RETRIEVAL OF A WIND PROFILE FROM THE GALILEO PROBE TELEMETRY SIGNAL JAMES B. POLLACK NASA Ames Research Center, Moffett Field, CA 94035, U.S.A. DAVID H. ATKINSON University of Idaho, Moscow, ID 83843, U.S.A. ALVIN SEIFF San Jose State University Foundation, San Jose, CA 95106, U.S.A. and JOHN D. ANDERSON Jet Propulsion Laboratory, Pasadena, CA 91109, U.S.A. Abstract. Ultrastable oscillators onboard the Galileo Probe and Orbiter will permit very accurate determinations of the frequency of the Probe's telemetry signal as the Probe descends from a pressure level of several hundred mb to a level of about 20 bars. Analysis of the time-varying frequency can provide, in principle, a unique and important definition of the vertical profile of the zonal wind speed in the Jovian atmosphere. In this paper, we develop a protocol for retrieving the zonal wind profile from the Doppler shift of the measured frequency; assess the impact of a wide range of error sources on the accuracy of the retrieved wind profile; and perform a number of simulations to illustrate our technique and to assess the likely accuracy of the retrieval. Because of unavoidably large uncertainties in the absolute frequencies of the oscillators, we use time-differenced frequencies in our analysis. Nevertheless, it is possible to recover absolute wind speeds as well as wind shears, since the Orbiter/Probe geometry changes significantly during the Probe relay link. We begin with the full relativistic Doppler shift equation. Through the use of power series expansions and a basis function representation of the wind profiles, we reduce the basic equation to a set of linear equations that can be solved with standard linear least-squares techniques. There are a very large number of instrumental and environmental factors that can introduce errors into the measured signal or to the recovery of zonal winds from the data. We provide estimates of the magnitudes of all these error sources and consider the degree to which they may be reduced by a posteriori information as well as the results of calibration tests. The most important error source is the a posteriori uncertainty in the Probe's entry longitude. The accuracy of the retrieved winds is also limited by errors in the Probe's descent velocity, as obtained from atmospheric parameters measured by several Probe experiments, and in the a posteriori knowledge of secular drifts in the oscillators' frequencies during the relay link, due, for example, to aging and radiation damage. Our simulations indicate that zonal winds may be retrieved from the Doppler data to an accuracy of several m s^-1. Therefore, it may be possible to discriminate among alternative models for the basic drive of the zonal winds, since they differ significantly in the implied zonal wind profile. 1. Introduction The Galileo spacecraft consists of a Probe that will enter Jupiter's atmosphere and an Orbiter that will make multiple close flybys of the Galilean satellites of Jupiter. The Probe contains six science instruments that will conduct the first in-situ measurements of the structure and composition of the Jovian atmosphere. The data from these measurements will be sent to a receiver on the Orbiter over approximately a 60-min interval that occurs soon after a close flyby of Io and shortly before commencement of orbit insertion maneuvers. The presence of ultra-stable oscillators on both the Probe and Orbiter will provide very accurate measurements of the frequency of the telemetry signal. As a result, it may be possible to derive useful information on the vertical profile of the horizontal winds. Such a profile could provide a valuable constraint on alternative models for the basic drive of the Jovian winds, as described shortly. In this paper, we describe an in-depth analysis of the methodology and practicality of retrieving a wind profile from the Probe's Doppler signal that the authors of this paper have been carrying out over the last 10 years. Although we will present a thorough discussion of this problem here, a much more detailed presentation can be found in the Ph.D. dissertation of one of us (Atkinson, 1989; henceforth Paper I). In the remainder of this section, we summarize past attempts to derive data on winds of planetary atmospheres other than that of the Earth, indicate the aspects of the Galileo Doppler data that makes it challenging to retrieve wind formation, and describe the organization of this paper. Estimates of wind speeds in planetary atmospheres have been obtained in a number of ways. These include tracking cloud features seen on images taken from space and the ground, measuring the Doppler shift and positional shift of entry Probes and balloons, and applying the thermal wind equation to measured temperature fields. All three techniques have been applied with considerable success to Venus' atmosphere (e.g., Counselman et al., 1980; Kerzhanovich et al., 1972; Marov et al., 1973; Preston et al., 1986; Limaye, 1985; Rossow et al., 1980; Sagdeev et al., 1986; Schubert et al., 1980; Taylor et al., 1980). To date, only the first and third of these techniques have been used to study winds in the Jovian atmosphere (e.g., Ingersoll et al., 1981; Conrath et al., 1981; Gierasch, 1976). The three approaches to deriving winds are complementary. Feature tracking can provide a good definition of the global distribution of wind speeds at one or at most several vertical layers in the atmosphere (namely, at the level of the first optically thick cloud layer). Measurements of the motion of entry Probes yield a detailed vertical profile of the horizontal wind speeds (and, in restricted circumstances vertical wind speeds) at isolated locations. Finally, winds obtained from measurements of three-dimensional temperature distributions can provide useful estimates of the vertical shear in the east-west (zonal) wind speed (and, in certain circumstances, the north-south or meridional wind speed) over a range of altitudes, especially ones above the cloud tops. These latter estimates are subject to the joint caveats of certain dynamical assumptions (e.g., geostrophic balance, which breaks down near the equator, or cyclostrophic balance, which breaks down near the poles) and the need to have independent estimates of wind speed at some reference altitude (e.g., the cloud tops) (Seiff et al., 1980). Wind speeds obtained from Probe tracking are valuable both in their own right (see below) and for providing 'ground-truth' for remote sensing techniques. With regards to the latter, Probe-derived winds help to define the average altitude to which the feature tracked winds pertain and help to test the accuracy of winds found from measured temperature fields. Because of Venus' proximity to the Earth, it has proved possible to obtain both accurate Doppler shift and angular displacement measurements of the time-varying positions of entry Probes and balloons. As a result, it was possible to determine the three-dimensional velocity field of the winds (especially, the usually much larger horizontal wind speeds) (e.g., Counselman et al., 1980; Preston et al., 1986). Unfortunately in the case of the Galileo Probe, it is possible only to measure the Doppler shift of the Probe's telemetry, but not its angular displacement, as a result of the much greater distance of Jupiter from the Earth and the Probe's antenna being pointed at the Orbiter (which puts the Earth in the distant lobes of the Probe's telemetry antenna). Nevertheless, the Doppler data can yield some very important and unique information on the variation of the zonal wind speed with altitude over a pressure range from about 100 mb to about 20 bars. The probe is expected to traverse the atmosphere at approximately 6.5 degrees north, which lies well within Jupiter's equatorial jet. In principle, all three wind speed components as well as the Probe's descent velocity and the Orbiter's motion can contribute to the observed Doppler shift. However, the Probe's descent velocity will be determined very accurately from analysis of temperature and pressure measurements made by the atmospheric structure experiment and the Orbiter's motion can be extremely accurately determined from other data. Among the three wind components, the zonal speed is expected to be the dominant contributor to the measured frequency shift in all, but the most pathological cases for the following reasons: (1) The projection of the Probe/Orbiter line of sight onto the local horizontal plane of Jupiter lies almost in an east-west direction, except at times very close to overflight. (2) At the level of the visible cloud layer, zonal wind speeds are typically more than an order of magnitude larger than the meridional wind speed, including at the latitude of the Galileo Probe (Ingersoll et al., 1981). (3) It is easy to show from the continuity equation that the vertical wind speeds associated with the large-scale zonal wind speeds near the cloud tops are more than three orders of magnitude smaller than the horizontal wind speeds (Holton, 1979). However, vertical velocities associated with thermal convection driven by the internal heat source can have somewhat larger values in the deeper portions of the atmosphere that will be examined by the Galileo Probe experiments (Gierasch, 1976). Thus, it would be possible to retrieve the vertical profile of the zonal wind speed from the time-varying frequency of the Probe's telemetry signal. Differences in the amount of sunlight absorbed at different latitudes provides the basic drive for the winds in the atmospheres of the terrestrial planets. However, a much wider suite of potential drives exists for the Jovian winds. They may be driven principally by differential solar heating, latent heat released by condensing water vapor, heat being advected upwards from the deep, hot interior, or heat generated from ortho/para transitions of molecular hydrogen (Stone, 1967; Gierasch, 1976; Ingersoll and Pollard, 1982; Conrath and Gierasch, 1984). Below, we briefly review the nature of these different drives and present schematic profiles of zonal winds that might result from each of them. The magnitude of Jupiter's internal heat source is nearly comparable to the amount of sunlight that the planet absorbs (Hanel et al., 1981). The thermal convective heat flux associated with this internal heat source almost totally compensates for the global-scale variation in absorbed sunlight from equator to pole so that the amount of thermal radiation emitted to space is almost constant on this length scale (Ingersoll and Porco, 1978). Consequently, no significant global-scale temperature gradient results from differential absorption of sunlight and hence no significant global-scale circulation pattern can result from this drive, contrary to some pre-Pioneer models of Jovian dynamics (e.g., Stone, 1967). Nevertheless, this compensating effect of the interior heat flux does not