Gravity/topography admittance inversion on Venus using niching genetic algorithms

Transcription

1 GEOPHYSICAL RESEARCH LETTERS, VOL. 30, NO. 19, 1994, doi: /2003gl017515, 2003 Gravity/topography admittance inversion on Venus using niching genetic algorithms Kristin P. Lawrence and Roger J. Phillips McDonnell Center for the Space Sciences & Department of Earth and Planetary Sciences, Washington University, St. Louis, Missouri, USA Received 11 April 2003; revised 6 August 2003; accepted 19 August 2003; published 9 October [1] We used niching genetic algorithms (NGAs) to invert localized Venus gravity/topography admittance over lowland regions Atalanta and Lavinia Planitiae, as well as volcanic rise Atla Regio for comparison. Assuming both top (topography) and bottom (mantle density anomalies) loads, we calculated theoretical admittance using thin elastic shell models. We inverted admittance for crustal thickness, elastic lithosphere thickness, mantle density anomaly thickness, and ratio (p z ) of mantle density anomaly to topographic load. NGA inversion provides an efficient means of finding globally optimal and sub-optimal solutions. Error analyses of all three regions show that p z is a robust estimate; there is significant trade-off between elastic lithosphere and mantle anomaly thicknesses, while crustal thickness is illconstrained. Optimal models suggest that mantle density anomalies are +1 to 2% underlying lowland regions and 3to 4% underlying Atla Regio. INDEX TERMS: 1227 Geodesy and Gravity: Planetary geodesy and gravity (5420, 5714, 6019); 3260 Mathematical Geophysics: Inverse theory; 5430 Planetology: Solid Surface Planets: Interiors (8147); 6295 Planetology: Solar System Objects: Venus. Citation: Lawrence, K. P., and R. J. Phillips, Gravity/topography admittance inversion on Venus using niching genetic algorithms, Geophys. Res. Lett., 30(19), 1994, doi: /2003gl017515, Introduction [2] In the absence of seismic data, combining gravity and topography information remains the primary way to explore the interiors of the terrestrial planets (save Earth). Previous studies employed either spatial modeling techniques [e.g., Smrekar and Phillips, 1991; Wieczorek and Phillips, 1997], or spectral admittance and coherence approaches [e.g., Forsyth, 1985; Phillips, 1994; McGovern et al., 2002] to examine subsurface planetary structure. However, both methods suffer from inherent nonuniqueness of estimating model parameters using potential fields. Few studies map out a multidimensional parameter space to find the range of acceptable solutions and correlations among estimated model parameters. Here, we introduce the application of the niching genetic algorithm (NGA) [Mahfoud, 1995; Koper et al., 1999] as an efficient means of exploring the parameter space of gravity/topography admittance spectra inversions. We apply the technique to large-scale basins on Venus and show that the ratio of mantle density anomaly load to topographic load is the only robust result of the inversion. Copyright 2003 by the American Geophysical Union /03/2003GL [3] Venusian lowland plains are often conceptually associated with mantle downwellings, though never explicitly modeled as such [e.g., Bindschadler et al., 1992; Banerdt et al., 1997]. The two regions chosen for this study are Atalanta Planitia, centered on 65 N, 165 E, and Lavinia Planitia (45 S, 335 E). To demonstrate the robustness of the NGA technique on potential field inversions, we also invert admittance at the volcanic rise Atla Regio (4 N, 200 E), which may reflect a mantle upwelling [e.g., Phillips, 1994; Stofan et al., 1995]. This provides a contrasting environment to that of the lowland plains. [4] Genetic algorithms (GAs) invert for a single optimal solution using an evolutionary framework, where evolutionary costs are associated with least-squares misfit of the model data [Goldberg, 1989; Koper et al., 1999]. Only models with low evolutionary cost will continue into future generations of the inversion simulation. A NGA is a simultaneous compound genetic algorithm, meaning that multiple populations, or demes, compete for multiple local and global error function minima [Mahfoud, 1995]. A deme is composed of a set of models, of which each model is defined by a set of parameters. In addition to mutation and crossover probabilities between