JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 96, NO. BI, PAGES 301-315, JANUARY 10, 1991

Comparison of Observed and Predicted Gravity Profiles Over Aphrodite Terra, Venus

MARTIN T. BLACK

Astronomy Program, University of Maryland, College Park

MARIA T. ZUBER

Geodynamics Branch, NASA Goddard Space Flight Center, Greenbelt, Maryland

DAVID C. MCADOO

National Geodetic Survey, Charting and Geodetic Services, National Ocean Service, NOAA, Rockville, Maryland

We compare observed Pioneer Venus orbiter (PVO) gravity profiles over Aphrodite Terra to profiles predicted from models of thermal isostasy, mantle convection, and Airy compensation. Similar approaches are used in order to investigate how well the models can be distinguished with the PVO data. Topography profiles across Aphrodite are compared to model spreading ridge profiles in order to further assess this model. Airy compensation depths and convection layer thicknesses are greater under eastem Aphrodite than western Aphrodite. Compensation depths in the east are greater than most estimates of lithospheric thickness, suggesting that this part of the ridge is dynamically supported. In parts of westem Aphrodite, the spreading ridge model gravity provides a better fit to the data than either Airy compensation or man fie convection. Best-fit spreading rates are between 0.3 and 1.6 cm/yr. Airy compensation and mantle convection cannot be distinguished in most places using only PVO data.

INTRODUCTION

Aphrodite Terra is the largest highland area on Venus. It is a long linear topographic feature running approximately east-west in the equatorial region from about 60 ø east longitude to 210 ø east longitude and reaching a height of over 4 km above the mean planetary radius [Pettengill et al., 1980]. Within Aphrodite are several distinct uplands, including Ovda Regio (spanning the range 80 ø < longitude < 110ø), Thetis Regio (120 ø < longitude < 140ø), and Atla Regio (195 ø < longitude < 210 ø) (Figure 2). Because of its size and distinctive morphological character, an understanding of the tectonics of Aphrodite is essential for an understanding of the global tectonics of Venus.

Two hypotheses for the origin and nature of Aphrodite have been put forward in the literature. The first, that Aphrodite is the surface expression of rising mantle plumes, has been advanced by several workers [Phillips et al., 1981; Phillips and Malin, 1983; Kiefer et al., 1986; Kiefer and Hager, 1988 ]. The high degree of correlation between gravity and topography on Venus is un- like the terrestrial case, where the long-wavelength geoid (spher- ical harmonic degrees 4-9) is highly correlated with subduction [Hager, 1984]. Isostatic compensation models cannot account for the observed relationship between the long-wavelength grav- ity and topography on Venus [Kiefer et al., 1986], suggesting that mantle convection is responsible for dynamically maintaining the anomalies. It has also been proposed that Aphrodite is a diver- gent plate boundary, analogous to terrestrial oceanic rises. Kaula and Phillips [1981] examined this hypothesis and concluded that plate tectonics is not an important heat transfer mechanism on

Copyright 1991 by the American Geophysical Union.

Paper number 90JB01853. 0148-0227/91790JB-01853505.00

301

Venus. Others [Crumpier et al., 1987; Head and Crumpier, 1987] have argued that Aphrodite displays morphological characteristics consistent with the divergent plate boundary hypothesis.

In this paper we develop an approach to the modelling of Pio- neer Venus orbiter (PVO) gravity profiles which allows us to test several models of isostatic compensation in Aphrodite in a uni- form way. Our objective is to determine whether we can distin- guish between the models using the PVO gravity and topography data and, if so, to determine which model best fits the data. We

present the results of modelling 96 orbital arcs over Aphrodite using models of crustal spreading, mantle convection, and local compensation. It is found that in parts of western Aphrodite the spreading ridge model provides a better fit to the PVO gravity data than the other models do, though most topographic profiles across Aphrodite do not resemble those expected for a spreading ridge like the Mid-Atlantic Ridge. Elsewhere in Aphrodite the three models cannot be distinguished using only the PVO gravity and topography data. We find evidence for dynamic support of topography east of 140 o .

DATA

The topography data were taken from the Project Magellan Topography Model (PMTM), Version 3.0, a 1/8 ø x 1/8 ø grid of the PVO radar altimeter data. This grid spacing, which is about 13 km at the equator, is smaller than the altimeter footprint size at all latitudes [Pettengill eta/., 1980], so the horizontal resolution of the underlying altimetry data is everywhere less than the PMTM grid spacing. The vertical uncertainty in the altimetry data is about 200 m, taking into account both statistical errors and systematic orbit determination errors [Pettengill et al., 1980].

The gravity data were derived from PVO line-of-sight (LOS) velocity measurements. The instantaneous velocity of PVO rela-

302 BLACK ET AL.: GRAVITY PROFILES OVER APHRODITE TERRA

rive to a receiving antenna on Earth, projected onto the Earth-PVO direction, was obtained from Doppler tracking of a radio signal between PVO and the ground station [Sjogren et al., 1980]. Resid- ual LOS velocities were obtained by removing the effects of the orbital motions of Earth and Venus, the rotation of Earth, gravity of the Sun and other planets, and the central mass component of the orbital motion of PVO [Phillips et al., 1979; Sjogren et al., 1983]. Time derivatives of the velocity residuals give the residual LOS accelerations, which are attributed to local variations in grav- ity along the spacecraft orbit. A velocity determination was made every 5 s during the 30 min before and after periapsis [Sjogren et al., 1980], which corresponds to a maximum along-groundtrack spacing between data points of about 45 km. Bills et al. [1987] concluded that although the acceleration measurements in adja- cent orbits are consistent at a 1-2 mgal level, a formal error of 3 mgal more accurately reflects the accuracy of each measurement when errors in the determination of the spacecraft orbit are taken into account. The limiting noise source in the individual veloc- ity determinations is the modulation of the radio signals by the interplanetary plasma, which can introduce uncertainties in the gravity determinations of up to about 3 mgal, though the noise along individual orbital arcs is often much less than this [Phillips et al., 1979].