occur on smaller length scales: the difference in the thermal radiation emitted by adjacent belts and zones is somewhat larger than the non-trivial difference in the amount of sunlight they absorb (Pirraglia, 1984). Consequently. differential solar heating could be an important drive for wind systems having length scales on the order of the bands. Significant absorption of sunlight occurs near the level of the uppermost, optically thick cloud layer, a putative ammonia ice cloud deck, whose bottom lies near the 700 mb level. This cloud layer typically has an optical depth of several and a non-unit single-scattering albedo in the visible (West et al., 1986). Additional absorption of sunlight takes place in the lower stratosphere (1-100 mb) due to gases (especially methane) and aerosols and at greater depths in the troposphere due to gases and other cloud particles. According to the calculations of Hunten et al. (1980), almost all the solar energy absorbed by Jupiter's atmosphere is deposited at altitudes above the 2 bar pressure level. Therefore, we approximate the winds resulting from differential solar heating as having a maximum slightly below the ammonia cloud deck and tending towards zero near the 2 bar level (cf. Figure 1; P. Gierasch, private communication). On the length scale of the Jovian bands, the cloud-top winds are approximately in geostrophic balance, i.e., the basic balance in the meridional momentum equation is between the Coriolis force and the pressure-gradient force (e.g., Gierasch, 1976). Consequently, the thermal wind equation can be used to place a constraint on the magnitude of the temperature difference between belts and zones that is needed to produce the observed zonal winds. This temperature difference scales inversely with the depth of the source region for the winds. For winds being driven by latent heat that is released by water vapor condensing near the 7 bar level of the atmosphere, the required temperature difference is comparable to the difference expected from wet and dry adiabats for solar abundances of water (Gierasch, 1976). Thus, this type of model envisions that water preferentially condenses in regions of large-scale rising motion. In this sense, the latent heat drive strongly reinforces any small differential rising and sinking motions. Note that although almost all the water condensation occurs within a fraction of a scale height of the water cloud base, the temperature difference between wet and dry adiabats persists throughout the convectively unstable region lying above the water cloud base. We represent the winds resulting from the latent heat drive as beginning at the base of the water clouds and reaching their maximum near the top of the convection zone (cf. Figure 1; P. Gierasch, private communication). Fig. 1. Vertical profiles of the zonal wind speed predicted by alternative models of the basic drive for the Jovian atmospheric circulation. Temperature differences on constant pressure surfaces in the convective region of the Jovian troposphere can also be produced by horizontal variations in the relative abundance of molecular hydrogen in its ortho and para states (Gierasch, 1983). Since the specific heat at constant pressure of these two states of hydrogen differ, variations in the ortho to para ratio lead to variations in the adiabatic lapse rate and hence temperature. As the interconversion of ortho and para hydrogen is spin forbidden, it is expected to take place slowly in the colder, upper region of the Jovian troposphere (Massie and Hunten, 1982). In fact, there is evidence from the Voyager IRIS data that significant horizontal variations in the ortho to para ratio exist in the uppermost troposphere (Conrath and Gierasch, 1984). However, analysis of ground-based spectra implies that only equilibrium hydrogen is present at deeper levels of the Jovian atmosphere (Cunningham et al., 1988). The wind profile in Figure 1 associated with horizontal variations in the ortho/para ratio was obtained by assuming that extreme differences in this ratio occur in neighboring bands, that temperature gradients vanish at great depths in the troposphere and at stratospheric altitudes, and that the circulation extends fairly deep within the troposphere (Glerasch, 1983). Thus, like all the other profiles in Figure 1, the one shown for an ortho/para drive is simply meant to be one plausible example of this class of dynamical model. In all the above models, the drive for the circulation is located at relatively shallow depths in a very deep molecular hydrogen envelope for the planet. An alternative viewpoint is to consider a much deeper circulation pattern that persists throughout the molecular hydrogen envelope and that is driven by the internal heat source (Ingersoll and Pollard, 1982). Theoretical and laboratory studies show that columnar convection cells, aligned with the spin axis, are the preferred mode of convective instability in rapidly and uniformly rotating, viscous, conducting fluids (e.g., Busse, 1976). A somewhat more generalized state of motion involves cylindrical cells that interact with a basic cylindrical, differential rotation to produce zonal winds near the top of the adiabatic portion of the fluid (Ingersoll and Pollard, 1982). We crudely represent the zonal wind profile for this type of model as having a modest wind shear throughout the region measured by the Galileo Probe and as having a non-zero velocity at the base of this region (cf. Figure 1; P. Gierasch, private communication). According to Figure 1, the 'representative' zonal wind profiles associated with the four dynamical models differ substantially from one another. Thus, one might hope to distinguish among them by comparing their predictions with zonal wind profiles derived from the Galileo Probe's telemetry frequency. However, there is a significant variance possible in the wind profiles corresponding to each of the dynamical models. Conceivably, data taken by a variety of Galileo Probe and Orbiter experiments will lead to well-constrained predictions for these various models in the post-Galileo era. For the purposes of this paper, we merely wish to show that winds can be recovered with sufficient accuracy from the Galileo Doppler wind experiment to provide useful constraints on the underlying drives for the winds. We also recognize that winds associated with eddy motions may contribute to the derived wind profile, although eddy wind speeds in the Jovian atmosphere are typically almost an order of magnitude smaller than the zonally average zonal wind speed near the cloud tops (Ingersoll et al., 1981). Given the experience in deriving wind speeds from Doppler data for probes into Venus' atmosphere and given the well-known and relatively simple form of the Doppler equation, it might be thought that its application to the Galileo Probe would be quite straightforward. However, this is not the case for several reasons. First, the geometry of the Probe and Orbiter positions changes significantly during the communication period. In particular, the angle between the line of sight of the probe and orbiter and the local vertical (probe aspect angle), which determines the contribution of horizontal wind speeds to the Doppler shift, varies from essentially 0 degrees at overflight to values that approach 10 degrees near the beginning and end of the communication period. As shown in the next section, such variations both complicate the analysis of the Doppler data and open up opportunities (e.g., it is possible to recover absolute wind speeds without knowing well the oscillators' zero-point frequency). A second reason that the analysis of the Galileo Probe Doppler data is less than straightforward is the need to do careful study of a wide range of possible error sources that make deriving winds for Jupiter's atmosphere more problematic than was the case for Venus' atmosphere. Except very close to the surface of Venus, the zonal wind speeds are much larger than the speed of the planet's solid body rotation. In contrast, Jupiter's rotational velocity near its equator has a value of approximately 10 km s^-1 which is about two orders of magnitude larger than the measured zonal wind speed near the ammonia cloud tops. Consequently, small inaccuracies in the a posteriori knowledge of the Probe's location might seriously compromise the ability to derive accurate relative wind speeds from the Doppler shift data. Other aspects of the spacecraft geometry and Jovian environment also make the retrieval of accurate wind speeds more challenging than was the case for Venus. These include the small values of the Probe aspect angle and the strong Jovian magnetosphere (the high flux of high-energy particles can cause drifts in the Orbiter and Probe oscillators). Despite this highly unfavorable situation for performing Doppler wind measurements, we will show that accurate wind retrievals are possible. In the next section of this paper, we describe the mathematical basis of our approach that begins with the relativistic Doppler shift equation and reduces it down to the solution of a set of linearized equations. In the subsequent section, we investigate the effects of a large number of possible error sources that limit the accuracy in the recovered winds and provide estimates of their magnitudes. Next, we present full-up simulations of our approach by attempting to recover the four classes of wind profiles given in Figure 1 in the presence of realistic errors in the data and our knowledge of critical parameters. Finally, we summarize the conclusions of this paper by assessing the accuracy with which a wind profile can be retrieved from the Galileo Probe's telemetry frequency, key sources of error, and the ability of the retrieved profiles to discriminate among the major candidate drives of the circulation. 