Representation of Global Earth Models Interoperability,
efficiency and prediction accuracy Bill Menke Lamont-Doherty Earth
Observatory Slide 2 Interoperability Being able to use someone
elses model in a quantitative way Slide 3 Goal providing the right
information to make your model interoperable Slide 4 even simple,
well known issues can cause substantial interoperability problems
heres an example Slide 5 down to center Geographical latitude not
equal to geocentric latitude Down not the same as direction to
center the world is not a sphere . X north pole, Z (X,Y,Z) Earth-
Centered, Earth Fixed Cartesian coordinates Slide 6 Recommendation
ECEF Coordinates You should always specify know to translate from
what ever coordinate system that youre using to earth-centered,
earth- fixed (ECEF) Cartesian coordinates, (X,Y,Z) (0,0,0) at
center of mass [0,0,1] points geographic north [1,0,0] points to
intersection of equator and prime meridian And you should describe
how to do it for any models that you publish or make widely
available Slide 7 Example: formula for converting geodetic
coordinates to EDEF Slide 8 ellipticity correction ~0.5s over
continental distances Slide 9 issues associated with ellipticity
1.Broad consensus to use geodetic lat, lons But what datum (model
for earths shape)? GPS satellite system: WGS-84 hundreds of others
in use Might effect: station locations event locations registration
of regional tomographic results with rest of world Slide 10
positioning errors of a few hundred meters are possible from using
wrong datum significant in earthquake location; somewhat
significant in global tomography Slide 11 issues associated with
ellipticity 2. How is depth defined? A problem here geodetic height
at a point is the distance from the reference ellipsoid to the
point in a direction normal to the ellipsoid so height is parallel
to local up/down, which makes intuitive sense Slide 12 down to
center Oops! depth defined this way misses the center of the earth
Yet if you define depth as straight-line distance to the earths
center, then geographic (lat, lon) varies along that line. center
Slide 13 issues associated with ellipticity 3. How so you ellipsify
a radial model of earth structure? You need to know how ellipticity
varies with depth. Kennett, B., Seismological Tables: AK135
ellipticity coefficients were made for the iasp91 model using the
algorithms presented by Doornbos (1988) with the density
distribution from PEMC model of Dziewonski et al (1975) and the
assumption that the ellipticity is nearly hydrostatic OK: So how do
I figure out what ellipticity vs. depth function Kennett used? I
need to know (r)! Why different? Slide 14 issues associated with
ellipticity 4. What is meant by a traveltime observation? A.
Traveltime B. Traveltime corrected for somebodys model of
ellipticity I suspect that the reference to iasp91 in the AK135
Tables means that traveltime data were de-ellipsified using before
being used Slide 15 recommendation Good Model prediction +
correction = observed data Bad Model prediction = observed data
correction bad because it tempts you to publish data that have been
corrected in some not-very-obvious way. corrections: ellipticity,
station elevation, anelasticity Slide 16 Case Study #1 of
Interoperabilty comparing continental scale models of P and S
velocity Interoperabilty includes having enough information to
understand the comparison Slide 17 P velocity: Phillips (2007)
isotropic model database of traveltimes 0.5 x0.5 grid, registered
on 0 , 50 km depth S velocity: Nettles & Dziewonski (2007)
Voigt average of anisotropic model Love & Rayleigh waveforms
0.5 x0.5 grid, registered on , 50 km depth Heres the biggest
problem Slide 18 My solution: interpolate S to 0 , registration
using bicubic splines Irony is that I know Nettles & Dziewonski
(2007) model was based on spherical splines, so exact values at
desired registration were theoretically available. Error probably
small, since grid appears to be oversampled, but its hard to be
sure. Slide 19 1:3 slope Slide 20 Clearly some systematic
disagreement in location western edge of craton what would we need
to know to understand it ? Slide 21 theres a rather lot of scatter
around dVs/dVp 3 expected from effect of temperature what would we
need to know to understand it ? Slide 22 Case Study #2 of
Interoperabilty how well does surface wave derived shear velocity
model predict S-wave traveltimes? Slide 23 Shear velocity: Nettles
& Dziewonski (2007) Voigt average of anisotropic model Love
& Rayleigh waveforms Crust 2.0 crust included in inversion but
not given 50 km depth intervals given form mantle 0.5 x0.5 grid,
registered on , Traveltime calculation 3D model with 0 ,
registration, 50 km depth slices raytracing by shooting in model
with tetrahedral splines Crust2.0 crust on top of Nettles &
Dziewonski (2007) mantle S-wave Traveltimes Database of North
America Regional Earthquake Traveltimes provided by W-Y Kim
(personal communication) and where did he get the corresponding
event locations? Slide 24 craton Western NA Map of ray exit points
Arrivals from Transition zone Slide 25 Another Oops (but for a
different reason) The surface wave model has negative lithospheric
velocity gradient over most of North America So there are no S-wave
rays turning in the lithosphere Slide 26 This problem is not
limited to Nettles and Dziewonskis (2007) North America model.