generations, the NGA assigns similarity costs to models in separate demes to maintain distinct solutions. When parameters of models from separate demes are too similar, an artificially high cost (similarity cost) is added to the evolutionary cost for that model, which further reduces its likelihood of reproduction for future generations. A NGA converges upon all global and local minima if it has a sufficient number of generations and demes. An advantage of NGA is that it finds the number of global and local optima empirically rather than assuming an a priori value. NGA improves upon the standard GA by inverting for all possible solutions, global and local, of the objective error function that describes the geophysical system. NGA and GA do not require a priori assumptions of models parameters and are therefore unbiased, which is one of their great advantages. For a comprehensive explanation of GAs and NGAs, refer to Goldberg [1989] and Koper et al. [1999], respectively. [5] Multiple solutions clearly exist for gravity/topography admittance modeling, as significant trade-offs frequently exist between model parameters such as elastic lithosphere thickness and crustal thickness. Fitting a single model admittance function to an observed admittance spectrum [e.g., Smrekar, 1994; Simons et al., 1997; McGovern et al., 2002] does not address the uniqueness of the estimated model parameters. McGovern et al. [2002] carried out an extensive trade-off analysis for Martian admittance spectra by contouring a least-squares norm against two model PLA 1-1

2 PLA 1-2 LAWRENCE AND PHILLIPS: GRAVITY/TOPOGRAPHY ADMITTANCE INVERSION parameters at a time. While this method addressed some uniqueness issues, it did not provide a systematic search for acceptable solutions in a multidimensional parameter space. A multi-parameter search using a Monte Carlo technique has been applied to admittance functions in the Atla Regio region [Phillips, 1994]. Such an approach, however, can be computationally expensive, as often millions of forward solutions must be calculated. Niching genetic algorithms can drastically reduce the number of forward solutions required by applying evolutionary rules as the solutions accumulate. 2. Method [6] To test the NGA approach for Venus, we calculated observed admittance (the spectral transfer function between gravity and topography), o F l, as a function of spherical harmonic degree, l, using a spatio-spectral localization technique [Simons et al., 1997] and employing a scalable window. Following Simons et al. [1997], the window is a scalable spherical cap, with a diameter proportional to twice the wavelength under consideration, centered at the region of interest. Localizing a function with a spherical cap window estimates the regional contribution to the spectral function. [7] We define our model admittance, m F l, using an analysis of lithospheric deformation [Banerdt, 1986] that accounts for both topographic top loading and mantle density anomaly bottom loading of a thin elastic spherical shell. This total load can be expressed in terms of its spherical harmonic coefficients: lm ¼ g r cðh lm N lm Þþr w lm Nlm c þ drlm M ; ð1þ q tot where l is degree and m is order of spherical harmonic coefficients, w lm are calculated coefficients of vertical displacement, N lm are coefficients of the observed geoid at the surface [Konopliv et al., 1999], N c lm are calculated coefficients of geoid at the crust-mantle boundary, r c is crustal density, r m is mantle density, r = r c r m, dr lm are spherical harmonic coefficients of the mantle density anomaly, M is the mantle density anomaly thickness (extending downward from the crust-mantle boundary), g is gravitational acceleration, and H lm are observed topography coefficients [Rappaport et al., 1999]. In order to solve the system for lithospheric deformation exactly (increase a system of 5 equations with 6 unknowns to 6 equations with 6 unknowns), we parameterized the mantle density anomaly load as a fraction, p z, of the topographic load: p z dr lmm r c H lm : Note that this parameterization differs from the standard bottom to top load parameterization as defined by Forsyth [1985]. For consistency with our thin-shell model, we might have used Banerdt s [1986] definitions for surface and interior load to define a spherical version of