2. Approach Beginning at about the 100 mb level in the atmosphere, when the Galileo Probe's descent velocity has become subsonic and its parachute systems have been deployed, the Probe's radio system will send telemetry signals to the Orbiter's radio system. This communication will last approximately 60 min, at which point the probe will have descended to about the 20 bar level in the Jovian atmosphere. The Probe's telemetry will be sent by two independent channels to help ensure the successful receipt of the results from the Probe experiments. One of these channels is controlled by an ultra-stable oscillator (USO) on the Probe. An identical USO on the Orbiter will provide a highly accurate measurement of the frequency of the carrier, f, whose nominal value is 1387 MHz. The received frequency will be measured every 2/3 s during the Probe/Orbiter relay link. We suppose for the moment that the contributions to the Doppler shift from all sources other than the atmospheric winds can be removed exactly, we consider only the effects of the zonal component of the winds for reasons given in the Introduction, and we use the classical form of the Doppler shift equation for planar geometry f_o = f_p + f^mis + f_p U/c sin psi cos alpha, where f_o is the received frequency at the Orbiter, f_p is the transmitted frequency at the Probe, f^mis is the Doppler shift due to all factors other than the zonal wind speed, U is the zonal wind speed, c is the speed of light, psi is the angle between the local vertical and the line of sight between the Probe and Orbiter, and alpha is the angle between the east-west direction and the projection of the line of sight on the local horizontal plane. The product sin psi cos alpha is simply the fraction of U that lies along the line of sight. Unfortunately, there are large uncertainties in the zero-point frequency of the Probe and Orbiter's oscillators. While the USOs are quite stable over the period of the communication link, they are expected to undergo a significant drift over the 6 to 7 year cruise interval from launch to arrival at Jupiter. Since there will be no 2 way links between the ground and the USOs following launch, there is no procedure for accurately determining the actual long-term drift. Ground-based tests that have been conducted on the USOs suggest that they may drift by about 500 Hz during the cruise period. A drift of this magnitude is equivalent to the Doppler shift produced by a 1000 m s^-1 zonal wind for the geometry of the Galileo mission! The solution to the problem posed by the large uncertainty in the USOs' zero-point frequency is to use differences in the measured frequencies: delta f_o - delta f^mis = f_p (delta[U])/c sin psi cos alpha + f_p U/c delta(sin psi cos alpha), where delta x means the difference in the value of parameter x at any time t during the relay link and some reference time, t_r. The left-hand side of the above equation contains a combination of the measured frequencies, now having a high accuracy, and Doppler shifts due to a variety of well-measured or known velocities. According to the above equation, the zonal velocity affects the measured frequency difference in two ways: through changes in zonal wind speed with depth in the atmosphere and through the changing geometry of the line of sight, as influenced by the time-integrated effect of the wind on the Probe's position. Figure 2 shows the variation of psi and alpha as a function of time after entry for the nominal Probe mission, with Orbiter overflight of the Probe occurring 21 min after Probe entry. The geometrical factors change significantly over the course of the relay link, making the magnitude of the second term on the right-hand side of the above equation comparable to or greater than that of the first term during much of the relay link. As a result, it is possible in principle to determine absolute wind speeds as well as wind shears from measured frequency differences. Fig. 2. Probe and Orbiter longitudes, Probe aspect angle (psi), azimuth angle (alpha), and projection factor (fraction of the zonal wind projected onto the line of sight) as function of time from the Galileo Probe's entry into the Jovian atmosphere. However, the above equation cannot be readily inverted to yield wind speeds directly. This last point is brought out even more clearly when it is realized that the winds significantly alter the longitudinal location of the probe (in a time-integrated fashion) and therefore significantly affect the values of the angular quantities appearing in these equations, as verified by our detailed calculations. For these reasons, we will perform a number of transformations on the Doppler shift equation, beginning with the more rigorous relativistic counterpart of the above equation, to reduce the problem to a set of linearized equations that can be solved with least-squares techniques. In more detail, the derivation of the final linearized set of equations given below follows the following path: (1) The full relativistic Doppler equation is expanded to second-order in velocity. (2) We express the resulting equation as an equality between the measured Doppler shift and accurately known quantities on one side of the equation and the velocity along the line of sight on the other side of the equation. (3) This velocity is expanded about a state of zero horizontal wind speed. This expansion is useful since the horizontal wind speeds are much less than the planet's rotational velocity. (4) The horizontal wind speed's dependence on atmospheric pressure is expressed as a Legendre polynomial expansion. This expansion is needed since the Doppler shift depends, in part, on the time integral of the wind speed, which affects the instantaneous location of the Probe (cf. Equation (2)). (5) The problem at this point has been reduced to a linear least-squares problem involving the coefficients of the Legendre polynomials. The singular value decomposition method is used to invert the relevant equations, reduce them to a form required by standard least-squares techniques, and insure good numerical stability. Since the magnitude of the relative velocity between the Probe and Orbiter, V_tot is much less than the speed-of-light, c, we can use a Taylor series expansion on the full relativistic Doppler shift equation and retain terms only up to order 1/c^2 (Kerzhanovich et al., 1969; Paper I). The resulting equation is given by f_o - f_p = - (f_p/c) [V_parallel + 1/c ((V_tot^2/2) - V_parallel^2) - cG_r], where V_parallel, is the projection of V_tot along the line of sight between the Probe and Orbiter and G_r is the general relativistic red shift in the weak field limit (Paper I). The first term on the right-hand side of this equation is the familiar classical Doppler shift term. Our sign convention in this equation and the ones below is that a positive value of V_parallel implies an increasing distance between the Probe and Orbiter and, hence, a red shift to the measured frequency. The gravitational red shift has a value of approximately 20.5 Hz for the nominal distances of the Probe and Orbiter from the center of Jupiter. However, it changes by only a few tenths of a Hz during the relay link and even this relatively trivial effect can be removed to very high accuracy. The second term on the right-hand side of the equation has a magnitude of several tens of Hz and varies by about this much over the relay link. Since a zonal wind speed of 1 m s^-1 produces a Doppler shift that averages about 0.5 Hz over the relay link, these higher-order terms are important and must be taken into account in the ultimate analysis. Fortunately, these higher-order terms are dominated by the projections of the Orbiter's motion and the planet's rotational velocity along the line of sight and, hence, these terms can be evaluated with high accuracy from the start of the inversion procedure. We rewrite the relativistic Doppler shift equation with 'knowns' on the left-hand side of the equation and the unknowns on the right-hand side: f_Dop' = -f_p (V_parallel/c) f_Dop' = f_o - f_p(1 + G_r) - 1/c^2 [(V_tot^2/2) - V_parallel^2]. We next cast the above equation in terms of the difference in the Doppler shift at an arbitrary time, t, during the relay link and some reference time, t_r: delta f_Dop' = -f_p ( delta V_parallel/c) Here, as earlier, delta x means the difference in the value of x at time t and its value at time t_r. It is useful at this point to introduce a spherical-coordinate system whose origin lies at the center of Jupiter, as illustrated in Figure 3. The positions of the Probe and Orbiter Fig. 3. The Probe/Orbiter geometry, in a Jovian centered coordinate system. Various basic angles used in the text are defined in this figure. are specified by a radial distance r from the origin and colatitudes theta and longitudes phi. Auxiliary angles in this coordinate system include gamma, the angle between the direction to the Probe and Orbiter, as measured from the origin, psi, the Probe aspect angle, and alpha, the line-of-sight azimuth. The mathematical relationships between the auxiliary angles gamma, psi, and alpha and the primary spherical coordinates and between V_parallel and these primary coordinates are given in Paper I. It is useful to write V_parallel in terms of a Taylor series expansion of the spherical coordinates and their time derivatives both to obtain a linear set of equations for the zonal wind velocities and to provide a mathematical basis for performing the error analysis discussed in the next section. For our present purposes, we will assume that all the relative motions contributing to V_parallel are either known with negligible uncertainties or are unimportant, except for the zonal wind speed. In this case, only the coordinate phi_p and its time derivative enter into the second term on the right-hand side of the Taylor-series expansion. We obtain the following linearized equations for the zonal wind speed, U, in terms of the measured Doppler shift and other, known velocities: Delta f_Dop = - f_p | dV_parallel dV_parallel | ------ Delta| ------------- delta phi_p + --------------- delta (phi_p)dot| c | dphi_p d(phi_p)dot | Delta f_Dop = Delta f_Dop' + f_p/c Delta V_parallel^0, delta phi_p(t) = delta phi_p(t_r) + integral from t_r to t of U(t')/(r_p(t') sin(theta_p(t'))) dt', delta (phi_p(t))dot = U(t)/(r_p(t) sin(theta_p(t))), where delta x_i is the difference between the true of x_i and the value determined from the best available information (e.g., Probe descent velocity, Orbiter trajectory, etc.) and subscript p refers to the Probe. As a final step in linearizing the equation relating measured and known quantities to the unknown zonal wind speed, we express the zonal wind speed in terms of an Nth order expansion in Legendre polynomials: U(t) = the sum from l = 0 to N of a_l P_l(z(t)), log(pr(t)) - log(pr(t_r)) z(t) = 2 ------------------------------- - 1, log(pr(t_f)) - log(pr(t_r)) where P_l is a Legendre polynomial of order l, pr is the atmospheric pressure at the Probe's location, and subscripts r and f denote the times of the first (or reference) and last Doppler shift data. The unknowns now are the constants a_l. Finally, we combine the above two sets of equations to obtain a matrix equation for the unknowns, a_l, which we now write as a vector "a" to simplify the notation: Ca = b, b = Delta f_Dop, C = - f_p | dV_parallel dV_parallel | ------ | ------------- (t_m)beta_1 + --------------- (t_m)beta_2 |, c | dphi_p d(phi_p)dot | beta_1 = delta t sum from m' = 1 to m of ------------------------------- P_l(z(t_m')), r_p(t_m') sin(theta_p(t_m')) P_l(z(t_m)) P_l(z(t_r)) beta_2 = --------------------------- - -------------------------- ; r_p(t_m)sin(theta_p(t_m)) r_p(t_r)sin(theta_p(t_r)) vector a has L elements ( = N + 1), corresponding to the number of terms used in the Legendre polynomial representation of the wind speed; vector b has M elements, corresponding to the number of discrete time points, t_m, at which the measurements are made (or a subset that have been appropriately time averaged over small time segments); matrix C is of dimension M by L, as are matrices beta_1 and beta_2. In the above equations, delta_t is the time between successive samples. There are many more time points, M, than unknowns, L. In principle.the above matrix equation could be solved by simply multiplying the equation by C^-1 on the left side of both sides of the equation, where C^-1 is the inverse of matrix C. In practice, such a procedure is susceptible to numerical problems that arise when dealing with matrices that may be nearly singular. For this reason. we express C in terms of the product of three matrices, U, W, and V, in accord with the method of singular value decomposition (Golub and Van Loan, 1983). The resultant equation for a is then (Paper I): a = [VW^-1U^T]b, where superscripts -1 and T means the inverse and transpose of a matrix, respectively. Given a set of values of b_m (m = 1 to M), obtained from Doppler shift measurements and the removal of well-measured values of velocity components other than the zonal wind speed, one can obtain a best estimate of the zonal wind speed coefficients a_l, (l = 0 to N) by using standard least-squares techniques for solving the above matrix equation. 