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An Analysis of Gravity-Field Estimation Based on Intersatellite Dual-1-Way Biased Ranging

The GRACE (Gravity Recovery And Climate Experiment) mission is designed to make global, highly accurate measurements of the Earth's gravity field with high spatial resolution. Ancillary GPS occultation measurements are also to be carried out for atmospheric monitoring. In the dual-1-way biased ranging of this mission, the range between two satellites separated by 100 to 200 km in nearly polar, coplanar, circular orbits, is measured to very high precision, to within an additive constant, through the exchange of K- and Ka-band sinusoidal signals. Such biased ranging data, along with GPS L-band range and phase data, can be processed and fit over successive multiday intervals to obtain accurate estimates of the Earth's gravity field. This report approximately models and analyzes this process, from the generation of the RF signals at the two satellites through the extraction of the geopotential. The steps include generation of the transmitted signals, processing the received signals to extract high-rate baseband phase, carrying out a dual-1-way combination of baseband phase to extract high-rate biased range for each band, combining K- and Ka-band ranges to correct for the ionosphere effect, and processing the resulting high-rate biased range values to extract three types of reduced-rate observables: biased range, range rate and range acceleration. The version of dual-1-way biased ranging developed by this report improves upon previous versions in a number of ways: highly accurate satellite-timetag corrections derived from concurrent GPS data, better baseband phase extraction using highly digital processing, highly accurate USO-rate calibration derived from concurrent GPS data, an improved method for extracting high-rate biased range from baseband phase, improved filtering for extracting reduced- rate observables from high-rate biased range, and parallel extraction of three observable types.

SINUSOIDAL SIGNALS↗

TRMM On-Orbit Performance Re-Accessed After Control Change

The Tropical Rainfall Measuring Mission (TRMM) spacecraft, a joint mission between the U.S. and Japan, launched onboard an HI1 rocket on November 27,1997 and transitioned in August, 2001 from an average operating altitude of 350 kilometers to 402.5 kilometers. Due to problems using the Earth Sensor Assembly (ESA) at the higher altitude, TRMM switched to a backup attitude control mode. Prior to the orbit boost TRMM controlled pitch and roll to the local vertical using ESA measurements while using gyro data to propagate yaw attitude between yaw updates from the Sun sensors. After the orbit boost, a Kalman filter used 3-axis gyro data with Sun sensor and magnetometers to estimate onboard attitude. While originally intended to meet a degraded attitude accuracy of 0.7 degrees, the new control mode met the original 0.2 degree attitude accuracy requirement after improving onboard ephemeris prediction and adjusting the magnetometer calibration onboard. Independent roll attitude checks using a science instrument, the Precipitation Radar (PR) which was built in Japan, provided a novel insight into the pointing performance. The PR data helped identify the pointing errors after the orbit boost, track the performance improvements, and show subtle effects from ephemeris errors and gyro bias errors. It also helped identify average bias trends throughout the mission. Roll errors tracked by the PR from sample orbits pre-boost and post-boost are shown in Figure 1. Prior to the orbit boost the largest attitude errors were due to occasional interference in the ESA. These errors were sometime larger than 0.2 degrees in pitch and roll, but usually less, as estimated from a comprehensive review of the attitude excursions using gyro data. Sudden jumps in the onboard roll show up as spikes in the reported attitude since the control responds within tens of seconds to null the pointing error. The PR estimated roll tracks well with an estimate of the roll history propagated using gyro data. After the orbit boost, the attitude errors shown by the PR roll have a smooth sine-wave type signal because of the way that attitude errors propagate with the use of gyro data. Yaw errors couple at orbit period to roll with '/4 orbit lag. By tracking the amplitude, phase, and bias of the sinusoidal PR roll error signal, it was shown that the average pitch rotation axis tends to be offset from orbit normal in a direction perpendicular to the Sun direction, as shown in Figure 2 for a 200 day period following the orbit boost. This is a result of the higher accuracy and stability of the Sun sensor measurements relative to the magnetometer measurements used in the Kalman filter. In November, 2001 a magnetometer calibration adjustment was uploaded which improved the pointing performance, keeping the roll and yaw amplitudes within about 0.1 degrees. After the boost, onboard ephemeris errors had a direct effect on the pitch pointing, being used to compute the Earth pointing reference frame. Improvements after the orbit boost have kept the the onboard ephemeris errors generally below 20 kilometers. Ephemeris errors have secondary effects on roll and yaw, especially during high beta angle when pitch effects can couple into roll and yaw. This is illustrated in figure 3. The onboard roll bias trends as measured by PR data show correlations with the Kalman filter's gyro bias error. This particularly shows up after yaw turns (every 2 to 4 weeks) as shown in Figure 3, when a slight roll bias is observed while the onboard computed gyro biases settle to new values. As for longer term trends, the PR data shows that the roll bias was influenced by Earth horizon radiance effects prior to the boost, changing values at yaw turns, and indicated a long term drift as shown in Figure 4. After the boost, the bias variations were smaller and showed some possible correlation with solar beta angle, probably due to sun sensor misalignment effects.

