1. Stable North American Reference Frame (SNARF): Version 1
SNARF Working Group
Presented by
Jim Davis and Tom Herring

2. Outline Jim Davis Tom Herring Problem, approach, initial results
Products, use, future work

3. Purpose
To define a geodetic reference frame for “stable” North America
SNARF will form a common geodetic reference frame for PBO/EarthScope studies

4. Definitions and Assumptions
Large parts of North American continental crust are currently not deforming (“stable”) from plate tectonic forces.
These parts are mostly east of the Rocky Mountains
The entire North American continent is deforming significantly due to glacial isostatic adjustment (GIA)

5. Ice-1
LT = 120 km
nUM = 0.8  1021 Pa s
nLM = 10  1021 Pa s

6. Definitions and Assumptions
Given a geodetic solution with site velocities VGPS at locations (l,f), we can describe the solution using
The velocity rotation and translation parameters are unknown and must be estimated as part of the SNARF definition

7. GIA Predictions: Requirements
A model for the Earth’s viscoelastic structure
Theory and code to calculate the time-dependent Green’s functions for deformation from surface loads
A history of the time-dependent ice load, starting preferably prior to the most recent glacial maximum ~20 kYr ago
Theory and code to convolve the time-dependent load with the viscoelastic Green’s functions, while simultaneously solving for effects due to the redistribution of the surface load (icewater)

8. GIA Predictions: Practical Issues
No consensus concerning viscosity structure
No consensus concerning ice history
Ice & Earth models are generally not independent (inversions nonunique)
Current Earth models for GIA are spherically symmetric, but lateral variations are important (Latychev et al., 2004)

9. SNARF Approach
Rather than adopt an unrealistic ice/Earth model pair that we know will introduce systematic errors, SNARF is investigating a novel approach
GPS velocities will be assimilated into an a priori GIA model based on a suite of predictions to yield an observation-driven model

10. Assimilation of GPS Data into GIA Models
Bayesian approach
We use a Kalman-filter to assimilate the GPS velocities into a prior GIA model
The GIA model is the “average model” based on a suite of GIA models spanning a range of Earth models
The variability of the GIA models is used to calculate a statistical distribution (and covariance matrix) for the starting GIA field
We estimate GIA deformations on a grid (2°2°)

11. GPS Data Assimilation
We simultaneously estimate six rotation and translation para-meters, and GIA velocities at n grid locations and at m GPS sites
At right, the parameter vector (u = east velocity, v = north, w = radial)
The observations consist of (u,v,w) for GPS sites
The GIA values at the grid locations are adjusted through the covariances calculated from the suite of model predictions
SNARF 1.0 solution: n = 1537, m = 99, # parameters = 4617

12. Assimilation Tests
For proofs-of-concept tests, we focus on radial motions
These tests will use no “real” GIA information (“no physics”
The starting GIA model is the null field w(l,f) = 0
We adopt a Gaussian covariance model:
Lij = w(li,fi) w(lj,fj) = s2 exp(-dij2/D2)
Where dij is the angular distance between the locations, D = 10°, and s = 1 mm/yr
In the first test, we assimilate a single GPS observation at the location of site Churchill, Hudson’s Bay, Canada (w = 9.1 ± 0.2 mm/yr)

14. Assimilation Tests
In the second test, we assimilate a subset of NA radial site velocities
The assimilated field (still “no physics”) has many features of a realistic GIA field

15. Assimilation
Ice model: Ice-1 [Peltier & Andrews, 1976]; gives slightly better results than Ice-3G [Tushingham & Peltier, 1991]
Earth models: Spherically symmetric three-layer, range of elastic lithospheric thicknesses, upper and lower mantle viscosities (see Milne et al., 2001)
Elastic parameters: PREM
GPS data set: Velocities from “good” GPS sites, recent NAREF solution from Mike Craymer
Placed in approximate NA frame by Tom Herring (unnecessary step but simpler)

16. Prior Correlation wrt Churchill

17. SNARF 1.0 GIA Field

18. Solution Statistics Prefit statistics: WRMS (hor): 1.22 mm/yr
WRMS (rad): mm/yr
WRMS (all): mm/yr
Postfit statistics:
WRMS (hor): mm/yr
WRMS (rad): mm/yr
WRMS (all): mm/yr

22. Conclusions (Part 1)
Current “accuracy” of SNARF 1.0 is ~1 mm (+30% radial/-30% horizontal)
Estimated rotations/translations (from “nominal” no-GIA NA-fixed)  1.5 mm/yr
GPS assimilation technique seems to work, may be useful for GIA work
Straightforward to assimilate other data types