We have subtracted a harmonic background gravity field of degree and order 4 [Bills et al., 1987], corresponding to a wave- length of about 9500 km, from the observed LOS accelerations in order to remove unmodelled dynamic orbit effects from the data. These effects are largest at long wavelengths, i.e., on the order of the planetary circumference, and less important for shorter wave- lengths. Orbit errors are small for wavelengths of a few thousand kilometers and less, which are of interest for this study. In order to check whether any unmodelled dynamic effects remained in the data, several of the passes were analyzed using the ORBSIM orbital simulation program [Phillips et al., 1978] and a simple model of local compensation. These results were compared with the results of our Airy compensation model (discussed in the next section) for the same passes and found to be consistent within the data uncertainty.

MODEI•

In this section we present the tectonic models used to produce the predicted LOS gravity profiles over Aphrodite Terra. We consider three models for the relationship between gravity and topography: (1) crustal spreading (thermal isostasy); (2) local

TABLE 1. Isostatic Models

Model Input Adjustable Parameters

Thermal isostasy Location of ridge Spreading rate, ridge axis, distance from half-width (vs, w) ridge axis

Airy compensation PVO topography Depth of compensation (t)

Dynamic compensation (free-free and free-rigid boundary conditions)

PVO topography Convecting layer thickness (z; y)

compensation; and (3) convection in the mantle. A model line- of-sight gravity profile is produced for each of these models and fit to the observed gravity profile by adjusting parameters, as described below. Table I summarizes the models and their

adjustable parameters. Figure 1 presents schematic drawings of the three models tested.

(b) Crustal Spreading

Reference Level

Airy Compensation

(c) Upper Boundary

Lower Boundary

Mantle Convection

Fig. 1. The three models of isostatic compensation tested in this analysis: (a) Crustal spreading, where v, is the half-spreading velocity and w is the ridge width. (b) Local compensation, where t is the depth of compensation. (c) Mantle convection, where d is the convection layer thickness.

Crustal Spreading

In a plate tectonic environment, new, hot material is added to lithospheric plates at spreading ridges and moves away from the ridge as the crust spreads outward. The lithosphere cools and contracts as it moves, causing the plate to subside with age. This produces a relationship between topography and age as well as between geoid height and age. If the ridge maintains a constant spreading rate, distance from the ridge is equivalent to age. For a ridge described by the simple half-space cooling model [Turcotte and Oxburgh, 1967; Parker and Oldenburg, 1973; Davis and Lister, 1974], the topography, b(x), and the geoid height, h(x), are given by [Haxby and Turcotte, 1978]

= 2•rGp,•a (T,,• - T.) n

X 1+ (pm--Pa) a' (2)

BLACK ET AL.: GRAVITY PROFILES OVER APHRODITE TERRA 303

where p,,, (=3330 kg m -a) is the density of the mantle, a (=3.1x10 -5 øC-a) is the volume coefficient of thermal expan- sion, T,,, (=1365 øC) is the temperature of the manfie, T• (=464 øC) is the surface temperature, p• (=63 kg m -a) is the density of the atmosphere, n (=8.0x10 -7 m 2 s -•) is the thermal diffusiv- ity of the lithosphere, G is the universal gravitational constant, g (=8.87 m s -2) is the acceleration of gravity at the surface, v., is the half-spreading rate of the ridge, w is the ridge width, and x is the distance from the ridge axis. We have introduced the parameter w in order that the Fourier integrals converge. The ridge width is the distance from the spreading ridge axis to the oldest crust that formed at the ridge axis; i.e., where the topographic expression of the ridge is defined to be zero. The ridge width is indepen- dent of the length of the gravity profile used in the parameter estimation and may be much larger than the profile. The sur- face temperature and atmospheric density are averages of in situ measurements from the Venera 7-12 spacecraft [Avduevskiy et al., 1983]. The Venus surface gravity is calculated from the known physical parameters of the planet. We have adopted the terres- trial values for the other parameters under the assumption that the composition of the Venusian lithosphere and the temperature of the mantle are similar to Earth. Increasing the mantle temperature by 125 øC, as is suggested by some convection and mantle flow models [Turcotte et al., 1979; Kaula and Phillips, 1981; Stevenson et al., 1983; Phillips and Malin, 1983; Sotin et al., 1989b], has a negligible effect on our model results. On Earth the depth of oceanic crust older than about 70 Myr diverges from that given by (1) and is better described by the plate model [Parsons and Sclater, 1977], which assumes an isothermal bottom boundary, though there are regional variations in the subsidence rate. We have used the half-space model in our analysis because its formu- lation is simpler than that for the plate model and our results are not significantly changed by considering only data from young lithosphere (as determined by our spreading velocity estimates).

In order to find the LOS gravity at spacecraft elevations, h(x) must be upward continued and converted to an expression for gravity rather than geoid height. This is accomplished most easily by working in the frequency domain. The Fourier transforms of the vertical gravity go (z) and the geoid height h(x) are simply related by [Chapman, 1979]

= allH() (3)

where uppercase letters indicate the Fourier transform of the corresponding lower case function and k=2a-/A is the wave number corresponding to a wavelength A. Upward continuation to an elevation a is accomplished by multiplication by an exponential term

(4)

The Fourier transform of the horizontal component of gravity, gn (x, a), is the Hilbert transform of Go (k, a) [Chapman, 19791

G•, (k, a) = i sgn (k) g (5)

The inverse transforms then yield expressions for vertical and horizontal gravity at an elevation a. For the half-space cooling model we find

(6)

2pma(Tm -T•)] = _4ap - +

x [tan_• (x) 1 •(w+x) I _•(w-x) • -]tan- +•tan (7)

The adjustable p•ameters for this model •e v,, the half-spreading velocity, •d w, •e •dge width.

