Glacier mass variations using satellite

Anthony Arendt

Geophysical Institute University of Alaska Fairbanks

1 / 27 Lecture Layout

Fundamentals I gravitational force and the geopotential I expressing the potential in I expressing changes in the geopotential as equivalent water mass

Satellite Gravimetry I description satellite gravity missions and measurement concepts I processing GRACE data (spherical harmonic and mascon solutions)

GRACE and Glaciology I correcting GRACE for non-glacier sources of mass change I analysis of time series trend and amplitude to determine glacier mass balance I issues of temporal and spatial resolution I comparison with geodetic and conventional measurements I pitfalls of GRACE time series analysis

Background 2 / 27 Why are GRACE estimates so different?

Mass balance estimates for the Greenland ice sheet. GRACE estimates vary by up to a factor of two. We will explore why the same sensor can produce such variable estimates. 100 80 60 40 20 0 -20 -40 -60 -80 -100 -120 -140 -160 RADAR Altimetry

-180 RADAR Altimetry -200 Net Balance (Gt/year) Balance Net -220 Laser Altimetry -240 Laser Altimetry -260 GRACE -280 Flux Imbalance -300 1990 91 92 93 94 1995 96 97 98 99 2000 01 02 03 04 2005 06 07 08 09 2010 Year Background 3 / 27 Fundamentals

I gravitational force exerted by attracting mass m on attracted mass (equal to unity); G is ’s gravitational constant, l is the distance between the masses: m F = G (1) l2

I define the gravitational potential as a scalar function: Gm V = (2) l I the total potential acting on a body in space∫ ∫ ∫is: dM V = G (3) l Earth

V

Background 4 / 27 Fundamentals

I the potential satisfies Laplace’s Equation: ∆V = 0 (4)

I Spherical harmonic functions form a solution Laplace’s Equation:

∑∞ ( ) ∑l GM R l+1 V (r, ϑ, λ) = P¯ (sin ϑ)(C¯ cos(m, λ) + S¯ sin(m, λ)) (5) r r lm lm lm l=0 m=0 I l and m are the degree and order of the spherical harmonic expansion I r, ϑ, λ are the spherical geocentric radius, latitude and longitude coordinates I C¯ and S¯ are dimensionless Stokes Coefficients I P¯ is the fully normalized associated Legendre polynomial

Background 5 / 27 Fundamentals

I Legendre’s polynomials satisfy the solution P of Laplace’s Equation in spherical harmonics P6 6 I the geometric P P4 representation of spherical harmonics illustrate 4 P2 how a particular field on a sphere can be P1 P3 represented, and how the resolution increases P5 P7 with increasing degree (l) and order (m) P 7 P 5 P3

zonal l = 6, m=0

tesseral sectorial l = 12, m=6 l = 6, m=6

Background 6 / 27 GRACE derived gravity field: degree/order = 4

Background 7 / 27 GRACE derived gravity field: degree/order = 8

Background 8 / 27 GRACE derived gravity field: degree/order = 12

Background 9 / 27 GRACE derived gravity field: degree/order = 100

Background 10 / 27 Fundamentals

Recall geopotential equation:

∑∞ ( ) ∑l GM R l+1 V (r, ϑ, λ) = P¯ (sin ϑ)(C¯ cos(m, λ) + S¯ sin(m, λ)) (6) r r lm lm lm l=0 m=0 We can express changes in the potential as changes in equivalent water mass:

∑∞ ∑l ( ) RρE 2l + 1 ∆σ(ϑ, λ) = P¯lm(cos ϑ)(∆C¯lm cos(m, λ) + ∆S¯lm sin(m, λ)) (7) 3 1 + kl l=0 m=0

I ρE is the average density of the solid Earth; kl are the load love numbers I this equation accounts for change due to the added surface density assuming a rigid Earth, as well as the resultant elastic yielding of the Earth that tends to counteract the additional surface density

Background 11 / 27 Satellite Gravimetry

I key requirements: uninturrupted tracking in 3-D; measure or compensate for non-gravitation forces; low orbit I early studies treated a satellite as a test mass in free fall in Earth’s gravitational field I however, satellite motion is determined r by gravitation but is also disturbed by non-gravitational surface forces I recall: attenuation is an inverse( function) of satellite altitude R l+1 (recall r term in geopotential) I goal: measure higher order terms in the geopotential to offset attenuation

Background 12 / 27 CHAMP

I launched July 2000; altitude 450 km I HIGH-LOW mode: one Low-Earth Orbiter receives positioning information from GPS constellation I a single measures (first derivative of geopotential) ∂V ∂V ∂V ∂x = ax ; ∂y = ay ; ∂z = az

Challengin Minisatellite Payload (CHAMP)

Background 13 / 27 GRACE

I The Gravity Recovery and Climate Experiment (GRACE) was launched March 2002 I two satellites in Low Earth Orbit (500 km) separated by 220 km I GRACE concept: a one-dimensional( gradiometer ) ∂2V − with a very long baseline ∂x2 = a2x a1x

