Experimental constraints and data selection

During the BSR experiment at Vesta, the Dawn spacecraft’s HGA was used to transmit telemetry data at X-band radar frequency (8.435 GHz, 3.55 cm wavelength) in RCP, while the three 70-m DSN antennas on Earth—at Goldstone (USA), Canberra (Australia) or Madrid (Spain)—were used to receive22. The three receiving systems have the same measurement requirements in terms of noise temperature, antenna gain, pointing loss, and polarization loss as listed in Supplementary Table 1. In two-way coherent downlink mode, Dawn’s transmission frequency is driven by the DSN’s uplink frequency and yields a Doppler stability (Δf/f) of 10−12 over 60-s measurements, i.e., a frequency drift of ~0.0001 Hz s−1 1. In one-way or non-coherent downlink mode—used when the uplink signal is expected to be unavailable, such as the minutes preceding and following an occultation—the transmission frequency is driven by an onboard internal auxiliary oscillator with a maximum frequency drift of 0.05 Hz s−1 37. One-way downlink mode contains too much Doppler noise to be used for gravity science23 but is sufficiently stable for BSR measurements since they are integrated over a much shorter timespan—a few seconds, as opposed to one minute for gravity science measurements—and because we are measuring the relative Doppler shift between the surface echo and direct signal, which are equally affected by slow Doppler changes.

Dawn’s HGA was almost constantly pointed at ground stations on Earth for communication1, so by continuously transmitting basic telemetry information, we had the opportunity to observe surface reflections of the radar signal just before and after each occultation of the Dawn spacecraft behind Vesta at highly oblique incident angles of ~89° as depicted in Supplementary Fig. 1, similar to geometry of the BSR experiment at Mars by the Mars Global Surveyor38. In order to analyze the resulting BSR data, we selected occultations that occurred specifically during Dawn’s lowest-altitude mapping orbit (LAMO) around Vesta, and using the HGA to ensure the strongest observable surface echoes. During LAMO, there were 16 unique orbits during which an occultation occurred, and therefore 16 individual entries and 16 individual exits.

Out of these events, several were discarded based on the following criteria: (1) if the radar amplitude data files containing RCP and LCP were copies of each other; (2) if the direct (carrier) signal’s power did not consistently exhibit 50 dB strength before or after its occultation; (3) if the surface echo was indistinguishable from the noise (i.e., < ~3 dB); (4) if the window of occultation entry or exit was so brief that the signal disappeared faster than our temporal resolution (e.g., if the carrier dropped by 30 dB within a few seconds, and usually showed no indication of a measurable surface echo); (5) if the δf was too small to distinguish the surface echo’s power from that of the carrier; or (6) if the surface echo occurs in a region of high topographic variability with respect to the incident HGA beam size, which is assessed below. In all, five entries and nine exits passed our criteria for analysis and measurement of σ (km2) and σ/σ max (dB). The following sections describe the method used to process and analyze the resulting BSR data. Supplementary Table 1 summarizes the acquisition parameters of the BSR experiment.

Processing DSN BSR data

Radio waves received by the DSN 70-meter antennas are recorded as amplitude (voltage) versus time, and are collected in two channels: RCP and LCP. Within each channel, amplitude data is recorded in two components, in-phase I and quadrature Q, which correspond to the real and imaginary parts of the complex voltage, respectively.

The raw modulated telemetry data collected at the DSN antenna is sampled at a rate of 16 kHz. In addition, a Doppler shift correction is applied to the X-band receiving frequency in order to counteract the calculated Doppler shift induced by Dawn’s orbit around Vesta. However, the DSN utilizes a local oscillator to subtract the ephemeride Doppler shifting, and what persists is a low frequency component of error, which we observe is as much as 300 Hz. The offset frequency of the carrier signal is observed to decrease as the spacecraft moves further away from Earth during occultation entry—and as expected, increases as the spacecraft moves closer toward Earth after leaving occultation. Given that most of the BSR observations of occultations were conducted in one-way mode, the accuracy of the predicted receiving frequency was also diminished in the time leading up to and following these unique orbital geometries. Hence, when radar amplitude data is plotted against time, a change is observed in the overall envelope frequency of the sinusoidal amplitudes over the course of, for example, 1 min, during which the carrier signal may shift from 200 to 100 Hz. Notably, this frequency offset does not impact the relative Doppler shift between the surface echo and direct signal.