Local compensation

To model the gravity due to locally compensated topography we have used the model of simple Airy compensation as presented by McKenzie and Bowin [1976]. In this model the mass anomalies due to topography variations are compensated by variations in the thickness of a constant density crust in such a way that the mass per unit area is constant everywhere. The frequency response (or admittance) between gravity and topography for this model, upward continued to an elevation a, is

Z (k)= 2a'G (pc - p,)e -k• (1- e -kt) (8)

where pc (=2800 kg m -z) is the crustal density and t is the depth of compensation. The adjustable parameter in this model is t.

Mantle Convection

McKenzie [1977] and Parsons and Daly [1983] used a sim- ple model of mantle convection to predict admittances between gravity and topography. Their model assumed uniformly viscous Newtonian convection in a mantle layer of thickness d. Effects of inertia, self-gravitation, and sphericity were ignored. The model is useful because it yields analytic solutions which are readily applied to calculations of gravity anomalies, while avoiding the complications of numerical modelling. Parsons and Daly [1983] derived admittances by convolving topographic and gravity ker- nels with simple structure functions for temperature. By using an assumed temperature structure rather than solving the coupled heat transport equation, they were able to derive analytic expres- sions for admittance. They found that the admittances are most sensitive to the temperature structure of the upper thermal bound- ary layer. The actual temperature structure of Venus is not known, so this simple approach is appropriate for our study. While the model admittances calculated here are probably overestimated at long wavelengths, the discrepancies are on the order of 10% or less for wavelengths shorter than the convection layer thickness [Parsons and Daly, 1983, Figure 7], showing that the analytic formulation adequately reproduces the effects of the top boundary layer on the admittance.

Modelling of the terrestrial geoid indicates that Earth's lower mantle is about 300 times more viscous than the upper mantle [Kiefer et al., 1986]. In contrast, Kiefer et al. [1986] can match the observed admittances on Venus with whole-mantle, uniform viscosity convection, though the models allow the lower mantle to be up to about 10 times more viscous than the upper mantle [Kiefer and Hager, 1988]. Constant viscosity convection results

304 BLACK ET At,.: GRAVITY PROFILES OVER APHRODITE ]"ERRA

in higher admittances than convection in which viscosity increases with depth. The model used here will thus tend to overestimate the admittance if viscosity in Venus's mantle does increase with depth, resulting in an underestimation of the convection layer thickness. Since the change in viscosity appears to be small, the assumption of uniform viscosity does not introduce large errors into our analysis.

For a temperature distribution T(k,s) which varies with depth s in a convecting layer of thickness d as

(9)

where Ti is a constant, the admittance is [Parsons and Daly, 1983, equation (C7)]

Z (k) = 2a'G (p,• - p,•) [• + (.e-•a •r(l+e-•a)] • - i• +/c•7 5 (10) where

•r 3 + 3a'k2d 2 _ 2k3daA

•' = ( 71'2 '"[- k2fl•2) 2 (11) a -a + 3a'k2d 2 _ 2kadaB

½ '-- ( •I"2 '"[- k2a2)2 (12) The constants A and B are determined by the boundary conditions at s=0 and s=d. A "free" boundary condition corresponds to a free-slip boundary on which shear stresses vanish, and a "rigid"

boundary condition corresponds to a no-slip boundary. Both free and rigid boundary conditions are considered for the lower boundary. For free-free boundary conditions, A=B=0. For free- rigid boundary conditions,

a'cosh kd A = (13)

(sinh kd cosh kd - kd)

B = (14) (sinh kd cosh kd - kd)

The results for a model with a rigid upper boundary were found to be intermediate to those of the free-free and free-rigid models, so they are not presented here. The adjustable parameter for the convection model is d, the thickness of the convecting layer. In what follows, the free-free convective layer thickness is denoted by z and the free-rigid convective layer thickness by y.

METHOD OF ANALYSIS

In a study of PVO and Arecibo radar data, Crumpier et al. [1987] divided western Aphrodite Terra into eight domains bounded by topographic and radar backscatter linearions which .they called cross-strike discontinuities (CSDs). Within the frame- work of these CSDs, Crumpler and Head [1988] reported a trend of bilateral topographic symmetry parallel to the CSDs. The cen- ters of bilateral symmetry are linear ridge segments, interpreted by Crumpler and Head [1988] to be analogous to terrestrial mid- ocean ridge axes. The centers of bilateral symmetry are offset

30

15

-3

-45

CSD DOMAINS--APHRODITE TERRA, VENUS

60 90 120 150 180

LongRude (Degrees)

Fig. 2. The nine CSD domains superimposed on a contour map of Aphrodite Terra [after Crumpier and Ilcad, 1988]. The solid lines (domains 1-8) are the CSDs and centers of bilateral symmeuy selected by Crumpier and Head [1988]. The dashed line marks the position of the ridge axis chosen by us for domain 9. The ground tracks of orbits 437 and 475 are also marked. Only those data which fell wilhin domains 3 and 8, respectively, were used in the analysis of these two orbits. The CSDs trend roughly NW-SE and the centers of bilateral symmetry trend roughly SW-NE.