Forsyth s load ratio ( f ) as a model parameter. However, use of this f generates singularities (near l = 45) in forward calculations by NGA, and so was inappropriate to use. Using our new parameterization and following Banerdt [1986], we derived ð2þ Table 1. List of Parameters Used to Calculate Theoretical Gravity Parameter Symbol Value Young s modulus E Pa crustal density r c 2900 kg/m 3 mantle density r m 3300 kg/m 3 Poisson s ratio n 0.25 gravity g 8.87 m/s 2 bulk planetary density r 5243 kg/m 3 planetary radius R 6051 km crust thickness c 4 50 km elastic lithosphere T e km mantle density anomaly thickness M km parameterization p z 5 +5 a solution for theoretical gravity in terms of four variable model parameters: crustal thickness, c, elastic lithosphere thickness, T e, M, and p z. We localized observed and theoretical gravity as well as topography [Simons et al., 1997] prior to calculating o F l and m F l. The short wavelength limit of m F l converges to the short wavelength limit of the equivalent Cartesian solution presented by Phillips [1994]. We successfully benchmarked the localized top-load only (dr lm =0) m F l spectra against previous models [Simons et al., 1997] for equivalent locations. Table 1 contains the parameters assumed in this study. [8] Each NGA inversion began by randomly producing a population of geophysical models with distinct model parameters (c, T e, M, and p z ). We calculated m F l for each model of the population and then determined the leastsquares error between m F l and o F l. The value of this error determined a model s likelihood of reproduction for future generations. From generation to generation, mutation and crossover probability statistics governed a guided search of the model space. The result of a single NGA inversion has an a model, or the model with the absolute minimum error occupying deme one. Deme two is characterized by the b model, which has the next lowest error with a low enough similarity cost to remain in the population. Subsequent demes with increasing error are represented by models g, d etc. In this manner, it is easy to invert for multiple optimal admittance models. 3. Results [9] We performed inversions for lowlands Atalanta Planitia, Lavinia Planitia, and volcanic rise Atla Regio (for comparison). Each inversion consisted of 10 generations with 10 demes per generation and 20 models per deme. The number of models and demes is empirically determined as the combination that maximizes model space and minimizes computation time. Inversions with increased generations negligibly decreased least-squares error of global and local optima. Evolutionary cost, or least-squares error, was calculated from harmonic degree 20 to 60. Degrees less than 20 contain contributions from gravitational loads that are larger than the geographical extent of the study regions. Due to spatial variation in gravity field resolution, admittance data for Atalanta Planitia above degree 60 are unreliable and for consistency the same cut-off is used for Lavinia Planitia and Atla Regio. [10] Table 2 shows the resulting best three minima of the NGA inversion and their associated model parameters and least squares error. a denotes the global minimum, while b

3 LAWRENCE AND PHILLIPS: GRAVITY/TOPOGRAPHY ADMITTANCE INVERSION PLA 1-3 and g are the two best local minima. We solved for the fractional density anomaly associated with each minimum, dr tot /r m using equation (2) and summing over all degrees and orders. As implied by the p z values, only negative dr lm result for Atla Regio and only positive values are produced for Atalanta and Lavinia Planitiae. Figure 1 represents observed and modeled admittances for these regions for the corresponding model parameters given in Table 2. [11] All models are defined by a particular combination of four parameters and an associated least-squares error. For mapping out the broader solution space (i.e., greater range of error) of Atalanta Planitia (results for Lavinia Planitia are similar but not shown), we retained all models having a least-squares error less than 10 mgals/km, or three times the mean least-squares standard deviation of our observed admittance. Our approach is not only to find the best solutions but to generate a continuous error surface by opening up the search to a larger error. The a, b, and g