Lunar and Planetary Science XLVIII (2017)
2199.pdf
WHAT GRAIL TEACHES US ABOUT ERROR AND BIAS IN PRELIMINARY GRAVITY FIELDS. P. B.
James1, J. C. Andrews-Hanna2, and M. T. Zuber3, 1The Lunar and Planetary Institute, Houston, TX 77058, USA; 2
Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ, 85721, USA; 3Dept. of Earth, Atmospheric and
Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
(
)(
)
l
(2)
obs
true
glm
(a) = glm
(r0 ) + Ilm (r0 ) ⋅ r0 a ⋅ nl
where nl is:
( )
(
) ()
1
2
2
2 ⎤2
⎡
(3)
nl = ⎢ g obs
g true + I ⎥
⎣
⎦
The observed gravity gobs typically has power comparable to gtrue (see Model B for exceptions). Gravity
power may be approximated with a power law:
(g )
true
lm
2
(4)
≈ Al β
The filter nl can then be re-formulated in terms of the
"degree strength" ls, the spherical harmonic degree at
which the power of noise surpasses that of the signal:
−1
1
0.5
0
−0.5
Lunar Prospector
−1
0
20
40
60
80
100 120 140
Spherical Harmonic Degree
160
180
Figure 1. Correlation of LP error and GRAIL gravity.
We will show that bias in the LP gravity field can
largely be explained using two models: random 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 admittance.
Model A: White noise at spacecraft altitude.
Gravity is detected at the altitude of the orbiting spacecraft (which, for simplicity, we represent as a single
radial position r=r0), and these detections are obscured
by random noise I. If I is spectrally "white", the power
spectral density is constant:
(1)
I (r )2 ≈ N
lm
The presence of noise in a gravity dataset generally
increases the power of an unconstrained gravity field,
and the suppression of this power (typically with a
Kaula constraint) can be approximated as a degreedependent filter nl. The observed gravity gobs at the
reference radius r=a is a downward continuation of the
filtered gravity and noise from the spacecraft altitude:
0
l
where Ilm(r0) is the spherical harmonic coefficient of I
at r=r0, and angled brackets indicate expectation across
spherical harmonic orders.
2
β
2(l−l )
⎡
⎛l ⎞ ⎛r ⎞ s ⎤
(6)
s
0
⎢
⎥
nl = 1 + ⎜ ⎟ ⎜ ⎟
⎢
⎥
l ⎠ ⎝ a⎠
⎝
⎢⎣
⎥⎦
The Lunar Prospector spacecraft orbited with a
mean altitude of 40 km at the end of its primary mission and a mean altitude of 30 km during the extended
mission, so we use an intermediate value of r0 = 35 km.
Error in the LP gravity approximately equals the amplitude of the gravity on the lunar nearside at spherical
harmonic degree 180, so we take this to be the degree
strength ls. For these parameter values, the filter nl
nicely reproduces the high-degree decline in the Lunar
Prospector gravity/topography correlation (Fig. 2).
This demonstrates the efficacy of Model A.
Gravity/Topography Correlation
Error Correlation
Introduction. The Gravity Recovery And Interior
Laboratory mission (GRAIL) determined the Moon's
gravity field to a high precision: the RMS amplitude of
gravity exceeds that of the error by a factor of more
than 10,000 at spherical harmonic degree l=180 [1,2].
The GRAIL gravity field may be treated as exact for
all practical purposes, and this allows us to retrospectively analyze past gravity fields in light of the "true"
gravity. In this study we analyze the LPE200 gravity
field from Lunar Prospector (LP) [3]. LP gravity is less
precise on the lunar farside than on the nearside due to
line-of-sight obstruction, so we focus on the nearside
using a bandwidth L=2 Slepian taper [4].
Error in LP gravity (i.e. the departure from GRAIL
gravity) is positively correlated with GRAIL gravity at
high spherical harmonic degrees (Fig. 1). We call this
"bias". Equivalently, the gravity/topography admittance spectrum is biased toward zero. Such a bias in
gravity/topography ratios is detrimental to a variety of
geophysical analyses, including determinations of
compensation depth and crustal density.