To seek surface echoes, the DSN I-and-Q amplitude time series data is converted into the frequency domain by taking the complex fast Fourier transform. The power frequency spectrum is generated in voltage-squared per Hz such that:

$${P_{{\rm{spec}}}} = \frac{1}{{{t_{{\rm{spec}}}}}}{\left| {{\rm{FFT}}\left( {{A_I} + i{A_Q}} \right)} \right|^2}$$ (1)

The duration t spec over which to generate each power spectrum is selected only after assessing the theoretical Doppler separation δf between the direct signal and any surface echoes, as this dictates the frequency resolution necessary to accurately distinguish each surface echo. Background noise is smoothed by averaging power spectra together, but surface-echo strength may be diminished in the process. The latter occurs when spectra are averaged over times that lack a surface-echo signal or while the echo has changed in frequency. The final parameters used to generate power spectra are outlined at the end of the following section.

Calculating the differential Doppler shift

In order to verify the detection of surface reflections within BSR power spectra, the theoretical differential Doppler shift δf is calculated between the direct signal and grazing surface-reflected echoes, and compared with measured values. Theoretical δf is estimated from the known positions and line-of-sight velocities between (1) the Dawn spacecraft (D) at the time of transmission; (2) the center coordinate of the radar-illuminated surface on Vesta (Vpt) when the signal reaches the surface; and (3) the receiving antenna on Earth (E) after accounting for light travel time; each of which are extracted from the reconstructed trajectory of Dawn’s orbit that is provided in SPICE ephemeride data28.

For occultation entry during orbit 355, Supplementary Fig. 2 shows the instantaneous total velocities v of Dawn, the surface point of reflection on Vesta (hereafter referred to as “the echo site” or “the site”), and the receiving antenna on Earth within the bistatic plane—defined as the plane containing all three bodies at a given moment12. All positions and velocities of D, Vpt and E are obtained with respect to Vesta’s inertial frame of reference, such that the origin of the Cartesian coordinate system is Vesta’s center of gravity and Vesta’s equator defines the xy-plane. The individual components of each body’s instantaneous velocity is listed in Supplementary Table 2 for occultation entry during orbit 355, where each velocity v A is given in m s-1 along the line of sight with respect to the position \({\hat r_{\rm{B}}}\). Velocity components are defined to be positive if body A is moving toward target B, and negative if away. For example, the notation \({\it v_{\rm{D}}}{\hat r_{{\rm{Vp\it t}}}}\) = −82.2 m s−1 (column 2, row 4) indicates that Dawn is traveling away from the echo site at a speed of 82.2 m s−1.

The differential Doppler shift δf between the surface echo and direct signal is calculated by:

$$\delta f = \Delta {f_{{\rm{echo}}}} - \Delta {f_{{\rm{direct}}}}$$ (2)

where the absolute Doppler shift Δf of the direct signal or surface echo depends on the relative velocity of the transmitting and receiving bodies along their line of sight as described by Simpson27. Supplementary Table 3 shows the mathematical definition and calculation of individual contributions to the total theoretical differential Doppler shift (δf total ) between the surface echo and direct signal, which are the result of motions along the line of sight between Dawn and Earth (column 1), Dawn’s orbital motion around Vesta (column 2), and Vesta’s rotation (column 3).