3. Error Analysis There are a very large number of factors that can influence the measured Doppler shift. In this section, we determine the uncertainties in the knowledge of all factors aside from the zonal wind profile and their impact on the measured Doppler shift to ascertain the feasibility and limitations on recovering the zonal wind profile in Jupiter's atmosphere from the frequency of the Probe's telemetry signal. In the next section of this paper, we will provide quantitative estimates of the errors in the zonal wind speed profile due to the cumulative effects of the uncertainties in the a posteriori knowledge of all the relevant factors that affect the Doppler shift. It is useful to place the factors of interest into several broad categories. These categories include the trajectories of the Probe and Orbiter during the relay link, as specified in terms of the six basic variables and their time derivatives; factors that influence the frequency of the USOs; propagation effects that influence the phase path of the Probe's signal; and a variety of miscellaneous factors that include such quantities as the effects of special and general relativity. In the discussion below, we emphasize those factors that can potentially generate the largest errors in the recovered winds. As formulated in the previous section, the primary variable used to determine the zonal wind speed profile is the difference in the frequency of the Probe's signal at some arbitrary time t during the relay link and that measured at some fixed reference time, t_r, that we take to be the first such measurement. These frequency determinations are made by a USO on the Galileo Orbiter. Thus, factors that cause a time-independent alteration of the measured frequency ('constant' errors) will have no effect on the recovered winds. Rather systematic distortions to the recovered winds can be introduced by factors that cause a time-varying error in the measured frequency. Here, we distinguish between time-varying errors having a consistent trend during the link ('systematic' errors) and ones having a random or periodic nature on time scales short compared to the duration of the relay link ('random' errors). We emphasize that it is only the uncertainty in the values of the various factors subsequent to the relay link that are of interest: we intend to remove the influence of all factors from the measured frequency difference to the degree that their values are known and their impact can be appropriately modeled. The influence of the zonal winds on the measured frequencies provides a convenient standard for assessing the importance of various error sources. Over the course of the relay link, the internal heat wind model of Figure 1 produces approximately a 100 Hz shift in the measured frequency due to the wind's effect on (phi_p)dot and about 100 Hz in the measured frequency due to its effect on phi_p. Comparable shifts apply to the other wind models of Figure 1. Thus, factors that produce time-varying errors in frequency of 100 Hz or more, 10-100, 1-10, and < 1 Hz would have lethal, serious, non-trivial, and very minor impacts, respectively. on the recovered winds. Finally, the measured frequency differences are linked to the cumulative effects of winds at all earlier times as a result of the changing geometry of the observations (cf. Equations (7)-(10)). As a result of this relationship and our use of a least-squares solution algorithm, it is possible to recover winds near the beginning of the relay link with comparable accuracy to those towards the end, despite our use of the initial frequency measurement as our reference frequency. This claim will be demonstrated when we show the frequency residuals of simulated cases. 3.1. TRAJECTORY UNCERTAINTIES Several sets of observations will provide an accurate determination of variables that define the positions and velocities of the Probe and Orbiter, aside from the horizontal winds. These include Earth-based tracking of the Orbiter's position prior to, during, and following the relay link; reconstructions of the Probe's trajectory following its release from the Orbiter and, hence, its entry location in Jupiter's atmosphere that are based on careful tracking of the Orbiter's location around the time of Probe release; determination of the Probe's descent velocity from data obtained by several Probe experiments. notably the Atmospheric Structure Instrument (ASI); and current knowledge of the planet's solid body rotation rate and its figure. As we will shortly see, errors in the descent velocity and Probe entry location, especially its entry longitude, constitute by far the most important error sources among the ones listed above and these are among the largest error sources of all the factors. We, therefore, begin our discussion by focusing on these two error sources. The descent velocity, (r_p)dot, will be derived from frequent measurements of pressure, P, and temperature, T, made by the ASI experiment by using the following equation: RT dP (r_p)dot = - ---------- ------ , g mu P dt where R is the universal gas constant, g is the acceleration of gravity, and mu is the mean molecular weight of the atmospheric gases (Seiff et al., 1980). This equation is based on the equation of hydrostatic equilibrium and the perfect gas law, relationships that should be obeyed very closely in the region of the Jovian atmosphere of interest. To evaluate the uncertainty in (r_p)dot, we consider errors arising from each of the four variables in Equation (19). Very accurate values of g (to better than 1 part in 10^4) may be obtained from Jupiter's mass, its radius at the 100 mb pressure level as a function of latitude, and the Probe's position relative to the 100 mb pressure level. The first two of these variables are currently known to a very high accuracy from past spacecraft missions to Jupiter, while the third variable can be inferred from the data of ASI (from (r_p)dot in a slightly iterative fashion). The mean molecular weight, mu, is expected to be measured to a relative accuracy of 2 parts in 10^3 by the Helium Abundance Experiment of the Galileo Probe. This uncertainty makes a non-trivial contribution to the overall uncertainty in the values of (r_p)dot derived from the above equation. It is useful to partition the errors in P and T into scale factor, zero point or offset, and random errors (Seiff et al., 1989). An extensive set of calibration tests of ASI indicate that the scale factor, offset, and random errors in the temperature measurements will be approximately 0.002, 0.15, and 0.087 K, respectively. Scale factor errors in pressure do not affect (r_p)dot. Offset and random errors in pressure vary among three intervals of pressure values spanned by the separate sensors of the ASI, but typically correspond to fractional errors of 0.002 and 0.001, respectively. Collectively, the errors in pressure, temperature, and mean molecular weight produce scale, offset, and random fractional errors in the inferred descent velocity of about 0.002, 0.0035, and 0.0016, respectively. When these separate components are combined in a root sum square fashion (RSS), a typical total fractional error in the descent velocity of about 0.0043 results. Thus, when the descent velocity is 50 m s^-1, as occurs at about the 5 bar pressure level, the corresponding uncertainty in this value would be about 0.2 m s^-1. Such an uncertainty may not seem to be very significant -- indeed it represents a real triumph in measurement accuracy! However, it is non-trivial since almost all of the descent velocity lies along the line of sight between the Probe and Orbiter and only about 0.1 of the zonal wind speed projects along the line of sight. Thus, errors in descent velocity can cause an order of magnitude larger error in the recovered zonal wind speed. Table I summarizes the effects of the errors in the inferred descent velocity, (r_p)dot by giving the resulting constant, systematic, and random errors in the measured frequency. A priori errors refer to ones expected in the absence of relevant data (e.g., measurements of the descent velocity by the Probe's ASI experiment) or the use of ground-based calibration data, while a posteriori errors refer to ones remaining after such information is employed. Thus, the magnitude of the a priori error provides an indication of the need to partially correct it, if possible, while the difference between the a priori and a posteriori errors show the degree to which the former can be eliminated. Table I also gives the corresponding errors due to a large number of other factors, most of which will be discussed in the remainder of this section. Another important error results from the uncertainty in the entry location of the Probe. Its trajectory can be reconstructed accurately from a knowledge of its relative velocity to the Orbiter at the time of separation, the Orbiter's trajectory, and the laws of celestial mechanics. (Note that the Probe does not have any active propulsion system.) The Probe release dynamics can be calculated by tracking the Orbiter prior to, during, and after Probe release, since the Probe's relative momentum at release is equal and opposite to the change in momentum of the Orbiter at release. Table II provides our current estimates of the 1 sigma uncertainties in the entry parameters for the Probe that result from uncertainties in the Orbiter's trajectory. Of these errors, the most important one is the uncertainty in the entry longitude of the Probe, since it produces a non-trivial error in the angle between the zonal direction and the line of sight. Figure 4(a) shows the change