BLACK ET AL.: GRAVITY PROFILES OVER APHRODITE TERRA 30_5

, ,

Domain

TABLE 2. Location of PVO Orbits

Number of Longitude Range PVO Orbits

West East

1 12 59.9 71.3

2 3 73.5 75.7

3 18 76.6 92.0

4 4 94.3 98.O

5 8 99.7 106.4

6 16 114.3 131.1

7 7 130.7 137.5

8 6 139.9 146.5

9 22 161.3 194.7

along the CSDs. We have taken their domains, along with a ninth one which we have defined in eastern Aphrodite, as a convenient framework for our analysis of PVO gravity and topography pro- files over Aphrodite (Figure 2). Since our analysis was done, Crumpler and Head [1989] have identified more CSDs in the area of domain 9.

Each PVO orbit which crosses Aphrodite was assigned to the domain in which it crosses the "ridge axis," as defined by the position of the center of bilateral symmetry. The orbital data used in this analysis were taken from orbits which passed through the area bounded by east longitudes 50 ø and 200 ø and latitudes -40 ø and 20 ø (except for domain 9, where all data were south of 0ø). In every case, only that part of an orbit whose groundtrack fell within the assigned domain was used to construct the gravity and topography profiles for the analysis of that particular orbit. The gravity data were projected onto a line perpendicular to the ridge axis to create the gravity profile for the analysis. The LOS angle and azimuth were calculated for each gravity datum in order to properly calculate the model LOS gravity at each point. For each gravity datum a corresponding topography value was obtained by adopting the value of the PMTM grid point closest to the

corresponding subspacecraft point. A regularly spaced topography sequence, with a sampling interval of 20 km in the direction normal to the ridge axis, was created by projecting each of these topography values onto the nearest point in this ridge-normal series. The position errors introduced by this approximate method of assigning topography values are smaller than the footprint size of the PVO altimeter [Pettengill et al., 1980], so are not significant. Projecting the gravity and topography data onto a line perpendicular to the ridge axis compensates for the fact that the orbital tracks cross Aphrodite at an oblique angle. After the LOS gravity profile and corresponding topography profile were constructed, each model was fit to the data by one of two least squares methods.

The half-space cooling model was fit to the data for each pass by calculating the model vertical and horizontal gravity components at each data point position along the PVO gravity profile using (6) and (7). The two components were projected onto the LOS direction and summed to give the model LOS gravity at this point. A model gravity profile was produced for all combinations of the two adjustable parameters within the ranges 0.1 cm yr -1 _< v _< 20.0 cm yr -• and 102 km _< w _< 104 km. The standard error of estimate of the fit, S (v, w), was calculated for each model profile; the best-fit parameters produce the minimum value of S (•, vo) [Spiegel, 1975, p. 262]. The observed topography profile was not used in the gravity fitting procedure because the model contains implicitly the ideal square- mot-of-distance topography profile (equation (1)). Because the observed topography was not used to construct the model gravity, it can be used as an independent assessment of the spreading model. A model topography profile was constructed for each pass by substituting the best-fit values of •8 and w into (1) and calculating the model elevation at the position of each topography datum. The standard error of estimate of the fit of this model

profile to the topography data profile, S(b), is a measure of how well the predicted profile fits the observed data. The half- space cooling model used here assumes that each spreading ridge segment has a constant profile and maintains a constant spreading velocity. On Earth, the topography predicted by the half-space model diverges from observed oceanic topography for lithosphere older than about 70 Myr [Parsons and Sclater, 1977]. For many of the PVO passes we analyzed, the length of the data profiles included in the fit is short enough that only lithosphere younger

Domain

1

2

3

4

5

6

7

8

9

Airy Compensation

t•

101

175

82

70

56

25

63

139

153

6t, km

12

18

5

6

6

6

19

30

18

'FABLE 3. Parameter Fits for Aphrodite . i , ,

Free-Free Free-Rigid Convection Convection

z, 6z, y, km km km km

415 45 940 54

583 60 1208 101

342 15 907 23

288 19 775 38

303 21 906 36

128 27 464 37

282 48 811 99

467 70 1071 121

481 46 1095 77

Spreading Ridge

rs, 6rs, w, cm/yr cm/yr km km

1.2 0.1 7769 554

1.1 0.1 8467 993

0.5 0.1 3011 93

0.9 0,1 6975 557

3.6 0.5 6125 761

5.8 1.7 1444 236

3.8 1.2 3629 788

1.1 0.8 7033 959

3.0 0.6 886 198 ,

306 BLACK ET AL.: GRAVITY PROFILES OVER APHRODITE TERRA

than this is included, given the estimated spreading velocities. The fits are not significantly changed by truncating those passes with longer profiles.

The Airy and convection models were fit to the data for each pass using digital finite impulse response (FIR) filters [Oppenheim and Schafer, 1975, p. 237ff]. For each model, the appropriate ad- mittance ((8) or (10)) was low-pass filtered and sampled at each data point position along the PVO gravity profile to create a dis- crete representation of the frequency response. A complex fast

nine domains shown in Figure 2. Table 2 lists the locations of these passes, grouped by domain. Column 2 lists the number of passes in each domain. Columns 3 and 4 list the longitude ranges of the ridge-crossing points of the orbits in each domain.

Table 3 contains a summary of the parameter fits. For each model the average of the best-fit values of the adjustable param- eter(s) in each domain is listed along with the associated uncer- hainties.