Bilanow, Steve↗

TRMM On-Orbit Performance Reassessed After Control Change

The Tropical Rainfall Measuring Mission (TRMM) spacecraft, a joint mission between the U.S. and Japan, launched onboard an H-I1 rocket on November 27, 1997, and transitioned in August, 2001, from an average operating altitude of 350 kilometers to 402.5 kilometers. Due to problems using the Earth Sensor Assembly (ESA) at the higher altitude, TRMM switched to a backup attitude control mode. Prior to the orbit boost TRMM controlled pitch and roll to the local vertical using ESA measurements while using gyro data to propagate yaw attitude between yaw updates from the Sun sensors. After the orbit boost, a Kalman filter used 3-axis gyro data with Sun sensor and magnetometers to estimate onboard attitude. While originally intended to meet a degraded attitude accuracy of 0.7 degrees, the new control mode met the original 0.2 degree attitude accuracy requirement after improving onboard ephemeris prediction and adjusting the magnetometer calibration onboard. Independent roll attitude checks using a science instrument, the Precipitation Radar (PR) which was built in Japan, provided a novel insight into the pointing performance. The PR data helped identify the pointing errors after the orbit boost, track the performance improvements, and show subtle effects from ephemeris errors and gyro bias errors. It also helped identify average bias trends throughout the mission. Roll errors tracked by the PR from sample orbits pre-boost and post-boost are shown in Figure 1. Prior to the orbit boost, the largest attitude errors were due to occasional interference in the ESA. These errors were sometime larger than 0.2 degrees in pitch and roll, but usually less, as estimated from a comprehensive review of the attitude excursions using gyro data. Sudden jumps in the onboard roll show up as spikes in the reported attitude since the control responds within tens of seconds to null the pointing error. The PR estimated roll tracks well with an estimate of the roll history propagated using gyro data. After the orbit boost, the attitude errors shown by the PR roll have a smooth sine-wave type signal because of the way that attitude errors propagate with the use of gyro data. Yaw errors couple at orbit period to roll with 1/4 orbit lag. By tracking the amplitude, phase, and bias of the sinusoidal PR roll error signal, it was shown that the average pitch rotation axis tends to be offset from orbit normal in a direction perpendicular to the Sun direction, as shown in Figure 2 for a 200 day period following the orbit boost. This is a result of the higher accuracy and stability of the Sun sensor measurements relative to the magnetometer measurements used in the Kalman filter. In November, 2001 a magnetometer calibration adjustment was uploaded which improved the pointing performance, keeping the roll and yaw amplitudes within about 0.1 degrees.

Bilanow, Stephen↗

How to observe the gyre to global-scale variability in satellite altimetry - Signal attenuation by orbit error removal

Formulas analogous to the frequency response functions for commonly used filters in orbit error removal are analytically derived to devise observational strategies for the large-scale oceanic variability and to decipher the signal contents of previous results. These include the polynomial orbit error approximations, i.e., the linear, bias-only and quadratic corrections, and the sinusoidal orbit error approximations (the purely sinusoidal correction, and the sinusoid-and-bias correction). It is shown that the frequency response function for a polynomial correction is a function of the ratio of wavelength/track length and to retain 90 percent or more of the signal at a certain wavelength, the ratio must be less than 0.65 (for the quadratic case), 0.90 (linear), and 1.54 (bias-only).