in relay link frequency near the start of the link as a function of the uncertainty in the entry longitude, delta phi_p. For the uncertainty in entry longitude given in Table II, there is a resulting uncertainty of almost 100 Hz in relay link frequency. The uncertainty in the entry longitude arises almost entirely from uncertainties in the time interval between Probe release from the Orbiter and entry into the Jovian atmosphere. delta phi_p strongly affects the analysis of the relay link frequency because of the error it causes in the projection of the planet's rotational velocity (which is shared by the Probe) along the line of sight. Fortunately, as illustrated in Figure 4(b), the impact of the entry longitude on the relay link frequency difference, the primary quantity used in our analysis, is considerably smaller than its impact on the absolute frequency. As summarized in Table I, there still results a systematic error of about 8 Hz in the frequency difference, making the error in Probe entry longitude perhaps the single most important source of error in the recovered winds. The Orbiter's trajectory during the relay link will be known to a very high degree of accuracy as a result of its being continually tracked by a series of techniques. The Orbiter's location and velocity will be determined from Earth-based Doppler tracking, ranging (signal propagation time), and angular position measurements made with a Differential Very Long Baseline Interferometer. Reconstruction of the Orbiter's trajectory will include allowance for stochastic accelerations due to spacecraft venting, gas leaks, and attitude reorientation maneuvers. Table III summarizes the estimated 1 sigma errors in the position and velocity of the Orbiter during the Probe relay link and their associated frequency errors. As can be seen, TABLE I Summary of Galileo error sources "a" Error Magnitude Constant Random Systematic (Hz) (Hz) (Hz) A priori A posteriori ------------------------------------------------------------------------------ Probe trajectory errors theta_p 0.006 deg 0.1 0.0 0.0 d/dt theta_p 10.0 m s^-1 1.1 0.14 "b" delta theta_p "c" 0.02 deg 1.53 NA "d" NA phi_p 0.231 deg 121.4 7.6 0.153 deg 81.6 4.6 "e" d/dt phi_p, wind 100-60 m s^-1 "f" 47.5 108.0 delta phi_p 0.2 deg "g" 0.0 109 d/dt phi_p, r_p "h" 4 km 0.0 0.3 r_p 4 km 0.2 0.4 d/dt r_p 0.2 m s^-1 "i" 2.4 1.8 Orbiter trajectory errors theta_0 6 x 10^-5 deg 7 x 10^-4 0.0 d/dt theta_0 5.5 x 10^-8 deg s^-1 7.7 x 10^-3 0.0 phi_0 4.7 x 10^-5 deg s^-1 0.025 0.0 d/dt phi_0 2.2 x 10^-9 deg s^-1 1.25 x 10^-3 0.0 r_0 0.094 km 2.5 x 10^-4 0.0 d/dt r_0 7.9 x 10^-3 0.036 0.0 ------------------------------------------------------------------------------ Error Magnitude"l" Constant Random Systematic (Hz) Delta f/f (Hz) (Hz) A priori A posteriori ------------------------------------------------------------------------------ Probe oscillator effects "k" Bus voltage +-2.3 x 10^-11 0 unknown 0.03 0.016 Grav accel. 8 x 10^-12 0 unknown 0.011 0.011 Spin accel. 4 x 10^-11 0 unknown 0.055 0.03 Temperature 1.2 x 10^-10 0 unknown 0.17 0.08 Pressure 10 ^-10 0 unknown 0.14 0.03 Magnetic field 3.6 x 10^-11 0 unknown 0.05 0.025 Short-term aging 4.3 x 10^-10 0 unknown 0.596 0.14 Allan variance "m" 2.0 x 10^-12 0 unknown 0.0 0.003 Radiation settling "n" 0 unknown 0.7 0.35 ------------------------------------------------------------------------------ Error Magnitude Constant Random Systematic (Hz) (Hz) (Hz) A priori A posteriori ------------------------------------------------------------------------------ Orbiter oscillator effects "o" Short-term aging 2 x 10^-11 0 unknown 0.027 0.027 Allan variance 2 x 10^-12 s^-1 0 unknown 0.0 0.003 Radiation settling 0 unknown -0.28 -0.28 ------------------------------------------------------------------------------ Error Magnitude Constant Random Systematic (Hz) (Hz) (Hz) A priori A posteriori ------------------------------------------------------------------------------ Probe spin "p" Spin axis offset 0 unknown +- 0.112 unknown Polarization 0 unknown +- 0.083 unknown Frequency measurement error "q" Measurement error 0.18124 0 0.052 "r" 0 0 Gravitational redshift Error Magnitude Constant Random"s" Systematic (Hz) (Hz) (Hz) A priori A post ------------------------------------------------------------------------------ Gravitational redshift -20.5 2.3 x 10^-4 -0.20 0.0082 ------------------------------------------------------------------------------ Atmospheric effects Error Magnitude Constant Random"u" Systematic (Hz) (Hz) (Hz) A priori A post ------------------------------------------------------------------------------ Neutral atmosphere "u" 0 0.04 0.30 0.043"v" Ionosphere "w" 2 x 10^-6 2 x 10^-7 0 0 ------------------------------------------------------------------------------ "a" At a probe aspect angle of 5 degrees, a 1 Hz error in frequency will lead to a 2.48 m s^-1 error in recovered zonal wind velocity, "b" Assuming a constant 10 m s^-1 meridional wind profile. "c" Due to integrated effect of meridional wind on Probe latitude after 30 min. "d" Not applicable. "e" Due primarily to time of entry uncertainty of 14.95 s. "f" Internal heat wind profile. "g" Due to integrated effect of internal heat wind profile on Probe longitude after 30 min. "h" Due to uncertainty in Probe radial location. "i" Assuming a fractional uncertainty in Probe descent velocity of about 0.005 (0.5 percent). "j" From Kenyon, 1986. "k" From Garriga, 1981; and Garriga, 1984. "l" 30-min drift rate. "m" Variance of the difference between consecutive 1 s USO frequency averages. "n" For nominal Probe entry radiation conditions. See text. "o" From Gussner, 1980. "p" Assuming Probe spin rate of 5 rpm. "q" Caused by digitization of Probe signal during frequency measurement process on the Orbiter. "r" Averaged over 1 s. "s" Due to uncertainties in Probe and Orbiter radial locations, and uncertainty in GM_J. "t" Random error based upon assumed 10 percent error in integrated electron density. "u" Based on Orton III atmosphere and refractive properties of H2, He, NH3, and CH4 mixtures at Jovian temperatures and pressures. See Atkinson and Spilker, 1989. "v" Random error based upon assumed 10 percent uncertainty in NH3 mixing ratio as measured by the Probe Neutral Mass Spectrometer, and 0.1 percent error in hydrogen and helium mixing ratios as measured by the Probe Helium Abundance Detector. See Hunten et al., 1986. "w" Assuming model ionosphere of constant electron density equal to 5 x 10^4 cm^-3 over a 4000 km layer. TABLE II Standard deviation in Probe entry parameters ------------------------------------------------------------------------------ Entry parameter 1-sigma uncertainty ------------------------------------------------------------------------------ Latitude 0.0064 deg Longitude 0.153 deg Time of entry 14.95 s Atmosphere relative velocity 1.212 m s^-1 Atmosphere relative entry angle 0.0205 deg Heading angle 0.0203 deg ------------------------------------------------------------------------------ TABLE III Standard deviation in Orbiter trajectory parameters "a" Parameter 1-sigma uncertainty 1-sigma frequency uncertainty (Hz) ------------------------------------------------------------------------------ r_0 94.32 m 0.000246 phi_0 4.712 x 10^-5 deg 0.0249 theta_0 6.011 x 10^-5 deg 0.000722 d/dt r_0 7.902 x 10^-6 km s^-1 0.0365 d/dt phi_0 2.207 x 10^-9 deg s^-1 0.00125 d/dt theta_0 5.534 x 10^-8 deg s^-1 0.0077 ------------------------------------------------------------------------------ "a" At time of Probe entry. these errors are extremely small. Furthermore. these values are primarily constant errors and so the systematic errors associated with the uncertainties in the Orbiter trajectory would be much smaller (cf. Table I). Uncertainties in the planet's solid body rotational velocity introduce errors in the absolute zonal wind speeds, i.e., the zero point of the wind speeds. They arise from two sources: the uncertainty in the planet's solid body rotation rate and the uncertainty in the Probe's distance from the planet's center. We assume that the planet's interior rotates at the same rate as the magnetic field. The fractional uncertainty of 10^-6 in the period of system III (Smith and West, 1982) produces an uncertainty in the planet's rotational velocity of about 1 cm s^-1 which is negligible compared to many other error sources. A larger uncertainty in the value of the rotational velocity results from the uncertainty in the radial distance of the Probe from the center of Jupiter. By combining occultation profiles obtained by the Voyager and Pioneer spacecraft, Lindal et al. (1981) have determined the gravitational equipotential surface of Jupiter at the 100 mb level. Based on the goodness of fit of the individual measurements to the proposed zenoid, the uncertainty in the radius of the 100 mb pressure level at the Probe's location will be +-4 km. This uncertainty produces a 0.6 m s^-1 uncertainty in the planet's rotational velocity at this level. Fig. 4. (a) Change in relay link frequency at 200 s after entry as a function of the offset in the Probe's entry longitude. (b) Change in the differential relay link frequency (instantaneous value - value at the reference time) as a function of time from entry for alternative values of the offset in the Probe's entry longitude. In addition to this constant error, there will also be a small systematic error in the rotational velocity due to errors in deriving the altitude separation between the 100 mb level and some deeper level, based on the ASI data. By the end of the relay link, the error in this altitude separation may reach about 0.05 km. Such an error produces a systematic error in velocity of about 0.9 cm s^-1, which is negligibly small compared to other error sources. Finally, two small changes in the measured frequency arise from the Probe's spin. First, the deliberate misalignment of the Probe's antenna and its spin axis results in a periodically varying velocity of the antenna along the line of sight and, hence, a periodic modulation of the measured frequency that depends on the spin rate, distance of the antenna from the central axis, and its offset angle. As shown in Table 1, a frequency modulation (peak to peak) of about 0.2 Hz results for a nominal spin rate of 5 rpm. The magnitude of this modulation is proportional to the spin rate and has a period equal to the spin rate. Conceivably by appropriately filtering the measured frequency, we may be able to recover the spin rate as a function of time during descent (it will vary slowly). Since the Probe's antenna transmits circularly polarized signals, the transmitted frequency is slightly altered by the antenna's rotation (Paper I). The observed frequency is shifted by an amount equal to the spin rate (0.08 Hz for the nominal rate). 