The best-fit Airy depths of compensation for each pass are Fourier transform (FlaT) algorithm was used to calculate the dig- plotted versus the ridge-crossing longitude of the pass in Figure 3. ital FIR filter coefficients. This filter was convolved with the The fits for which S(t)<3.0 mgal are denoted by filled circles and

topography data series to produce the model gravity at this data point position. The model gravity was projected onto the LOS direction to give the model LOS gravity at this point. This proce- dure was carried out at each point along the data profile to create a model LOS gravity profile. A standard error of estimate was calculated for the fit of each model profile to the data profile. The standard errors of estimate for the Airy, free-free convec-

the fits for which for which S(t)>3.0 mgal are denoted by open circles. The 3.0 mgal demarcation value was chosen because it represents the degree of uncertainty in the data, as explained in the next paragraph. The maximum upper bound on t is 500 km because parameter space was only searched this far to find the best-fit value, with the rationale that any compensation depth larger than this would not be physically meaningful. The best-fit

tion, and free-rigid convection models are given by S(t), S(z), and flee-free convection layer depths are shown in Figure 4 and the S(y), respectively. The best-fit parameters for each model for each best fits for free-rigid convection layer depth are given in Figure pass are those that minimize the standard errors of estimate. The 5. In Figures 4 and 5, as in Figure 3, filled circles denote fits adjustable parameters for each model were varied over the fol- lowing ranges (for reasons discussed later): 5 km _< t _< 500 km, 25 km_< z _< 1000 km, 25 km <_ •t _< 2000 km.

RESULTS

with standard errors less than 3.0 mgal and open circles denote fits with standard errors greater than 3.0 mgal. The maximum upper bounds on z and y are 1000 km and 2000 km, respectively, with the exception of the one or two passes on each plot for which the best-fit value was larger than this limit. The best fits for half-spreading velocity and ridge width are shown in Figure

The analysis described above was carried out on 96 passes of 6. Again, the fits for which $ (v, w)<3.0 mgal are denoted by PVO gravity data that crossed Aphrodite Terra through one of the filled circles and those for which $ (v, w)>3.0 mgal are denoted

500

400

:300

200

100

0

- o S(t)>3.0 mgal

- ß S(t)<3.0 regal

T

60 80 100

I ! I I I I

120 140

longitude 160 180 200

Fig. 3. Airy depth of compensation versus longitude. Filled circles are fits with S(t)<3.0 mgal; open circles are fits with S(t)>3.0 mgal.

1000 • 8OO

6OO

400

2OO

I I I

o S(z)>3.0 mgal ß S(z)<3.0 mgal

, I , 60 80 100 120 140 160 180

longitude

Fig. 4. Free-free layer thickness versus longitude. Filled circles are fits with S(z)<3.0 regal' open circles are fits with S(z)>3.0 mgal.

2OO0

-•1500

1000

500

-1 o S(y)>3.0 mgal ß S(y)<3.0 mgal

i I I I I

6O 80 100 120 140 160 180

longitude

Fig. 5. Free-rigid layer thickness versus longitude. Filled circles are fits with S(y)<3.0 mgal; open circles are fits with S(y)>3.0 mgal.

200

[ --

200

308 BLACK ET AL.: GRAVITY PROHLES OVER APHRODITE TERRA

2O

10000

SO00

6000

•)4000

2000

o

- o StY,w)>3.0 mgal - ß Sty,w)<3.0 mgal

80 100 120

longitude

I 1

140

' I '

160

I I i

180

I I

60 80 100 120

longitude

I I I t t

o S(v,w)>3.0 mgal

ß S(v,w)<3.0 mgal

'1

200

Fig. 6. (a) Half-spreading velocity versus longitude. Filled circles are fits with S (l•, w)<3.0 mgal; open circles are fits with ,5' (v, zv)>3.0 mgal. (b) Ridge width versus longitude. Filled circles are fits with $ (v, w)<3.0 mgal; open circles are fits with S (v, iv)>3.0 mgal.

140 160 180 200

BLACK ET AL.: GRAVrrY PROHt.ES OVER APHRODITE TERRA 309

_ _

o

_ o _ o

10-- o o -- 0 0 --

Oq• o _

o f<o oo o o o o o _ _

o o o _ o o o _

o

5-- o o _ o•O o o

_ O0 O0 0 (•0 0 - ....... 9. .......... ...ø•o..O. o ......... .ø..•.o ........................ :o .......................

•0 0 0 0

0 I,,,I,,,I,,,I,,,I,,,I,,,I,,, 60 80 100 120 140 160 1BO 200

longitude

15

10

1111111111111111111111111111 _

0 0 -

0 0 - o

o o o

o ooJ o• - o

o o o - o o o o o o_

o

- 0 0 O0 O0 - 5-- 6•o o o o --

- o o9%00 ø_ ø øøø• o - i .......... ; ......... ...... .... ........ .............. øo o o• •I• o _ _

0 I'''l'''l'''lf ,,,I,,,I,,,I,,, 60 80 100 120 140 160 180 200

longitude

10

o o o - o - o

- o o - oOO o

o

_ o o ø ø - _ o o øo o -

o

o o o_ - o o o o _ •) o o _

o o 0 __ co C9o oO --

(b Oo o ø co o - o o

o o ., Oo o _• o

- øo o d • • 'ø•o - _ o• o _

60 80 1DO 120 140 160 180 200

longitude

15

(d)- _

-

-- o o -

o Ooo o _

o

_0•0 0 J 0 0 _ - 0 0

O 0 0 0 -- 0

0 0

_ o o f o o o oo _ ....... •,. .... ,.e-oo• ................... 9. ..... o-.o ...................... 9. .... • ............... 0o_ O0 0 •0 0 -

- Oo TM OOoøo •ø•Oo•_fo o•o - _ 0 o • •od o•u _

I, ,,-,•1, ,o•ø?•o,, [•-,, t,,,, I,,, I , , , 60 80 100 120 140 160 180 200

longitude

Fig. 7. Standard errors of estimate of (a) Airy depth, (b) free-free depth, (c) free-rigid depth, and (d) the spreading model versus longitude. The dotted line in each plot marks the 3 mgal error level (see text).

by open circles. The upper limits on v• and w were taken to be 20 cm/yr and 10000 km, respectively.