The absolute Doppler shift of the direct signal Δf direct is first calculated from the combination of Dawn’s instantaneous line-of-sight velocity toward the receiving antenna on Earth (\({\it v_{\rm{D}}}{\hat r_{\rm{E}}}\)), and Earth’s line-of-sight velocity toward Dawn’s position (\({v_{\rm{E}}}{\hat r_{\rm{D}}}\)). The differential Doppler shift due to Dawn’s orbital motion (δf orbit ) is then calculated from the Doppler shift contributed by \({v_{\rm{D}}}{\hat r_{{\rm{Vp\it t}}}}\) and its difference from Δf direct . In turn, the differential Doppler shift contributed by the rotation of Vpt (δf rotation ) on Vesta’s surface is calculated from the Doppler shift contributed by \({v_{{\rm{Vp\it t}}}}{\hat r_{\rm{D}}}\) and \({v_{{\rm{Vp\it t}}}}{\hat r_{\rm{E}}}\) and their difference from Δf direct . The combined Doppler shift contributions of δf orbit and δf rotation yield the total theoretical δf total , which is calculated to range from ~2 Hz, as listed in Supplementary Table 3, to as much as 20 Hz when considering uncertainties in spacecraft position and Vesta’s rotational velocity (detailed further below in our error analysis). Hence, the surface echo during occultation entry of orbit 355 is calculated to have a frequency shift that ranges from ~2 to 20 Hz higher or lower than that of the received direct signal.

This calculation confirms that in the configuration of grazing incidence during occultation observations of Vesta by Dawn, and due to the spacecraft’s low orbital velocity of 200 m s−1, the frequency separation δf between surface echoes and the direct signal will be small. Higher frequency spectral resolution is therefore necessary to distinguish the Doppler-shifted surface echoes from the direct signal at Vesta. However, this requires longer integration time of the observation. We chose 2.5-s integration time to obtain a frequency spectral resolution of ~0.4 Hz as a tradeoff between SNR, frequency drift and resolution. Our final frequency spectral analysis averages two 2.5-s looks, and repeats this calculation shifting the start time of the averaging by 1 s. The resulting spectra are exemplified for occultation entry and exit during orbit 355 in Supplementary Fig. 3, which shows the power received in both RCP (reproduced from Fig. 2) and in LCP.

Calculating the radar cross section at high incidence

In order to assess the surface’s geophysical properties that contribute to the observed surface echoes, the radar cross section σ of Vesta’s surface is calculated in m2 from the BSR equation12. This parameter is defined as the effective surface area that isotropically scatters the same amount of power as the echo site on Vesta, such that larger values of σ are associated with stronger surface echoes. Assuming (1) each echo site has approximately the same surface area illuminated by radar and (2) are observed at the same geometry (89° incidence), relative differences in σ imply differences in geophysical properties of the surface.

The latter assumption is supported by excluding surface echoes that occurred in regions of high topographic variability with respect to the incident HGA beam diameter, where the illuminated surface area is estimated using a first-order spherical approximation to Vesta’s surface. Notably, this approximation excludes the effects of shadowing and diffraction, which are difficult to quantify at grazing incidence, and does not account for deviations of Vesta’s shape from the sphere within the large area illuminated by the HGA beam. Our first-order estimation yields an elongated radar footprint of ~51 km along the line of sight between Dawn’s HGA and Earth’s receiving antenna, and ~11 km in diameter perpendicular to the line-of-sight. Topographic variability is assessed by calculating the root-mean-square height h rms using elevations from Vesta’s digital terrain model39 within an 11.5° by 11.5° grid (~51 × 51 km) centered on each echo site’s coordinates, and must be sufficiently smaller than the incident 11-km beam diameter. Calculated h rms range from 0.002 to 0.13 km. All but one echo site had h rms < 0.11 km (i.e., topographic variability of 1% of the incident beam diameter). The echo site of occultation entry during orbit 377 exceeded this criteria and is therefore excluded from the following analyses.

Since HGA transmissions are measured continuously throughout the minutes that precede and follow an occultation of the spacecraft behind Vesta, only engineering data is included in the raw telemetry, and DSN receiver calibration measurements are not included with the raw radar data set. Because the BSR data are not calibrated in absolute voltage, calculating σ therefore requires calibrating the measured power to a known reference. This is made possible by calculating the theoretical received power of the direct signal P r dir|calc in watts, and comparing with measured received power P r dir|meas in data units. P r dir|calc is calculated from the one-way radar equation and depends on the transmitted power P t , gain of the transmitting G t and receiving G r antennas, the distance R DE between the transmitter aboard Dawn and the receiving antenna on Earth, and summed losses L. The one-way radar equation for P r dir|calc (W) is then as follows12:

$${P} {_{{\rm{r}} \,{\rm{dir|calc}}}} = \frac{{{P_{\rm{t}}}{G_{\rm{t}}}{G_{\rm{r}}}{\lambda ^2}}}{{{{\left( {4\pi {R_{{\rm{DE}}}}} \right)}^2}L}}$$ (3)

where the nominal range of each parameter—except for time-dependent R DE —is provided in Supplementary Table 1. Note that losses contributed by the DSN 70-m antennas are published in the telecommunications parameters of the Deep Space 1 mission40.

P r dir|meas is evaluated by measuring the area under the curve in non-logarithmic units during a time when the direct signal is not obstructed by Vesta. Since the data is discrete, P r dir|meas is the sum of the power in each frequency bin multiplied by the width of each frequency bin (subtracted by the noise power in the same bandwidth):

$${P_{{\rm{r}}\,{\rm{dir|meas}}}} = \left( {{\sum} {{P_{\rm{i}}} \cdot \Delta {f_{{\rm{step}}}}} } \right) - \overline {{P_{\rm{N}}}} \cdot {f_{{\rm{BW}}}}$$ (4)

where frequency step Δf step is the spectral resolution of ~0.4 Hz, as previously determined when calculating the differential Doppler shift; P i is the non-logarithmic power in data units of each discrete point measured within a 10-Hz bandwidth (f BW ) of the direct signal peak; and \(\overline {{P_{\rm{N}}}} \) is the average noise power (data units Hz−1) in the spectrum. The conversion factor between watts and power measured from BSR data units is therefore:

$${C_{{\rm{ToWatts}}}} = \frac{{{P_{{\rm{r}}\,{\rm{dir}}\left| {{\rm{calc}}} \right.}}\left[ {{\rm{Watts}}} \right]}}{{{P_{{\rm{r}}\,{\rm{dir}}\left| {{\rm{meas}}} \right.}}\left[ {{\rm{Data}}\,{\rm{Units}}} \right]}}$$ (5)

and the BSR equation, solved for σ, is then:

$${\sigma _{\left( {{{\rm{m}}^{\rm{2}}}} \right)}} = \frac{{{{\left( {4\pi } \right)}^3}{R_{\rm{t}}}^2{R_{\rm{r}}}^2L}}{{{G_{\rm{t}}}{G_{\rm{r}}}{\lambda ^2}}}\left( {\frac{{{P_{{\rm{r}}\,{\rm{echo}}|{\rm{meas}}}}}}{{\left( {1 - {X_{{{\rm{P}}_{\rm{t}}}}}} \right){P_{\rm{t}}}}}} \right)$$ (6)

where X Pt is the fraction of incident power that has been reduced due to partial obstruction of the HGA beam by Vesta’s surface, and P r echo|meas (W) =P r echo|meas (Data Units) × C ToWatts . By measuring the received direct signal P r dir|meas at a time close to each measured surface echo—10 s before a given occultation entry, and 10 s after an occultation exit—we minimize variations in the transmitting and receiving system characteristics, including changes in the HGA’s pointing accuracy, and potential differences in DSN receiver losses due to the use of different receiving stations with different system temperatures and atmospheric conditions. Hence, the measurement of the radar cross section σ (m2) becomes independent of P t , G t , G r , L and λ, such that:

$${\sigma _{\left( {{{\rm{m}}^{\rm{2}}}} \right)}} = \frac{{{{\left( {4\pi } \right)}^3}R_{{\rm{DVpt}}}^2R_{{\rm{EVpt}}}^2}}{{\left( {1 - {X_{{{\rm{P}}_{\rm{t}}}}}} \right)R_{{\rm{DE}}}^2}}\left( {\frac{{{P_{{\rm{r}}\,{\rm{echo|meas}}}}}}{{{P_{{\rm{r}}\,{\rm{dir|meas}}}}}}} \right)$$ (7)

where R DVpt is the distance between the transmitter aboard Dawn and the echo site on Vesta’s surface at the time of the observed surface echo; R EVpt is the distance between the receiving antenna on Earth and the echo site at the time of the observed surface echo; and R DE is the distance between Dawn’s HGA and the receiver on Earth at the time when P r dir|meas is measured (10 s before occultation entries, and 10 s after occultation exits).

