WHAT GRAIL TEACHES US ABOUT ERROR AND BIAS IN PRELIMINARY GRAVITY FIELDS

Peter B. James1,2 J. C. Andrews-Hanna3, and M. T. Zuber4 ! LPSC 48 – March 2017! 1The Lunar and Planetary Institute, Houston, TX, 77058, USA ([email protected]); 2Department of Geosciences, Baylor University, Waco, TX, 76798, USA; 3Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ, 85721, USA; 4Dept. of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.

The problem: biased gravity fields Model A – White noise at orbit altitude What this method is, and what it is not

The Gravity Recovery And Interior Laboratory mission (GRAIL) determined the Moon's The presence of noise increases the unfiltered gravity power by a factor nl: The described gravity enhancement technique (specifically Model B) relies on an gravity field to a high precision: the RMS amplitude of gravity exceeds that of the error by a factor extrapolation of the true gravity RMS to high spherical harmonic degrees using a power − 1 β 2(l−l ) 2 s (5) law, so this technique should be used cautiously for a low-resolution gravity field (l < of more than 10,000 at spherical harmonic degree l=180 [1,2]. The GRAIL gravity field may be ⎡ ⎛ l ⎞ ⎛ r ⎞ ⎤ max n ⎢1 s 0 ⎥ l = + ⎜ ⎟ ⎜ ⎟ treated as exact for all practical purposes, and this allows us to retrospectively analyze past ⎢ l a ⎥ 50). The technique also does not account for spatial variations in degree strength. ⎢ ⎝ ⎠ ⎝ ⎠ ⎥ gravity fields in light of the "true" gravity. In this study we analyze the LPE200 gravity field from ⎣ ⎦ There is no trick that allows us to remove noise from a gravity dataset; note that Lunar Prospector (LP) [3]. We focus on the nearside using a bandwidth L=2 Slepian taper [4]. while we can improve admittance, we cannot improve the gravity/topography correlation. r0 – a! Error in LP gravity (i.e. the departure from GRAIL gravity) is positively correlated with GRAIL However, gravity enhancement allows us to reverse the biasing effect that results from gravity at high spherical harmonic degrees (Fig. 1). We call this "bias". Equivalently, the gravity/ data noise, an effect which hinders certain geophysical analyses. Additionally, topography admittance spectrum is biased toward zero. Such a bias in gravity/topography ratios enhancement is expected to come at a cost of higher RMS errors (since the field is detrimental to a variety of geophysical analyses, including determinations of compensation amplitude increases). depth and crustal density. 1

0.9 An optimized gravity field is a 1 balance between minimized 0.8 0.5 error and unbiased estimation. “bias” Minimized 0.7 Published gravity fields often Unbiased 0 RMS Error 0.6 GRAIL prioritize the reduction of RMS Estimation −0.5 GRAIL + Noise error, but “enhanced” gravity Lunar Prospector 0.5 Lunar Prospector Error Correlation aims to minimize bias. −1 0 20 40 60 80 100 120 140 160 180 Gravity/Topography Correlation 0.4 40 60 80 100 120 140 160 180 Spherical Harmonic Degree Spherical Harmonic Degree Figure 1. Correlation of Lunar Prospector error with true gravity HOWEVER, even though the enhancement technique prioritizes unbiased estimation over Figure 2. Decline in LP correlation predicted using Model A! error reduction, Figure 5 shows that we can actually reduce RMS error in the Lunar Prospector gravity field by applying enhancement (Note that this is generally not expected to The solution: gravity “enhancement” Model B – Underestimation of power be the case):

) 1 We will show that bias in the LP gravity field can largely be explained using two models: random 2 a) b) data noise and an underestimation of gravity power. Furthermore, we propose an analysis technique called "enhancement", the purpose of which is to reduce bias a gravity field. When applied to the LP gravity field, this technique reduces RMS error and improves the high-degree 0.1 admittance. Gravity is detected at the altitude of the orbiting spacecraft (which, for simplicity, we GRAIL represent as a single radial position r=r0), and these detections are obscured by random noise I. Lunar Prospector