The standard errors of estimate for the best fit of all four models

to each pass over Aphrodite are plotted versus the ridge-crossing longitude of each pass in Figures 7a-7d. The horizontal dotted line marks the 3 mgal error level. Error values below this line indicate model gravity profiles which differ from the observed profile, in an rms sense, by an amount less than the inherent uncertainty of the data. Figure 8 shows S(b) versus longitude for the best fit of model spreading ridge topography to the observed topography of each pass. The horizontal dotted line marks the 200 m error level. Values of S(b) below this line indicate passes whose topography profiles differ by an amount less than the data uncertainty from the profile expected for the spreading ridge producing the best fit to the observed gravity profile for that pass. A spreading ridge would be expected to show some variation from an ideal profile due to contributions to the topography from other sources, such as crustal thickness variations. Grimm and Solomon [1989] found that profiles across the Mid-Atlantic Ridge extending to an average distance of 1500 km from the ridge axis, comparable to the lengths of our Aphrodite profiles, show an rms variation from the best- fitting spreading ridge profiles of up to about 800 m, though most of their profiles showed variations of less than 600 m. If we take a variation of 600 m to be applicable to a venusian spreading ridge and add it quadratically to the 200 m data uncertainty, we find that this variation dominates the measurement uncertainty, so the total uncertainty is about 600 m, marked by the horizontal dashed line in Figure 8. Values of S(b) below this line indicate

passes whose topography profiles, within the accuracy of the data, fit the spreading model as well as profiles across the Mid-Atlantic Ridge. It can be seen that, in general, the topography does not fit the spreading model as well as profiles across the Atlantic. Sotin et al. [1989b] suggest that the failure of the observed topography to fall off as age -m could be due to crustal thickness variations of about 15 km as a function of age with the thicker crust in the central part of the ridge, a ridge jump [Mammerickx and Sandwell, 1986], or tectonic processes unrelated to crustal spreading. Head and Crumpler [1990] suggest that Ovda and Thetis Regiones are plateaus similar to Iceland resulting from the presence of elevated upper mantle temperature, which produces thicker crust and thus increased isostatic topography. Our analysis has not allowed for crustal thickness variations.

The results for two representative PVO passes over Aphrodite are shown in Figures 9 through 11. The ORBSIM model results for pass 475 (in domain 8) are shown in Figure 9 along with both the filtered (degree and order 4 gravity field subtracted from the data) and unfiltered PVO LOS accelerations for the same

pass. The ground track for pass 475 is shown in Figure 2. The model profiles are consistent with each other within the 3 mgal uncertainty in the data. The removal of the background field from this pass has removed almost 10 mgal from the signal and gives a residual profile very similar to that obtained using the ORBSIM program. The observed topography profile for pass 475 is plotted along with the model spreading ridge topography in Figure 10a. The observed and model gravity profiles for pass 475 are plotted in Figure 10b. Topography and gravity profiles for pass

310 BLACK ET AL.: GRAVITY PROFILES OVER APHRODrrE TERRA

o o o

o o - o o 0(•) o o o

o o• b o o øo o • o • o o o o

- Ooo o - o o o oo o oO o 00 Oo •o oo• o •o o øo _ - o øo o o • oo o o oo o _Q•o__

• ¸ 0 0

....... t ....... , ....... i ...... i ...... t ....... i ...... i ...... i ...... t ....... i ...... i ...... i ....... t ...... i ...... i ...... : ....... t ...... i ...... i ....... i ...... t ...... i ...... : ....... i ...... t ...... i ....... , ....... i ...... 60 80 100 120 140 t60 180 200

longitude Fig. 8. Standard error of estimate of topography versus longitude for fits of spreading ridge topography to data. The dotted line marks the 200 m error level and the dashed line marks the 600 m error level (see text).

437 (domain 3) are plotted in Figures 1 l a and 1 lb, respectively, with the topography here. This effect is also present with the and its ground track is marked in Figure 2. Note that both the removal of a degree and order 2 field. vertical and horizontal scales are different in Figures 10b and 1 lb. The vertical scales are the same in Figures 10a and 11a, but the horizontal scales are different. The observed topography does not resemble the model spreading ridge topography in either case. All the compensation models fit the data for pass 475 with standard errors less than 3.0 mgal; none of the models do for pass 437. There is no significant difference between the fits for pass 475. The difference in standard errors of fit between the best-fitting model (Airy compensation) and the worst-fitting model (free-rigid convection) is only about 0.8 mgal. The spreading model fits the pass 437 data significantly better than do the other models, though it does not fit well. S (v, w)=3.41 mgal, more than 3 mgal less than the standard error for the next-best-fitting model (Airy compensation).

Several aspects of the results bear mention. None of the models fit the data east of 180 ø (Figures 3-8). It is probable that both Atla Regio (the topographic high lying on the equator between 195 ø and 210ø; Figure 2) and the rolling uplands to the north of this part of the ridge, which are morphologically different from Aphrodite [Ehmann and Head, 1983], are contributing significantly to the gravity here in such a way that the gravity signal cannot be ascribed to a single simple process.