Tai, Chang-Kou↗

Physiological mechanisms of the nystagmus produced by rotations about an earth-horizontal axis

The physiological basis of the nystagmus produced by rotation about an earth-horizontal axis is investigated with particular emphasis on the unidirectional nystagmus attributed to a bias component. Eye movement recordings were made with dc electro-oculography in alert squirrel monkeys and afferent responses were recorded from semicircular canals and otolith receptors of anesthetized animals upon rotations in the pitch and yaw planes. The eye-movement recordings show the rotation responses in the squirrel monkey to resemble those of other species, including persistent horizontal and vertical nystagmuses during yaw and pitch rotations, respectively, a unidirectional nystagmus at low rotation speeds, and postrotatory responses of relatively small amplitude and duration. The vestibular nerve recordings do not show a directionally specific dc response that can account for the bias component, but instead exhibit sinusoidal responses of peak amplitudes 0-15 and 25-75 spikes/sec for the canals and otolith, respectively. Results thus indicate that the dc signal to the oculomotor centers responsible for the nystagmus is of central origin, most likely based on some transformation of the otolith signals.

Goldberg, J. M.↗

Directions of Moving Plaids is Biased by Asymmetric Viewing Windows

Directionally selective V1 neurons are tuned to particular spatio-temporal frequencies and respond to local 1-D edge motion. At least some MT neurons however appear to respond to the actual velocity of moving 2D patterns. To better understand how the 1D local motion information available from VI is integrated to derive a 2D velocity signal we investigated human perception of moving plaids, 2-D patterns composed of the sum of two 1-D sinusoidal gratings of different orientations. We measured the effect of the shape of the viewing window on the perceived direction of plaid motion. The plaids were spatially windowed by 2-D Gaussians with unequal standard deviations (sigma 1, sigma 2). Four observers indicated perceived direction by adjusting a pointer. Direction errors were measured as a function of the difference between window orientation and true plaid direction (DELTA THETA) for several grating spatial frequencies (SF = 0.3, 0.6, 1.2 c/d) and window aspect ratios (AR = sigma 1/sigma 2 = 1, 1.4, 2, 4). Observers showed systematic errors in perceived direction (approx. 15 for AR = 4 and SF = 0.6 c/d) that peaked at DELTA THETA approx. 40. The errors increased for increasing aspect ratio and decreased for increasing spatial frequencies (or number of cycles). These results show that despite the unambiguous motion of the plaids, under these conditions human misperceive the motion. These data constrain models of motion integration from V1 to MT and, in particular, are inconsistent with algorithms that use either the Intersection of Constraints rule or cross correlation to compute the perceived direction of motion.

Beutter, B. R.↗

Direction of Moving Plaids is Biased by Asymmetric Viewing Windows

Directionally selective V1 neurons are tuned to particular spatiotemporal frequencies and respond to local 1-D edge motion. At least some MT neurons however appear to respond to the actual velocity of moving 2D patterns (Movshon et al., EBR, 11:117, 1986). To better understand how the 1D local motion information available from V1 is integrated to derive a 2D velocity signal we investigated human perception of moving plaids, 2-D patterns composed of the sum of two 1-D sinusoidal gratings of different orientations. We measured the effect of the shape of the viewing window on the perceived direction of plaid motion. The plaids were spatially windowed by 2-D Gaussians with unequal standard deviations (sigma l, sigma 2). Four observers indicated perceived direction by adjusting a pointer. Direction errors were measured as a function of the difference between window orientation and true plaid direction (delta theta) for several grating spatial frequencies (SF = 0.3, 0.6, 1.2 c/d) and window aspect ratios (AR = sigma 1/sigma 2 = 1, 1.4, 2,4). Observers showed systematic errors in perceived direction (approx. 15 degrees for R = 4 and SF = 0.6 c/d) that peaked at delta theta approximately 40 degrees. The errors increased for increasing aspect ratio and decreased for increasing spatial frequencies (or number of cycles). These results show that despite the unambiguous motion of the plaids, within asymmetric windows, human can systematically misperceive plaid direction. These data constrain models of motion integration within extrastriate cortex and, in particular, are inconsistent with algorithms that use either the Intersection of Constraints rule or cross correlation to compute velocity.

Beutter, Brent Robert↗