3.2. OSCILLATOR ERRORS Much effort has been expended in constructing extremely stable oscillators for use on the Probe and Orbiter, in shielding them, and in calibrating their behavior, especially for frequency shifts induced by a variety of environmental factors. As a result, frequency drifts of the USOs on the Probe and Orbiter should not seriously compromise the retrieval of winds from the Doppler data (indeed, this experiment influenced the design of the USOs), but these drifts can still have some effect on the accuracy of the retrieved winds. Here, we first summarize pertinent characteristics of the USOs and then enumerate the drifts they can undergo due to a variety of factors, with emphasis on the largest drifts. The heart of the USO is an 'SC' cut quartz crystal that is extremely stable in its resonance frequency and that is much less sensitive to g-loading, thermal disturbances, and radiation than the 'AT' cut used on the Voyager spacecraft. Furthermore, the Galileo crystals have been radiation hardened by exposure to a one Mrad dose of gamma rays from a cobalt-60 source to minimize their susceptibility to radiation-induced drifts. The housing for the crystal includes a double-insulated temperature control oven to maintain a highly stable operating temperature. The Orbiter USO is identical in terms of the crystal and its housing to the Probe USO. Various environmental factors and the intrinsic properties of the USOs will cause their frequencies to change by small, but not totally trivial amounts over the course of the relay link. A series of calibration tests on the flight USOs have provided estimates of the magnitudes of these drifts as well as calibration curves that will enable them to be partially modeled, thereby reducing somewhat their impact on the wind recovery. As shown in Table I, the separate and the cumulative errors due to time variations of pressure, temperature, bus voltage, acceleration, and orientation of the magnetic field produce very small systematic errors in the Probe's carrier frequency. Even in the absence of all known perturbations, the frequency of the oscillators will undergo some small drift in frequency. These effects are referred to as 'aging'. The calibration tests indicate that the USOs undergo secular frequency drifts of 0.6 Hz over a 30-min period, with an uncertainty of 0.14 Hz (Paper I). On much shorter time scales, the frequency experiences random variations of approximately 0.003 Hz ('Allen variance'). Jovian magnetospheric particles can alter the secular drift rate of the frequency of the Probe and Orbiter's USOs, thereby degrading the Doppler wind experiment. Figure 5(a) shows an estimate of the radiation dose rate (mostly bremsstrahlung X-rays) that the Probe's USO will experience as it passes through the inner magnetosphere on its way to the planet. This figure was derived from an empirical model of the Jovian magnetosphere, based on data from the Pioneer and Voyager spacecraft (e.g., Armstrong et al., 1981; Galileo Project, 1982), the nominal Probe trajectory, and the shielding provided by the spacecraft and oscillator housing (P. Garcia, private communication). Since the radiation is highest in the equatorial plane of the magnetic field, which is tilted Fig. 5. (a) Nominal and worst case dose rates that will be experienced by the Probe's ultrastable oscillator as a function of time before entry and distance from the center of Jupiter in units of the planet's equatorial radius. These rates include an allowance for shielding by the spacecraft and oscillator housing. (b) Model for the radiation-induced drift rate derived from the calibration test results. (c) Comparison of the predictions of the radiation model (solid curve) with a test result involving a dose rate of 1 rad s^-1 applied for 1200 s. (d) Predicted radiation-induced drift rate for the Galileo Probe USO as a function of time after entry for the nominal and worst case radiation models of Figure 5(a). by 11.5 degrees to the rotational equatorial plane, the actual radiation seen by the probe will vary sinusoidally with time. The curve labelled 'nominal' in Figure 5(a) refers to conditions that have been averaged over system III longitude, while 'worst case' shows the dose rate for system III longitudes that place the Probe closest to the magnetic equator. According to Figure 5(a), the Probe's USO experiences a maximum dose rate of about 1.4 rad s^-1 at 2 R_J from the planet's center, which occurs about 50 min before entry, where R_J is the planet's equatorial radius. A somewhat smaller, but more time constant dose rate will be experienced by the Orbiter's USO, which will be located at about 4.0 R_J during the relay link. In 1981, calibration tests were performed on the flight USOs to determine the effect of radiation on their frequency. Of greatest interest was the alteration to the secular drift rate of this frequency. The tests were performed following radiation hardening. They involved exposing the USOs to gamma rays from a cobalt-60 source. In the case of the Probe's USO, the dose rate and the total dosage were varied. In addition, one test involved the use of a time-varying dose rate. Less extensive tests were performed on the Orbiter's USO, in which the dose rate, but not the total dosage was varied. For most of these tests, the dose rates used (0.5 to 3 rad s^-1) were comparable to those expected in the Jovian environment. We have developed an empirical model of the frequency drift induced by radiation by simulating the radiation calibration tests. Figure 5(b) illustrates this model and Figure 5(c) shows its ability to reproduce one of the tests. In both cases, a constant dose rate was applied for a limited time interval. Clearly, the model approximately reproduces the data. In actuality, especially for the Probe, the applied dose rates will vary significantly as the Probe approaches the planet (cf. Figure 5(a)). We can apply the model of Figure 5(b) to this situation by appropriately averaging, NOT summing, the response curves produced by a series of discrete and constant dose rates that approximate the actual time-varying behavior. Based on this approach and the dose rates of Figure 5(a), we estimated the radiation induced drifts that the Probe's oscillator will undergo during the relay link, as shown in Figure 5(d). The kinks in the predicted curves are artifacts of our discretizing the dose rate curves for use in our model. Averaged over the relay link, the nominal radiation-induced drift would produce a frequency change of about 0.4 Hz over a 30-min period. The corresponding curve for the 'worst case' dose profile yields a drift that is about a factor of 2 higher than the nominal case near the start of the link and fairly similar to the nominal case near the end of the link. A somewhat smaller drift will be experienced by the USO on the Orbiter. This USO will experience a more constant radiation dose rate of about 1 rad s^-1. Tests on the Orbiter USO suggest that it will experience a radiation induced drift of about 0.07 Hz during the relay link. 3.3. PROPAGATION EFFECTS The frequency of the Probe's signal, as measured at the Orbiter, can be affected by its passage through the Jovian atmosphere and ionosphere. First, the propagation time is altered by the signal's passage through a refracting medium. More precisely, the phase path length will be retarded by the neutral atmosphere and advanced by the ionosphere. Since the phase path length varies with time due to these effects, the measured frequency will be altered from the value that would be found in the absence of a refracting medium along the path. Second, the direction of the signal is slightly altered by refraction. This induces a small, time-varying, change to the Probe aspect angle for the ray that reaches the Orbiter, thereby altering the Doppler shift produced by the horizontal winds and the planet's rotational velocity. These effects, especially that due to the neutral atmosphere, can be quite large. But, fortunately, they can be allowed for with high precision in the a posteriori analysis, since the indices of refraction along the path length will be known quite well. Based on the formalism developed in Paper I and Atkinson and Spilker (1990) and the characteristics of the Probe's descent profile during the relay link, we find that refraction by the neutral atmosphere can be expected to produce a systematic variation of less than a Hz in the measured frequency over the relay link (Table I). Furthermore, almost all of this variation can be removed a posteriori by using the highly accurate profiles of pressure and temperature, obtained by the ASI experiment, in conjunction with compositional information obtained from several Probe experiments. In a similar vein, frequency drifts due to passage through the Jovian ionosphere can be estimated from the formalism of Paper I and Atkinson and Spilker (1990). Using electron density profiles derived from past spacecraft occultation experiments, we find that the frequency shifts are quite small, even a priori (cf. Table I). Even smaller shifts occur during the passage of the signal through the Jovian magnetosphere. 3.4. MISCELLANEOUS EFFECTS The measured frequency is also affected by the gravitational red shift of general relativity, the time dilation effect of special relativity, and other higher-order terms that appear when the exact form of the Doppler shift equation is used. Here, we simply provide estimates of the magnitudes of these effects to assess the importance of including them in the final analyses. All of these can be treated quite accurately and so need not generate significant errors in the recovered wind. The gravitational red shift produces a large constant frequency shift and a very small systematic one, both of which can be reduced to negligible amounts by appropriately calculating them (Paper I; cf. Table I). Errors introduced by neglecting higher-order terms in the equation of special relativity (cf. Equation (3)) and by truncating the Taylor series expansion of the Doppler frequency residuals (cf. Equation (7)) result in systematic frequency errors on the order of 10^-2 Hz (Paper I). Thus, they can be safely neglected. 