In the longitude range 121ø-131 ø in Thetis Regio, the removal of the degree and order 4 field from the data has reduced the magnitudes of the gravity anomalies to close to zero, indicating that the sources of these anomalies are long-wavelength (of order 10000 km), and presumably deep, processes not closely correlated

In the central part of Ovda Regio (88 ø < longitude < 106 ø) there is a discrepancy in the fits for t, z, and y between the lower- numbered PVO orbits and the higher-numbered orbits that crossed the ridge with almost the same ground tracks about a year later. This discrepancy has been identified in other studies of Aphrodite gravity [Sotin et al., 1989a], but has not been explained. It could be due to a systematic problem in the reduction of the raw LOS data. The discrepancy does not affect any of our conclusions.

There is a sharp increase in Airy compensation depths in Thetis Regio between 130 ø and 140 ø (Figure 3). Compensation depths east of 140 ø, the eastern edge of Thetis, are generally deeper than compensation depths to the west. West of 140 ø Airy compensation depths vary with longitude in a manner similar to that found by Herrick et al. [1989], who derived Airy compensation depths using a three-dimensional point mass model. Both free-free and free-rigid convection layer thicknesses show the same pattern of greater depths east of 140 ø (Figures 4-5) and follow a pattern roughly parallel to that for Airy depths west of 140 ø.

Half-spreading velocity is low (most values of rs<2.0 cm/yr) everywhere the fits are well-constrained (Figure 6a). This is con- sistent with the conclusion of Kaula and Phillips [1981] that if Venus has terrestrial-style plate tectonics the spreading velocities are between 0.5 cm/yr and 5.0 cm/yr. Sotin et al. [1989b] com- bined data from many PVO passes over a narrow zone of western Aphrodite to produce a synthetic gravity profile perpendicular to the ridge at about 88 ø. They used a static mass model of the topography under this profile to model the gravity and obtained

Bt,^cI• ET AL.: GRAVITY PROFILES OVER APHRODITE TERRA 311

o t•

2O

10

-10

o

,

ß

ß

ß

o

_ /' 0...4•---0--(•

/'/a' • • ,•..• .... •'• .... •,...,• o c •

_

- ?• • filtered pvo los accelerations ,, .

•' ---•--.unfilLered pro los acceleraLions - _ - ...... e ....... airy compensaLion model, L=155 _

• orbsira compensaLion model, L=155 km

2 0 -2 -4 -6 -8 -10 latitude

Fig. 9. Comparison of the fits obtained for simple topographic compensation by ORBSIM and in this work for PVO pass 475. Also shown are the filtered and unfiltered PVO LOS data.

lower and upper bounds on the half-spreading rate of 0.3 cm/yr and 0.5 cm/yr, in agreement with our half-spreading rate of 0.4 cm/yr at 88 ø.

The fits for ridge width (Figure 6b) show quite a bit of scatter, but all of the well-constrained fits are at values of w less than

about 4000 km. This corresponds, for 0.3 cm/yr<v•<l.6 cm/yr, to an age between 250 Myr and 1300 Myr, if the spreading rate has been constant. The younger end of this age range is comparable to the oldest oceanic crust on Earth. There are a few fits with

$ (v, w)<3.0 mgal for which w is greater than about 8000 km. This implies ages of a few billion years for v•<l.0 cm/yr, which is an order of magnitude older than the oldest terrestrial oceans. However, the uncertainties on these fits are large and allow values of w such that the age would be on the order of hundreds of millions of years, which is comparable to terrestrial ocean ages. Any age estimates for the spreading center based on the ridge width fits are therefore quite speculative. In addition, Grimm and Solomon [1989] found that the CSD system cannot be due to plate divergence around a single, static Euler pole. If Aphrodite is indeed a spreading center, its history has not been simple and its velocity has probably not been constant, rendering any age estimates based on a single Euler pole spreading geometry also highly uncertain.

DISCUSSION

A comparison of Figures 7a-7d shows that for most of the passes between 68 ø and 107 ø the spreading model fits the data significantly better than the other models. However, S(b)<600 m

for only one of the passes in this part of Aphrodite. Aphrodite does not have the shape of spreading ridge like the Mid-Atlantic Ridge, though the gravity data can be fit very well over part of the ridge by the spreading ridge model. The models cannot be distinguished anywhere else in Aphrodite, i.e., there is no significant difference in how well the gravity and topography data are fit by Airy compensation, mantle convection, and the spreading ridge model.

Four of the topography profiles with S(b)<600 m are in eastern Aphrodite between about 165 ø and 175 ø (Figure 8). All of the compensation models fit the data well for many of the passes in this longitude range, but they cannot be distinguished from each other and the good fits here are probably not significant given the poor fits for adjacent passes (Figure 7). Crumpler and Head [1989] have identified more CSDs in eastern Aphrodite, which may affect model fits here, as the single long "ridge axis" we have used for domain 9 would be broken up into several smaller ones.

Though we cannot determine the tectonic processes responsible for Aphrodite Terra using only the gravity and topography data, our analysis does put some constraints on possible mechanisms for the development of this region.

The large compensation depths in eastern Aphrodite imply that there must be a dynamic component of topographic support here, unless the topography is very young, since both thermal models [Phillips eta!., 1981] and the widths of Dali and Diana chas- mam in central Aphrodite [Zuber, 1987; Banerdt and Golombek, 1988] imply that the base of the lithosphere in this region is no deeper than about 50-100 km. Mantle material below this depth

312 BLACK ET AL.: GRAvrrY PROFreES OVER APHRODITE TERRA

-2

-4

PVO Pass 475

[] observed topography

....... e ....... model topography, rms diff.=0.763 km

0 -2 -4 -6 -8 -10 latitude

20

o

(b) -lo

PVO Pass 475

o filtered pvo los accelerations ....... o ....... airy model, t=155 km --h--free-free model, z=550 km

, free-rigid model, y=1250 km ---•---spreading model, vs=0.6 cm/sec, w=1100 km

-2 -4 -6 latitude

Fig. 10. (a) Observed and model topography profiles for PVO pass 475. (b) Model gravity fits for pass 475 plotted along with the observed Pioneer Venus LOS accelerations.