solutions are points on this surface. All of these models share a common value of p z of approximately 1.75, so we present only the space of the other three parameters. The same analysis applied to the other two regions also found p z to be quite stable. Figure 2a shows error as a function of c and T e for Atalanta Planitia. The application of a NGA provides an efficient, simultaneous multi-parameter admittance inversion unseen in prior studies. As such, it is important to realize that Figures 2a and 2b are not simple contour plots of two parameters in isolation (with the other two held fixed) but are a result of the NGA considering all four-model parameters simultaneously. T e has one broad minimum centered at 70 km; however, c is ill constrained throughout the trough defined by T e. We also found similar behavior for Lavinia Planitia. The error space defined by T e and M (Figure 2b) exhibits a linear trade-off between T e and M, with a global minimum occurring near M = 200 km and T e = 70 km. Figure 1. (a) Observed (solid line) and model admittances, F l, for Atla Regio. Vertical lines represent 1-s errors on the admittance. Localization centered on 4 N, 200 E. Longdash, dash-dot, and short-dash lines are distinct models that represent the global and two local minima determined by NGA inversion. Corresponding model parameters are in Table 2. (b) Same as (a) for Atalanta Planitia, centered on 65 N, 165 E. (c) Same as (a) for Lavinia Planitia, centered on 45 S, 350 E. 4. Discussion and Conclusions [12] To evaluate a more traditional loading ratio [e.g., Forsyth, 1985] than p z, we calculated (for each region), from the best-estimate parameters of c, T e, M, and p z,an l-dependent f value based on the ratio of net bottom load to net top load. Strictly, this estimate of f is only valid over the model range of 20 l 60, but approaches low- and highdegree asymptotes at harmonic bounds. At low harmonic degree, f approaches 1, implying that at long wavelengths there is mass balance between mantle density anomaly and a combination of perturbed Moho relief and surface topography. Beyond l = 20, f departs negatively from 1, indicating increasing flexural resistance to Moho deformation and mass imbalance. The f parameter asymptotically approaches p z with increasing l, indicating negligible deflection as wavelengths become small relative to T e. [13] In each regional case: (i) the associated density anomalies extend into the mantle 200 km beneath the crustal boundary, (ii) the anomalies, while contributing to overall mechanical equilibrium, are (except for low l) out of mass balance with the surface topography, and (iii) areof opposite sign and thus (except for low l) overcompensate the corresponding topographic load. Such results for topographic highs on Venus have previously been interpreted in terms of dynamic compensation and envisioned as mantle upwellings [e.g., Phillips and Malin, 1983]. By analogy, the results for Atalanta and Lavinia Planitiae can be interpreted in terms of mantle downwellings. Inferences of subsurface load buoyancy are themselves robust, since the sign of dr lm (positively buoyant or negatively buoyant) is dependent on H lm and p z (equation (2)). Positive topography and a negative value for p z (as modeled for Atla Regio) produces a negative mantle density anomaly and hence a positively buoyant load. The topography of Lavinia and Atalanta Planitiae is primarily negative; therefore, the same sign for p z results in a negatively buoyant load. The observed consistency in sign for p z implies a unique representation of the buoyancy of the bottom load. Table 2. Inversion Results for Atla Regio, Atalanta Planitia, and Lavinia Planitia Model Error (mgals/km) c (km) T e (km) M (km) p z dr tot /r m Atla a % b % g % Atalanta a % b % g % Lavinia a % b % g %