If I is spectrally "white", the power spectral density is constant: 0.01

Power Spectral Density (mgal 40 60 100 180 (1) Spherical Harmonic Degree I (r )2 ≈ N lm 0 l Figure 3. Comparison of GRAIL power and LP power! where Ilm(r0) is the spherical harmonic coefficient of I at r=r0, and angled brackets indicate expectation across spherical harmonic orders. The presence of noise in a gravity dataset Additionally, LP gravity power is underestimated by a factor pl: 1 1 generally increases the power of an unconstrained gravity field, and the suppression of this 2 2 2 2 (6) p = gobs gtrue 0 5 10 15 20 25 30 l ( lm ) ( lm ) power (typically with a Kaula constraint) can be approximated as a degree-dependent filter nl. Error amplitude (mgal) The observed gravity gobs at the reference radius r=a is a downward continuation of the filtered We approximate pl for LP above l=120, where gravity power breaks from its Figure 5. a) Error in the LP gravity field expanded to l=140, with an RMS amplitude of 9.25 mgal across the lunar nearside; b) Error of the enhanced LP field, with an RMS amplitude of 8.59 mgal. gravity and noise from the spacecraft altitude: log-linear trend. The denominator is an extrapolation of the best-fit power

l (2) law for l=80–119, and the numerator is the power law fit for l=120–160. gobs(a) gtrue(r ) I (r ) r a n lm = lm 0 + lm 0 ⋅ 0 ⋅ l Takeaways ( ) ( ) where n is: l 1 Results (Model A + Model B) We’ve introduced a technique called “enhancement”, which improves gravity data ⎡ 2 2 2 ⎤ 2 n = gobs gtrue + I (3) for certain applications via a simple degree-wise scaling of the data.! l ⎢ ( ) ( ) ( ) ⎥ The enhanced gravity equals the observed gravity divided by n and p : ⎣ ⎦ l l Here’s when to use it: The observed gravity g typically has power comparable to g (see Model B for exceptions). obs true genhanced = gobs n ⋅ p lm lm ( l l ) (7) Gravity power may be approximated with a power law: Do you want the most accurate estimate of Existing gravity fields are 120 2 (4) gravity at an arbitrary point on the surface? generally acceptable! gtrue ≈ Al β ( lm ) 110 I.e., do you want to minimize RMS error?! (Fig. 5 notwithstanding) ! The Lunar Prospector spacecraft orbited with a mean altitude of 40 km at the end of its primary 100 mission and a mean altitude of 30 km during the extended mission, so we use an intermediate 90 Do you want an UNBIASED estimate of the value of r = 35 km. Error in the LP gravity approximately equals the amplitude of the gravity on An “enhanced” gravity ! 0 80 gravity amplitude such as for gravity/ field will be better! the lunar nearside at spherical harmonic degree 180, so we take this to be the degree strength ls. 70 topography comparisons?! For these parameter values, the filter nl nicely reproduces the high-degree decline in the Lunar GRAIL 60 Lunar Prospector Prospector gravity/topography correlation (Fig. 2). This demonstrates the efficacy of Model A. Admittance (mgal/km) 50 Enhanced LP Noise amplifies the power of a gravity field, so the process of creating a gravity field References typically involves suppression of high-degree power, often with a Kaula rule (β = –2). However, 40 40 60 80 100 120 140 160 180 excessive or deficient suppression can cause the resulting gravity power to differ from the true Spherical Harmonic Degree [1] Goossens S. et al. (2014) LPSC, #1619. [2] Park R. S. et al. (2014) AGU, #G22A-01. [3] Han S.-C. power. The correction of gravity power is described by Model B. Figure 4. GRAIL gravity, LP gravity, and the enhanced LP field et al. (2011) Icarus 215, 455–459. [4] Wieczorek M. A. & Simons F. J (2007) J. Four. An. 15, 665–692.