-10

BLACK ET AL.: GRAvrr¾ PROFILES OVER APHRODITE TERRA 313

(a)

-2

-4

PV0 Pass 437 -

•'00-000 ra. ' -

- [] observed topography -

....... •> ...... model topography, rms diff.=l.288 km -- _

5 0 -5 latitude

-10 -15

2O

10

-10

(b) -20

PVO Pass 437

-- .0.0.1• '0'0' _

_

__

- - ' % - - • ;o o -

[] filtered pvo los accelerations ...... <• ...... airy model, t=75 km - --•---free-free model, z=325 km -

,. free-rigid model, y=950 km - ---•--spreading model, vs=0.4 cm/sec, w=1600 km -

5 0 -5 -10 -15 latitude

Fig. 11. (a) Observed and model topography profiles for PVO pass 437. (b) Model gravity fits for pass 437 plotted along with the observed Pioneer Venus LOS accelerations.

314 BLACK ET AL.: GRAVITY PROHLES OVER APHRODITE TERRA

would not be resistant to viscous deformation on geological time scales. Turcotte [1989] has proposed a heat pipe mechanism for the lithospheric transfer of heat on Venus that would allow the lithosphere to be greater than 150 km thick, but east of 140 ø Airy compensation depths are up to 250 km and it is unlikely that the mechanical lithosphere could be this thick. The depths of convec- tion implied by the fits here are quite deep (about 600-1000 km for free-free convection and about 1500-2000 km for free-rigid convection). The core-manfie boundary is expected to be at about 2800 km depth [Phillips and Malin, 1983], so these layer thick- nesses imply convection extending down through a large fraction of the mantle. Convection in a mantle in which viscosity increases with depth requires deeper convection to produce gravity anom- alies of the same magnitude as the constant-viscosity model we have used. Kiefer and Hager [1988] can reproduce the observed geoid and topography in Aphrodite using variable-viscosity nu- merical models in which convection extends from the surface to the core-mantle boundary.

If Aphrodite is a terrestrial-style divergent boundary, it must be slowly diverging and enduring, based on our spreading velocity and ridge width results. The slow spreading rate implies a lower rate of heat transfer through plate creation than on Earth. Kaula and Phillips [1981] estimated that venusian plate tectonics could deliver at most 15% of the total surface heat flux, assuming that Venus has a heat generation per unit volume 90% of Earth's. This conclusion was based on inferred spreading velocities of about 1 cm yr -•, comparable to our results. This is in contrast to the 62% of the terrestrial surface heat flux carried by plate tectonic processes [Sclater et al., 1980]. It thus appears that plate tectonics cannot presently be the major mode of heat transport on Venus, though it is possible that this mechanism may contribute to the global heat loss budget.

In Thetis Regio (121 ø < longitude < 131ø) the gravity signal associated with Aphrodite is small, implying shallow compensa- tion of the local topography and hence significant along-strike variations in the processes which give rise to Aphrodite. The variations with longitude of Airy compensation depths and con- vection layer thicknesses west of 121ø, as well as the dichotomy between western and eastern Aphrodite, also imply a segmented ridge, as opposed to a ridge produced by a single process. West of 121ø most of the Airy compensation depths are less than 150 km and many are less than 100 km, so much of the topography here could be locally supported and due, for example, to crustal thick- ness variations. East of 140 ø, however, the compensation depths are too great for the topography to be statically compensated, im- plying some fundamental difference between eastern and western Aphrodite. The convection layer fits follow a similar pattern in eastern and western Aphrodite, supporting this interpretation.

CONCLUSIONS

1. In western Aphrodite, between 68 ø and 107 ø, the spreading ridge model provides a significantly better fit to the PVO gravity data than do models of Airy compensation or mantle convection. The models cannot be distinguished from each other in any other part of Aphrodite.

2. In most areas of Aphrodite the topography does not look like that expected for a Mid-Atlantic-type spreading ridge.

3. None of the models fit the data in far eastern Aphrodite (longitude > 180ø).

4. Shallow compensation depths in Thetis Regio (121 ø < longitude < 131 o) are consistent with static support, such as by crustal thickness variations.

5. Deep Airy compensation depths and convection layer thick- nesses indicate there must be some dynamic component to the support of topography in Aphrodite east of 140 ø .

6. The model fits indicate that it is likely that more than one mechanism is responsible for the formation of Aphrodite.

Acknowledgments. Thanks to Peter Ford for providing PVO data tapes, especially the PMTM data set, as well as much help with the data and many useful discussions of them. We thank the Magellan project for supplying their digital topography model. We also thank Bruce Bills for helpful discussion and suggestions for improving this work. The Lunar and Planetary Institute generously provided computer time and access to their Geophysical Data Facility, through which ORBSIM was used. This work was taken from a thesis submitted to the graduate school, University of Maryland, by M.T.B. in partial fulfillment of the requirements for the doctoral degree in physics and astronomy.

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M. T. Black, Astronomy Program, University of Maryland, College Park, MD 20742

D.C. McAdoo, National Geodetic Survey, Charting and Geodetic Services, National Ocean Service, NOAA, Rockville, MD 20852

M. T. Zuber, Geodynamics Branch, Code 621, NASA Goddard Space Flight Center, Greenbelt, MD 20771

(Received November 27, 1989; revised June 6, 1990;

accepted August 16, 1990.)