1
0.9
0.8
0.7
GRAIL
GRAIL + Noise
Lunar Prospector
0.6
0.5
0.4
40
60
80
100
120
140
Spherical Harmonic Degree
160
180
Figure 2. Gravity/topography correlation for GRAIL data,
GRAIL data plus white noise (Model A), and Lunar Prospector data.
Lunar and Planetary Science XLVIII (2017)
2199.pdf
where pl is:
pl =
(g )
obs
lm
2
1
2
(g )
true
lm
2
1
2
(8)
Power Spectral Density (mgal2)
We estimate this filter pl for LP above l=120, where
gravity power breaks from its log-linear trend (cf. Fig.
3). The denominator in Eq. 8 is calculated using an
extrapolation of the best-fit power law for l=80–119,
and the numerator is the power law fit for l=120–160.
1
0.1
Lunar Prospector
60
100
Spherical Harmonic Degree
180
Figure 3. Log–log plot of gravity power spectral density for
GRAIL and Lunar Prospector data.
Practical application. The "enhanced" gravity
field is simply the observed field divided by nl and pl:
enhanced
obs
(9)
glm
= glm
nl ⋅ pl
(
)
Unless gravity power is over-estimated (pl>1), this
operation results in a degree-wise amplification of the
gravity field.
a)
b)
0
5
120
110
100
90
80
70
10
15
20
Error amplitude (mgal)
25
30
Figure 4. 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.
GRAIL
60
Lunar Prospector
50
Enhanced LP
40
40
GRAIL
0.01
40
Enhancement reduces RMS error of the LP field by 7%
across the lunar nearside (Fig. 4). It is noteworthy that
we successfully reduced RMS errors despite increasing the amplitude of the gravity field (an action which
simultaneously amplifies noise).
More significantly, enhancement of the Lunar Prospector improves the admittance spectrum (Fig. 5). If
we define the spatial resolution of the gravity data to
be that for which admittance error is less than 10%, the
enhancement described here improves the LP gravity
resolution from l=119 (spatial block size ~46 km) to
l=171 (~32 km).
Admittance (mgal/km)
Model B: Underestimation of gravity power.
Noise amplifies the power of a gravity field, so the
process of creating a gravity field typically involves
suppression of high-degree power, often with a Kaula
rule (β = –2). However, excessive or deficient suppression can cause the resulting gravity power to differ
from the true power by a factor pl:
obs
true
(7)
glm
= glm
⋅ pl
60
80
100
120
140
Spherical Harmonic Degree
160
180
Figure 5. Gravity/topography admittance for GRAIL, Lunar
Prospector, and the Enhanced Lunar Prospector field.
What this technique is, and what it is not: The
described gravity enhancement technique (specifically
Model B) relies on an extrapolation of the true gravity
RMS to high spherical harmonic degrees using a power law. Power laws are generally more imprecise at
low spherical harmonic degrees, so this technique
should be used cautiously for a low-resolution gravity
field (lmax < 50). The technique also does not account
for spatial variations in degree strength, so enhancement should only be applied to regions of planetary
surfaces with relatively consistent degree strength.
There is no trick that allows us to remove noise
from a gravity dataset; note that while we can reduce
RMS error and improve admittance, we cannot improve the gravity/topography correlation. However,
gravity enhancement allows us to reverse the biasing
effect that results from data noise, an effect which hinders certain geophysical analyses. When applied to
other planets, this technique will allow us to characterize densities and compensation states at higher resolutions than was previously possible.
References: [1] Goossens S. et al. (2014) LPSC,
#1619. [2] Park R. S. et al. (2014) AGU, #G22A-01.
[3] Han S.-C. et al. (2011) Icarus 215, 455–459.
[4] Wieczorek M. A. & Simons F. J (2007) J. Four.
An. 15, 665–692.