4 PLA 1-4 LAWRENCE AND PHILLIPS: GRAVITY/TOPOGRAPHY ADMITTANCE INVERSION Figure 2. (a) Model errors for Atalanta Planitia as a function of c and T e for all models with an error less than 10 mgals/km. (b) Model errors as a function of T e and M. [14] The models parameters, T e, M and p z are comparable to those depicted in previous studies [Phillips, 1994; Smrekar, 1994]. For example, optimal model results (Table 2) in this study and the results of Phillips [1994] and Smrekar [1994] all indicate T e values well above the 20 km estimate for a top-loading only model presented by Simons et al. [1997]. In addition, given a difference in model geometries, estimates of M are comparable between the studies. Phillips [1994] and Smrekar [1994] considered the depth to the subsurface load for Atla Regio rather than thickness of applied load (M) as used here. Finally, the positively buoyant mantle load we find for Atla Regio agrees with previous studies [Phillips, 1994; Smrekar, 1994]. However, these authors estimated a buoyant bottom load about five times that of the top load when fitting admittance spectra across the available spectral band, while our results do not indicate a bottom load quite as large. This discrepancy is partly due to the fact that prior studies used a different parameterization of bottom-to-top loading. [15] Our results for Atla Regio parameters can be interpreted as a significant mantle upwelling, with a mantle density anomaly of 3%, depending on model parameters. For a canonical volumetric coefficient of thermal expansion of K 1, this implies a rather large temperature contrast of K. Thus, it is likely that there is an upper mantle compositional component to the compensation or a component of crustal thickening. As a result of using this static model of compensation, the elastic lithosphere thickness estimates that best represent Atla Regio admittance curves are most likely over-estimated. In the absence of dynamic compensation, the large mantle density anomalies (3%) must be supported by a thicker elastic lithosphere [Banerdt, 1986]. [16] The robust p z values for Atalanta and Lavinia Planitiae ( 1.8 and 2.2, respectively) indicate positive mass loads in the upper mantle. The NGA globally optimal solution for T e centered at 70 km contrasts with slightly smaller values found for other regions on Venus [Phillips et al., 1997]; this may reflect the environment of a cold, young mantle downwelling. Lavinia and Atalanta Planitia have mantle density anomalies of 1% and 2%, implying, for an end-member interpretation, negative thermal anomalies of 350 and 700 K, respectively. The larger value is bordering on extreme, and a compositional and/or crustal component to the density anomalies is probable. [17] This study demonstrates that the NGA is a powerful technique for inverting gravity/topography admittance functions due to its systematic search of a multi-dimensional model space. In the context of a four-parameter model, the method presented quantifies mantle density anomalies for Atalanta and Lavinia Planitiae as 2% and 1%, respectively, suggesting significant positive mass anomalies in the upper mantle that probably have both thermal and compositional or crustal components. In contrast, for Atla Regio we produce results consistent with mantle upwelling, with a density anomaly of 3%. Future geodynamical studies will benefit from meaningful constraints from robust parameter estimates obtained by NGA inversion. [18] Acknowledgments. This work was supported by grant NAG from the NASA Planetary Geology and Geophysics Program. We thank Mark Simons for use of his localization code and Keith Koper for use of his NGA code. 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5 LAWRENCE AND PHILLIPS: GRAVITY/TOPOGRAPHY ADMITTANCE INVERSION PLA 1-5 Rappaport, N. J., A. S. Konopliv, and A. B. Kucinskas, An improved 360 degree and order model of Venus topography, Icarus, 139, 19 31, Simons, M., S. C. Solomon, and B. H. Hager, Localization f gravity and topography: Constraints on the tectonics and mantle dynamics of Venus, Geophys. J. Int., 131, 24 44, Smrekar, S. E., Evidence for active hotspots on Venus from analysis of Magellan gravity data, Icarus, 112, 2 26, Smrekar, S. E., and R. J. Phillips, Venusian highlands: Geoid to topography ratios and their implications, Earth Planet. Sci. Lett., 107, , Stofan, E. R., S. E. Smrekar, D. L. Bindschadler, and D. A. Senske, Large topographic rises on Venus: Implications for mantle upwelling, J. Geophys. Res., 100, 23,317 23,327, Wieczorek, M. A., and R. J. Phillips, The structure and compensation of the lunar highland crust, J. Geophys. Res., 102, 10,933 10,943, K. P. Lawrence and R. J. Phillips, McDonnell Center for the Space Sciences & Department of Earth and Planetary Sciences, Washington University, St. Louis, MO, USA. (portlek@levee.wustl.edu)