaga, Isostatic response of the Australian lithosphere: Estimation of effective elastic thickness and anisotropy using multitaper spectral analysis, J. Geophys. Res., 105, , Smith, D., G. Neumann, and R. Simpson, Mars Global Surveyor Laser Altimeter Initial Experiment Gridded Data Record, NASA Planetary Data System, MGS-M-MOLA-5- IEGDR-L3-V1.0, Wieczorek, M. A., and R. J. Phillips, Potential anomalies on a sphere: Applications to the thickness of the lunar crust, J. Geophys. Res., 103, , Zhong, S., and M. T. Zuber, Degree-1 mantle convection and the crustal dichotomy on Mars, Earth Planet. Sci. Lett., 189, 75 84, Zuber, M. T., S. C. Solomon, R. J. Phillips, D. E. Smith, G. L Tyler, O. Aharonson, G. Balmino, W. B. Banerdt, J. W. Head, C. L. McGovern, G. A. Neumann,

12 12 D. D. Rowlands, and S. Zhong, Internal structure and early thermal evolution of Mars from Mars Global Surveyor topography and gravity, Science, 287, , M. Kido, JST Corp. Domestic Res. Fellow at JAMSTEC, Natsushima-cho 2-15, Yokosuka-shi, , Japan, ( D. A. Yuen, Dept. of Geology and Geophysics and Minnesota Supercomputer Inst., Univ. of Minnesota, 1200 Washington Ave. S., Minneapolis, MN , USA, ( Received January 3, 1998; revised February 27, 1998; accepted March 31, 1998.

13 13 Figure 1. Original (a) gravity and (b) topographic field for Martian surface. Waveletized (c) gravity and (d) topography, and patially localized (e) admittance and (f) correlation for l w = 8, 16, 32, and 64 are also shown. Corresponding lengthscales for l w are 424, 212, 106, and 53 km, respectively. In (g), the admittance has been masked by black where absolute value of the correlation is extremely small.

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