Background 14 / 27 GOCE

I The European Space Agency launched the Gravity field and steady-state Ocean Circulation Explorer (GOCE) March 2009 I a single Low Earth Orbit (250 km) satellite with aerodynamic design to minimize atmospheric drag I equipped with a 3-dimensional gradiometer to measure all components of geopotential variations over very short (50 cm) baselines ∂2V − ∂2V − ∂2V − ∂x2 = a2x a1x ; ∂y2 = a2y a1y ; ∂z2 = a2z a1z

Background 15 / 27 Processing GRACE

I The position of an orbiting body is defined by a series of orbital elements describing the shape of an ellipse (semimajor axis, eccentricity, inclination, etc.) √ ( ) 2a2 a p a˙ = b GM eS sin v + r T √ [ ( ) ] b a r+p er e˙ = a GM S sin v + r cos v + p T ...

I Perturbations in these orbital elements occur due to small variations in the geopotential, and also due to non-gravitational surface forces I components of the orbital elements are measured and expressed as perturbations to the Stokes coefficients in the geopotential equation

∑∞ ( ) ∑l GM R l+1 ¯ ¯ ¯ V (r, ϑ, λ) = r r Plm(sin ϑ)(Clm cos(m, λ) + Slm sin(m, λ)) l=0 m=0

I The direct measure of orbital elements above comprise the GRACE level 1 product. The GRACE Level 2 product is a series of C¯lm and S¯lm to degree/order 120.

Background 16 / 27 Two primary GRACE processing techniques

Spherical Harmonics I begin with the GRACE Level 2 product I calculate changes in the /surface mass density I mask out the region of interest and apply spatial and temporal smoothing algorithms mascons I begin with the GRACE Level 1 product (KBRR observations) I process only those data collected over region of interest (short-arc data reduction) I estimate the effects of the static gravity field on orbital parameters I compare this with the observed orbital changes I residual is the time variable gravity component of interest Background 17 / 27 spherical harmonic solution procedure

I use of an exact averaging kernel results in “ringing” at the kernel boundaries. This is addressed using an approximate (e.g. Gaussian) averaging kernel. I smoothing is also necessary due to errors

at orbit resonal orders 15 and 16 Gibbs Phenomenon

Background 18 / 27 mascon solution procedure

Recall: ∑∞ ∑l ( ) RρE 2l + 1 ∆σ(ϑ, λ) = P¯lm(cos ϑ)(∆C¯lm cos(m, λ) + ∆S¯lm sin(m, λ)) (8) 3 1 + kl l=0 m=0 Now solve for the delta Stokes Coefficients:

add uniform layer of 1 cm water

Earth

Background 19 / 27 mascon solution procedure

GRACE Level 1 KBRR Best estimate Time-variable gravity: Observations of static gravity all non-glacier sources eld of mass change

models/observations

Orbital Model

StokesOrbital coe€cients Model (express gravity as satellite obs)

Background 20 / 27 mascon solution procedure

scale factors on the set of dierential Stokes coe€cients for each mascon

Background 21 / 27 correcting GRACE for non-glacier sources of mass change

snowfall atmospheric pressure

r vegetation ie tent c LIA glacier ex la accumulation pack t g ocean snow ren onal cur seas

groundwater storage Earth crust ablation/calving Earth tides Earth mantle

Viscous mantle response to glacier unloading

Background 22 / 27 glaciological interpretation of GRACE time series

Latest mascon solution for all Alaska/NW Canada glaciers

400

300 A = (B s + B w )/ 2 200

100

0

-100 Cumulative mass balance (Gt)balance mass Cumulative -200 B = Bcumul /∆t -300 2003 2004 2005 2006 2007 2008 2009 2010 Year Background 23 / 27 Comparing GRACE with other glaciological datasets

I recall the fundamental limitations on GRACE resolution: higher degree terms become attenuated so solutions are truncated. I the spatial scale associated with a particular Stokes coefficient is about 20,000 km divided by its angular degree l. So for l 5 100, the length scales are about 200 km or larger 145°W 140°W I comparison 62°N 62°N 6 7 with individual glaciers only valid if 61°N that glacier represents a larger region 61°N I problem with mascon 60°N 60°N

regional comparisons: transfer Alaska Yukon 59°N Territory 59°N of glacier mass between mascons Study Area 10 58°N 58°N

145°W 140°W

Background 24 / 27 Background 25 / 27 GRACE validation in Alaska

Background 26 / 27 Summary

I there are numerous ways to convert satellite gravimetry data to Earth mass changes I regardless of the method, it is important to take care of local and global gravitation effects not associated with the geophysical parameter you are studying I trend analysis of short duration GRACE data series can be problematic, especially when fitting is done over non-integer numbers of years I GRACE has a fundamental spatial resolution limited by its orbital altitude, therefore regional analysis is more accurate than smaller scale analysis

Background 27 / 27