4. Simulations In this section, we use the linear least-squares algorithm developed in the approach section to recover wind profiles from simulated data. The simulated data were generated for each of the four wind profiles of Figure 1. In most cases, we included the more prominent errors of Table I to make the simulations realistic. These simulations are directed at addressing and answering several basic questions about our approach and the feasibility of recovering a wind profile from the Galileo Probe's telemetry: Are Legendre polynomials a useful set of basis functions for the zonal winds of Jupiter? How would we detect and analyze regions of sudden shears? What accuracy in the recovered wind profile is possible, given plausible error sources? What are the limitations on the vertical resolution of the recovered profile? We carried out simulations for the nominal Probe and Orbiter trajectories for a time interval ranging from 200 to 300 s after Probe 'entry'. Probe entry occurs when it reaches an altitude of 450 km above the 1 bar pressure level. During the time interval of the simulations, the Probe traverses pressure levels ranging from 800 mb to 25 bars. In reality, the relay link could begin as early as 120 s after entry, when the Probe is at the 100 mb level. The conclusions drawn based on the chosen period of simulation are equally valid for other possible periods. Figure 2 shows the variation of the Probe aspect angle, psi, and azimuth angle, alpha, with time from entry. These angles define the relationship between the zonal direction and the line of sight and hence the projection of the zonal wind speed on the line of sight. At the start of the simulated relay link, psi has a value of 6 degrees; it reaches a minimum value of 1.6 degrees at 21 min, the time that the Orbiter crosses the Probe's meridian; and it increases to a value of nearly 10 degrees by the end of the link. Throughout almost all this time, alpha has a value close to 0 degrees, since the Probe and Orbiter have nearly coplanar locations, but it goes through 90 degrees at the time that the Orbiter crosses the Probe's meridian. The first step in performing a simulation is to construct two frequency profiles. We refer to the first profile of frequency as a function of time as a 'true' profile, in the sense that it corresponds to what would really be measured and conveyed from the Orbiter to the Earth for analysis. It is generated by including the influence of the candidate wind profile on the measured frequency, by using the 'real' trajectories, and by including a variety of effects that would change the USOs' frequencies from their ground-based measured values, including oscillator aging, probe spin, atmospheric turbulence, measurement errors at the Orbiter, and the Gravitational red shift. The second profile, the 'nominal' profile, is obtained by neglecting winds (they are not known a priori), using the ground-based frequency of the oscillators, and trajectory information that differs from the 'real' trajectories by 1 sigma errors based on Table I. Table IV summarizes the errors introduced into the two profiles that were used in the simulations discussed below. TABLE IV Doppler wind recovery simulation errors and uncertainties ------------------------------------------------------------------------------ Parameter Magnitude Reference ------------------------------------------------------------------------------ Probe USO zero-point frequency 500 Hz RLIT report "a" Probe spin 20 rpm RLIT report Probe buffeting 0.36 cos(24t) "b" RLIT report Atmospheric turbulence 0.08 cos(5.6t) "b" RLIT report Fractional Probe USO drift 1.0 x 10^-9/30 min Paper I Probe oscillator Allan variance sigma = 2.0 x 10^-12 s^-1 Paper I Gravitational redshift -20.53 to -20.73 Hz Paper I Magnetometer boom interference 0.05 sin(7.9t) "b" RLIT report Frequency measurement error sigma = 0.0523 Hz Paper I Probe longitude 0.231 deg Paper I Probe latitude 0.006 deg Paper I Probe radius 21.4 km Paper I Orbiter longitude 4.7 x 10^-5 deg Paper I Orbiter latitude 6.0 x 10^-5 deg Paper I Orbiter radius 0.094 km Paper I Probe descent velocity 0.0043V_d "c" Paper I ------------------------------------------------------------------------------ "a" Galileo Probe-Orbiter Relay Link Integration Team Report (Bright, 1984). "b" Time after Probe entry (s). "c" Probe descent velocity m s^-1. Once the two frequency profiles have been generated, we subtracted the nominal profile from the true one to generate frequency residuals. Next, we subtracted from the residuals at all times t the residuals at the reference time, t_r, which we take as the initial time of the link. These operations produce the Doppler data, Delta f_Dop. We apply the least-squares algorithm to this Doppler data to recover the zonal wind speed profile and compare the recovered profile with the 'true' profile to assess the accuracy of the recovery. This assessment is done by plotting the two wind profiles as well as evaluating their root mean square (r.m.s.) difference and their mean difference. The latter provides a measure of the systematic offset between the two profiles. In order to keep the computational time at a reasonable level, we generated data every 10 s during the relay link (280 samples altogether). It might seem that the time of overflight offers a better choice for the reference time, t_r, since the zonal winds do not affect the measured frequency at overflight. Indeed. such a choice might neatly yield the Probe USO's zero-point frequency, if the meridional and vertical wind speeds are sufficiently small. Unfortunately, the 0.23 degree uncertainty in the Probe's descent longitude, when combined with the planet's rotational speed of 10 km s^-1, leads to a 26 m s^-1 error in the Probe-Orbiter range rate. This offset is essentially constant throughout the relay link so that it does not provide a marker of the actual time of overflight. Thus, the uncertainty in the Probe's position prevents a useful determination of the USO's zero-point frequency from being made. But, the near constancy of the resulting error permits its impact to be substantially reduced by using frequency differences as the primary data to be analyzed. We first assess the adequacy of a sum of Legendre polynomials to represent the wind profiles by determining the r.m.s. error in the recovered wind profile as a function of the order of the polynomial expansion for the four profiles of Figure 1. For all four wind profiles of Figure 1, an asymptotic error of a few m s^-1 is closely approached by expansions of order 5 and larger (Paper I). Below, we perform further simulations using an expansion of order 5. We conclude that the Legendre polynomial expansion provides a good representation of the wind profiles of Figure 1, since only a small number of terms are needed and since the r.m.s. error is comparable to that expected from the error sources of Table IV. We next assess the influence of random errors in the measured frequency on the accuracy of the recovered wind profile. For this purpose, we used covariance analysis. The mathematical procedure used to relate variances in the measured frequency to variances in the recovered wind speed is given in Paper I. Using the random errors given in Table I for the Allen variances of the Probe and Orbiter USOs and for the digitization error, we find that the total random error in the measured frequency is approximately 0.053 Hz. The corresponding random errors in the recovered winds at several times during the relay link are given in Table V for the latent heat wind profile of Figure 1. Similar results pertain to the other wind profiles. The variances in the wind speed shown in Table V are less than 1 m s^-1. Thus, our least-squares formalism is stable for reasonable choices of random errors and the implied variances in wind speed will not significantly degrade its recovery. The application of this formalism to our standard model in which the entry longitude of the Probe was added as an unknown showed that it was not possible to also derive this key parameter from the Doppler data: very large variances in wind speed were obtained in this case. We next present the results of full-up simulations that incorporate the ensemble of constant, random, and systematic errors given in Table IV. Figures 6(a), 7, 8, and 9 TABLE V 1-sigma deviation in recovered winds ------------------------------------------------------------------------------ Time after entry (s) sigma_vh "a", "b" (m s^-1) 200.0 0.088 1000.0 0.264 2000.0 0.384 3000.0 0.474 ------------------------------------------------------------------------------ "a" 1 sigma-deviation in horizontal wind velocity at Probe location. "b" Includes Probe and Orbiter USO Allan variances, and frequency measurement error. show comparisons of the recovered wind profiles with the 'true' profiles for the four standard wind models. Table VI summarizes the r.m.s. and mean wind errors for these 4 simulations. Finally, Figure 6(b) shows the corresponding least-squares residuals as a function of pressure in the atmosphere for the first of these recoveries. Similar residuals pertain to the other cases. The frequency residuals are the instantaneous differences between the frequencies derived from the recovered winds and the 'true' values. (In both cases, the frequency at the start of the relay link has been subtracted out.) As anticipated earlier in the paper, it is possible to recover not only the shape of the wind profiles, but also their absolute values. The residual profiles do not show any large biases over limited regions of pressure, indicating that the least-squares algorithm is working well. To evaluate the relative importance of the various error sources of Table IV in affecting the accuracy of the wind recovery and the possible impacts of sources not considered in this table, we have carried out a series of sensitivity studies. For this purpose, we used the latent heat wind profile as a base profile. As a standard of comparison, we first performed a wind recovery with all the errors set equal to 0. As shown in Table VI, the recovered winds are not in exact agreement with the true ones. This difference is attributable to a combination of the truncation of the polynomial expansion at order 5 and the finite number of data points used in our analysis. We next conducted several numerical experiments in which alternately only errors due to frequency effects, Probe descent velocity, and Probe entry longitude were considered. These three types of errors are the most significant ones, according to Table I. As shown in Table VI, the error in entry longitude has the biggest impact on the accuracy of the wind recovery. The large impact of an error in Probe longitude on the accuracy of the recovered winds prompted us to evaluate the errors in these winds that resulted when the errors in Probe entry longitude and latitude were systematically varied. According to the results shown in Table VI, both the r.m.s. error and the mean error increase significantly, but not catastrophically, as the errors in these two coordinates are separately increased. So far, we have considered only the nominal wind profiles of Figure 1, all of which Fig. 6. (a) Comparison of the retrieved wind profile with the actual profile of the zonal wind speed of the internal heat wind model for simulations in which a nominal set of errors was used. (b) Residuals as a function of pressure for the retrieval of (a). vary comparatively slowly with pressure. Conceivably, there could be localized regions of the Jovian atmosphere where much steeper shears exist. To simulate this possibility we added a shear to the latent heat wind profile. The shear region was placed in