Typically, the radar cross section is then normalized to the areal extent illuminated by the radar (σ 0), and is reported in regime of diffuse backscatter due to the use of Earth-based radar antennas as both transmitter and receiver for many observations (e.g., ref. 29)—where at increasingly high angles of incidence, the diffuse component of radar backscatter dominates the received signal14. BSR observations at Vesta are conducted in the forward scatter regime, however, whereby radar waves predominantly scatter in the forward direction and almost entirely within the plane of incidence12. In this regime, the polarization of a circularly transmitted wave is also conserved in major part even after reflection from the target’s surface25. Dawn’s measurements of σ 0 on Vesta’s surface are therefore not directly comparable with those observed in the backscatter regime on other planetary bodies.

While the lack of comparability might be overcome by deriving surface roughness from σ 0, two sources of uncertainty remain: (1) the absolute surface area contributing to forward-scattered surface echoes is difficult to quantify due to the effects of shadowing and multiple scattering that become important at such high, grazing incidence38; and (2) there is no appropriate scattering model to address the impact of wavelength-scale surface roughness on radar reflections at grazing angles of incidence approaching 90°41.

Instead, we calculate relative σ across Vesta’s surface with respect to the strongest observed surface reflection σ max by measuring σ when the direct signal is ~25 dB above the noise level for all surface echoes, and employ the assumption that the illuminated surface area is approximately equal for each site. Since incident power is also assumed equal for all surface echoes (and therefore X Pt assumed constant for Equations 6 and 7), we estimate that at least 50% of the HGA beam is obscured behind Vesta’s surface (X Pt = 0.5) and report σ (km2) as a lower limit for each echo site.

Under the above assumptions of equal incident power and equal surface area illuminated at 89° incidence, the relative strengths of surface echo reflections (σ/σ max ) can then be attributed to differences in the relative reflectivity of the surface material itself or variations in the roughness of the surface at the scale of the radar wavelength14, 26. Potentially greater obstruction of the incident power would result in an increase of σ (km2), but assuming equal obstruction for all surface echoes, this does not change the relative radar cross section (σ/σ max ).

Uncertainty in the differential Doppler shift

The primary sources of uncertainty in theoretical δf include the positions and velocities of (1) the spacecraft and (2) the radar-illuminated surface echo site on Vesta as listed in Supplementary Table 4. We assume that uncertainty in the position and velocity of Earth is negligible at such distances. For a given parameter Y ± ΔY that depends on multiple variables X i ± (ΔX) i , we calculate ΔY by summing in quadrature the partial derivative of each contributing variable (∂X i /∂Y) multiplied by its uncertainty (ΔX) i .

The position of the Dawn spacecraft is provided in Cartesian coordinates from SPICE ephemerides with an uncertainty of ± 3 m in the radial, along-track and cross-track directions from the reconstructed trajectory of LAMO28, while the error in the position of the echo site on Vesta’s surface is calculated from (1) uncertainty in the radius of the echo site from Vesta’s center ± ~0.5 km—due to topography within the large surface area illuminated by the HGA beam at grazing incidence—and (2) uncertainty in the latitude and longitude of the echo site center by ± ~0.2°. Uncertainties in the geodetic coordinates of the echo site are then converted to Cartesian coordinates at a height above or below a reference triaxial ellipsoid—which is defined using the best-fit ellipsoid derived from Hubble light-curve observations of Vesta, where R x = 289 km, R y = 280 km and R z = 229 km28, 42. Uncertainties in the x, y and z coordinates of the echo site for occultation entry of orbit 355 are on the order of ~0.7 km for the occultation entry of orbit 355.