the 2.3 to 4.4 bar location. which the Probe traverses in about 250 s. Thus, about 25 data points out of a total of 280 were affected by the shear. Figures 10(a) and 10(b) show the retrieval and residuals obtained for a case in which a shear of tens of m s^-1 Fig. 7. Same as Figure 6(a) for the latent heat wind model. Fig. 8. Same as Figure 6(a) for the solar energy deposition wind model. was imposed on the nominal wind profile (wind model of Figure 10(a)). The corresponding r.m.s. and mean errors are given in Table VI. Even in this extreme case a reasonably accurate recovery is possible, except of course in the shear zone itself. Furthermore, the residuals show an anomalous behavior in the region of the shear. In such a case, we will use a different type of analysis than the one used for the large-scale winds, one similar to the classical Doppler method. We are currently developing the protocol to recover both the shear and the large-scale wind profile in an iterative fashion. Fig. 9. Same as Figure 6(a) for the ortho/para hydrogen conversion wind model. Finally, we considered the impact on the recovery of a time-independent meridional wind. A constant 10 m s^-1 north-south wind had a negligible impact on the errors of the recovered wind, while even a 100 m s^-1 north-south wind did not increase the r.m.s. TABLE VI Errors in the wind recoveries ------------------------------------------------------------------------------ Model Input errors r.m.s. errors Mean errors (m s^-1) (m s^-1) Latent heat None 1.52 -0.03 Internal heat Nominal "a" 2.92 -2.65 Latent heat Nominal 3.53 -2.70 Solar energy Nominal 4.86 -3.33 Ortho/para Nominal 3.14 -2.91 Ortho/para Only frequency 1.80 -0.99 Ortho/para Only desc. vel. 1.72 0.40 Ortho/para Only entry long. 2.41 -1.82 Ortho/para Delta lat. = 0.05 deg 2.27 1.21 Ortho/para Delta lat. = 0.10 deg 2.73 1.93 Ortho/para Delta lat. = 0.20 deg 3.89 3.38 Ortho/para Delta lat. = 0.50 deg 8.02 7.78 Ortho/para Delta long. = 0.05 deg 1.85 0.09 Ortho/para Delta long. = 0.10 deg 2.05 1.08 Ortho/para Delta long. = 0.20 deg 3.53 1.55 Ortho/para Delta long. = 0.50 deg 3.84 3.24 Latent + shear Nominal 6.46 -2.72 Latent + meridional (10 m s^-1) Nominal 3.54 -2.63 Latent + meridional (100 m s^-1) Nominal 4.42 -4.02 ------------------------------------------------------------------------------ "a" Includes frequency errors, and uncertainties in Probe and Orbiter trajectories as listed in Table IV. Fig. 10a-b. Same as Figures 6(a) and 6(b) for the latent heat wind model with a strong shear added to it in a localized region of the atmosphere. and mean errors by very much (Table VI). This insensitivity is due primarily to the small values of alpha during much of the descent sequence, which minimizes the projection of the meridional wind on the line of sight. In addition, the recovery is not too sensitive to the latitudinal location of the Probe. The vertical resolution of the retrieved wind profile is fundamentally limited by the finite response time of the Probe to changes in the zonal wind speed. According to the derivations given in Seiff et al. (1980) and Paper I, the Probe's zonal wind speed adjusts to a change in the atmospheric value on a time-scale tau_h = w_p/g, where w_p is the vertical descent speed of the Probe and g is the acceleration of gravity. During this characteristic time scale, the Probe moves vertically a distance, l_min = w_p tau_h. Table VII gives estimates of the Probe's descent velocity, response time tau_h and minimum vertical distance, l_min, for a range of positions in the Jovian atmosphere. As indicated above, the relay link will begin when the Probe is at the several hundred mb level and end when the probe is at about the 25 bar level. According to Table VII, the TABLE VII Probe response time characteristics ------------------------------------------------------------------------------ Pressure Density w_p "a" tau_h "b" l_min "c" (bar) (kg m^-3) (m s^-1) (s) (m) 0.01 0.0019 770.55 32.72 25212.4 0.1 0.0244 213.66 9.20 1965.7 0.5 0.1020 104.50 4.50 470.2 1.0 0.1650 82.26 3.54 291.2 5.0 0.4990 47.25 2.03 95.9 10.0 0.8110 37.07 1.60 59.3 20.0 1.3190 29.06 1.25 36.3 ------------------------------------------------------------------------------ "a" Probe terminal velocity. "b" Time required for Probe to recover 1 - e^-1 of step wind gust. "c" Minimum detectable wind structure, equal to w_p x tau_h. Probe adjusts to changes in the zonal wind speed on time scales ranging from almost 10 s at the start of the link to slightly more than 1 s at its end. The corresponding minimum distances range from on the order of 1 km to several tens of meters. Thus, the Probe responds quickly enough to changes in the zonal wind speed so that good vertical resolution is possible during the entire relay link. Near the end of the relay link the response time approaches the sampling time of 2/3 s that separates successive measurements of the Probe's frequency made by the Orbiter's relay radio system. 5. Conclusions We have developed a practical approach for estimating the zonal wind speed profile within Jupiter's upper troposphere from its influence on the frequency of the Galileo Probe's telemetry signal. This was achieved by linearizing the relativistic Doppler shift equation about a state of zero wind speed, expanding the zonal wind speed in a Legendre polynomial series, and reducing the resulting equations to a linear least-squares problem. Since the instantaneous Doppler shift depends on both the instantaneous zonal wind speed and a time integral of this speed, it is possible to recover not only the vertical wind shear, but also the absolute wind velocity without an accurate knowledge of the USOs' zero-point frequencies. There are a very large number of factors that can introduce errors into the wind profile retrieved from the Probe's Doppler-shifted frequency. Fortunately, the collective impact of these error sources appears to be sufficiently mild so that the uncertainty in the retrieved winds should be only a few m s^-1. The largest error source is the uncertainty in the Probe's entry location. Consequently, it is essential that the Orbiter's trajectory be carefully monitored before, during, and after Probe release to provide as accurate a determination of the Probe's trajectory subsequent to release and hence as accurate an estimate of its entry location as possible. Other potentially major error sources include uncertainties in the descent velocity and frequency errors, especially those produced by systematic drifts (e.g., radiation-induced drifts and aging). Finally, random variations in frequency can produce r.m.s. errors in the wind speeds of several tenths of a m s^-1 as determined by co-variance analysis. If the error sources have magnitudes comparable to or smaller than the ones given here, the retrieved wind speed profiles can be used for a number of important purposes. These include placing a key constraint on the basic drive for the atmospheric circulation as well as anchoring winds derived from orbiter images to well-defined vertical positions in the atmosphere. With regard to the first of these objectives, it might seem a little naive to expect to learn something fundamental about the atmospheric circulation from a single vertical profile. Certainly, the retrieved profile will be influenced by eddy motions on a variety of scales and thus will not be an accurate representation of the time and longitudinal average profile at the entry latitude. However, experience with Venus indicates that the time and zonally-averaged zonal winds are the dominant component of zonal winds measured at a single entry site (Councelman et al., 1980). Finally, as mentioned earlier, data taken by a variety of Galileo experiments should help to flesh out the various wind models of Figure 1, leading to more precise predictions that can be compared to and, hopefully, distinguished among by the wind profile retrieved by the Galileo Doppler wind experiment. References Armstrong. T.P., Paonessa, M.T., Brandon, S.T., Krimigis, S.M., and Lanzerotti, L.J.: 1981, J. Geophys. Res. 86, 8343. Atkinson, D.H.: 1989, 'Measurement of Planetary Wind Fields by Doppler Monitoring of an Atmospheric Entry Vehicle', Ph.D. Thesis, Washington State University, Pullman, WA. Atkinson, D.H. and Spilker, T.R.: 1991, Radio Sci. (in press). Born, M. and Wolf, E.: 1964, Principles of Optics, MacMillan Co., New York. Budden, K.G.: 1961, Radio Waves in the Ionosphere, Cambridge University Press, Cambridge. Busse, F.H.: 1976, Icarus 29, 255. Condran. L.D.: 1983, Final Test Report for Galileo Probe System Test Drop No. 2, General Electric Report 83 SDS 2148. Conrath, B.J. and Gierasch, P.J.: 1984, Icarus 57, 184. Conrath, B.J., Flasar, F.M., Pirraglia, J.A., Gierasch, P.J., and Hunt, G.E.: 1981, J. Geophys. Res. 86, 8769. Councelman. C.C. III. Gourevitch, S.A.. King, R.W., Loriot, G.B., and Ginsberg, E.S.: 1980, J. Geophys. Res. 85, 8026. Cunningham. C.C., Hunten, D.M.. and Tomasko, M.G.: 1988, Icarus 75, 324. Galileo Project Document JP-590: 1982. Gierasch, P.J.: 1976, Icarus 29, 445. Gierasch, P.J.: 1983, Science 219, 847. Golub, G.H. and Van Loan, C.F.: 1983, Matrix Computations, Johns Hopkins University Press, Baltimore, MD. Hanel, R.A., Conrath, B.J., Herath, L.W., Kunde, V.G., and Pirraglia, J.A.: 1981, J. Geophys. Res. 86, 8705. Holton, J.R.: 1979, An Introduction to Dynamic Meteorology, Academic Press, New York. Hunten, D.M., Tomasko, M., and Wallace, L.: 1980, Icarus 43, 143. Ingersoll, A.P. and Pollard, D.: 1982, Icarus 52, 62. Ingersoll, A.P. and Porco, C.C.: 1978, Icarus 35, 27. Ingersoll, A.P., Beebe, R.F., Mitchell,J.L., Garneau, G.W., Yagi, G.M., and Muller, J.P.: 1981, J. Geophys, Res. 86, 8733. Kerzhanovich, V.V., Gotlib, V.M., Chetyrkin, N.V., and Andreev, B.N.: 1969, Kosmicheskie Issledovaniya 7, 592 (translated). Kerzhanovich, V.V., Marov, M. Ya., and Rozhdestvensky, M. K.: 1972, Icarus 17, 659. Limaye, S.S.: 1985, Adv. Space Res. 5, 51. Lindal, G.F. et al.: 1981, J. Geophys. Res. 86, 8721. Marov, M. Ya. et al.: 1973, Icarus 20, 407. Massie, S.T. and Hunten, D.M.: 1982, Icarus 49, 213. Orton, G.S.: 1981, JPL Report 1625-125. Pirraglia, J.A.: 1984, Icarus 59, 169. Preston, R.A. et al.: 1986, Science 231, 1414. Rossow, W.B., Del Genio, A.D., Limaye, S.S., Travis, L.D., and Stone, P.H.: 1980, J. Geophys. Res. 85, 8107. Sagdeev, R.Z. et al.: 1986, Science 231, 1411. Schubert, G. et al.: 1980, J. Geophys. Res. 85, 8007. Seiff, A. and Knight, T.C.D.: 1992, Space Sci. Rev. 60, 203 (this issue). Seiff, A. et al.: 1980, J. Geophys. Res. 85, 7903. Smith, R.E. and West, G.S.: 1982, Space and Planetary Environmental Criteria Guidelines for Use in Spacecraft Development, Vol. II, NASA Technical Memorandum 82478. Stone, P.H.: 1967, J. Atmospheric Sci. 24, 642. Taylor, F.W. et al.: 1980, J. Geophys. Res. 85, 7963. West, R.A., Strobel, D.F., and Tomasko, M.G.: 1986, Icarus 65, 161.