With regard to spacecraft velocity, the SPICE ephemerides containing Dawn’s state vector were released by the optical navigation team in 201228 before the peer-reviewed publication of Vesta’s gravitational solution in 201411. Hence, Dawn’s reconstructed trajectory does not include variations in orbital velocity due to the heterogeneous gravity field that exhibits accelerations between −1000 mGal and +2000 mGal (−1 cm s−2 and +2 cm s−2) relative to the homogeneous model11. Since each frequency spectrum produced in our BSR analysis is averaged over 5-s observations, unpredicted gravitational accelerations contribute –5 to +10 cm s−1 uncertainty in Dawn’s velocity vector.

The rotational velocity of the echo site on Vesta’s surface depends on the distance of the site from Vesta’s center and the rotation period, the latter of which is known to high precision11. Given the large extent of surface area illuminated by the HGA beam on Vesta, we use the radius at the center of the echo site to calculate a representative rotational velocity. During occultation entry of orbit 355, uncertainty of ± 0.5 km in the radius yields an uncertainty of ± 16 cm s−1 in the echo site’s rotational velocity.

Using the above-derived uncertainties in the positions and velocities of each body, we calculate the propagation of error into each line-of-sight velocity between Dawn, the surface echo site on Vesta, and the antenna on Earth that contribute to the calculation of theoretical δf. Uncertainty in the velocity of body A projected along the line of site with body B (\({\it v_{\rm{A}}}{\hat r_{\rm{B}}}\)) is calculated from the propagation of error in (1) the velocity vector of A, (2) the position of A, and (3) the position of B. Hence, the theoretical differential Doppler shift is calculated to range between ~2 and 20 Hz, and is consistent the observed δf of –12 Hz (Fig. 1b) for the surface echo measured during orbit 355, occultation entry.

Uncertainty in the absolute and relative radar cross section

The primary sources of uncertainty in the radar cross section include (1) HGA pointing error, (2) uncertainty in the measurement of received power, (3) uncertainty in the position of the Dawn spacecraft, and (4) uncertainty in the position of the echo site center—where the latter two uncertainties have been previously quantified above.

When outside of occultation, the measured power received from the direct signal P r dir|meas varies as a result of HGA antenna pointing inaccuracy due to unpredicted uneven gravitational torques on the spacecraft’s large solar panels while in Vesta’s microgravity environment43. The spacecraft’s reaction wheels are used to counteract accumulated spacecraft pointing errors, but one of the four wheels failed prior to Dawn’s arrival at Vesta43. Furthermore, corrections only occurred once every 1-3 days during LAMO, such that antenna pointing error increased steadily from zero to ~0.4° between corrections, and even exceeded 1.0° on a few occasions43. To quantify fluctuations in the direct signal power on the order of tens of seconds near the time of each surface echo observation, we measure the variation of P r dir|meas over 30 s preceding an occultation entry (and over 30 s following an occultation exit). We find that P r dir|meas varies less than ±2% for 11 of the 14 occultation observations but as much as ±8% before occultation entry of orbit 719.

The standard error in the measurement of P r echo|meas and P r dir|meas are quantified by deviations of noise power from the mean. We compute the standard deviation of noise in a given spectrum over frequencies where no signal is present (from −6 to −1 kHz and from 1 to 6 kHz), and find that the standard deviation of a given spectrum ranges from ~0.14 \(\overline {{P_{\rm{N}}}} \) to 0.17 \(\overline {{P_{\rm{N}}}} \). We use the upper limit of 0.17 \(\overline {{P_{\rm{N}}}} \) as a conservative estimate for all uncertainties in power measurement.

Together, the above errors in HGA antenna pointing, measurement of received power, spacecraft position and echo site position amount to uncertainties in σ (km2) that range from 1% to 10% depending on the surface echo—see Table 1. Subsequent error in the relative radar cross section (σ/σ max ) range from zero dB for the strongest surface echo reflection to ± 0.5 dB for the weakest surface echo reflection.

Data availability

Raw telemetry data from Dawn’s orbital BSR experiment at Vesta were generated from receiver output at stations of the NASA DSN and are managed by the NASA Jet Propulsion Laboratory of the California Institute of Technology. The unprocessed time-domain BSR amplitude data used in this study are available upon request from the corresponding author.