Kutowskis (2007) Eurasian model has regions with strong negative
velocity gradients, too. Slide 27 The Nettles and Dziewonski (2007)
model and body wave travel times are inherently interoperable Why,
we dont know maybe Earth looks different at very long periods or
S-arrivals are not real rays (Thybo & Perchuc, 1997) About the
best that one can do is to use it to motivate the construction of a
model that would be useful for body waves Slide 28 Example of
motivating the construction Nettles & Dziewonski (2007)
Vs-voigt at 50 km depth, scaled to Vp, and used to tweak velocity
up & down uniformly through-out an ak135 lithosphere Travetime
residuals predicted by model Traveltime residuals observed for
Gnome explosion Slide 29 many different model representations
spherical harmonics voxels splines, with varying arrangement of
nodes interpolant etc. Slide 30 How should models be converted
between representations? Guiding principle a model is a way of
summarizing the data therefore the conversion should try to
reproduce an idealized version of the data used to create the model
idealized in the sense that one might not know exactly what data
were used to create the model Slide 31 The figure at the left shows
a hypothetical layer cake crustal model. Many such models have been
published on the basis of active- source seismic refraction
surveys. Suppose that we want to switch to a representation that
uses linear splines. We use as the guiding principle that the
layer-cake model probably fits the first arrival traveltime data
well, in the distance range of a typical refraction experiment (say
200 km). crust mantle Example Slide 32 crust mantle Layer Cake
Model Traveltime Prediction calculated from simple formula Slide 33
crust mantle solid tts from layer-cake model Traveltime
observations and predictions calculated from inversion grey tts
from linear spline model Comparison of layer-cake and linear spline
models The fit to the data is excellent, r.m.s.=100 ms, even with
the crust being represented with only two linear splines. The
down-side is that an inverse problem needed to be solved to
computer the new model representation. Slide 34 recommendation
Model representations always be converted to preserve idealized
data, Even though this requires solving an inverse problem However,
the in hard cases the sense of idealization can be broadened to
include preserving quantities that are merely data-like, rather
than exactly data. Slide 35 Example traveltime data traveltime-like
data: integrals of slowness along a suite of ray paths, whose shape
is itself determined by the model traveltime-like data: integrals
of slowness along a suite of prescribed curves that look something
like ray paths Slide 36 efficiency and prediction accuracy Some of
my experiences with a ray-based traveltime tomography - earthquake
location algorithm that uses a 3D model representation based upon
linear tetrahedral splines Slide 37 Choice of linear splines and
tetrahedra motivated by efficiency Slide 38 Advantages of
tetrahedral models Ray paths known function (arc of circle) within
tetrahedron Important ray integrals can be performed analytically,
e.g. traveltime, T Where v is velocity, g is its gradient and Slide
39 Disadvantages of linear tetrahedral models Curved surface of
approximated as surface of polyhedron (but node spacing gives
reasonably accurate 100 ms - traveltime). Velocity gradient
discontinuous between tetrahedra (makes geometrical amplitudes
rather rough) Not obvious how to generalize method to anisotropic
earth models Slide 40 A few examples Slide 41 Slice through a 3D
model represented with linear tetrahedral splines Low Velocity Zone
(LVZ) Curved interface approximated as surface of polyhedron Slide
42 Some sample ray paths Slide 43 Map view of ray exit points for
source at rays loop in LVZ Moho triplication upper crustal
triplication Fan array in next slide Slide 44 Fan array Y-distance
traveltime, s Traveltime plot for fan array geometry LVZ
reverberations delayed by LVZ Slide 45 Efficiency issues Given an
irregular tetrahedral grid.. 1. How do you figure out whether a
given (x,y,z) point is in a given tetrahedron ? 2. How do you
figure out which tetrahedron a given (x,y,z) point is in ? Slide 46
1. Is point in a given tetrahedron? Yes or No Its is if for each of
the 4 faces of the tetrahedron it is on the same side of the face
as the excluded vertex* of the face One of the 4 faces Its excluded
vertex This point is on the same side as the excluded vertex This
ones not * Three of the four vertices of the tetrahedron define a
face. The fourth vertex is the excluded vertex. Slide 47 The
inside/outside test needs to be very efficient, because it is used
so often Thus information needed to perform it (e.g. the outward
pointing normal of each face) needs to be pre-computed and stored.
Slide 48 2. Which tetrahedron is a given (x,y,z) point in ? The
probability is overwhelmingly high that its in the same tetrahedron
as the last point that you focused upon because most operations are
spatially localized An if not, then its probably in a tetrahedron
that is close-by Slide 49 Finding the tetrahedron by walking toward
it New point vector connecting old and new points Always move to
tetrahedron adjacent to face with outward pointing normal is most
parallel to the line connecting test point and new point test point
current point Slide 50 Deciding which tetrahedron is adjacent to a
given face of a given tetrahedron needs to be very efficient,
because it is used so often Thus adjacency (nearest 4 neighbors)
information needed needs to be pre- computed and stored. Slide 51 1
2 3 4 5 6 7 1 2 3 4 5 6 1 2 3 6 5 4 9 8 7 10 16 15 14 13 12 20 19
18 17 28 27 26 25 24 23 22 21 29 38 37 36 35 34 33 32 31 30 48 47
46 45 44 43 42 41 40 39 55 54 53 52 51 50 Point is in Hash Cell
(5,5,0) which overlaps 5 tetrahedra 16, 19, 20, 21, 26 ECEF
Cartesian Hash Table can be used to facilitate finding tetrahedron
that encloses a point Slide 52 Station-specific 3D Traveltime
Tables that allow multiple arrivals Rays shot at suite of angles
from station Slide 53 Ray tubes with traveltime increments
tabulated And hashed onto underlying tetrahedral model Slide 54
Multiple ray tubes allowed Point tetrahedra ray tubes overlapping
this tetrahedron walk each ray tube to find segment inclosing point
Slide 55 Some Closing Remarks Slide 56 Interoperability Growing
need to use each others models in quantitative ways Other people to
be able to understand your model representation in considerable
detail Many subtle issues related to model representation make
interoperability difficult Models are not as interoperable as they
might seem, for reasons that go beyond mere differences in
representation Slide 57 Conversion between Representations
Conversions should try to preserve the original predictions of a
model, not merely the model itself. Conversion process should match
idealized data, a process that itself requires inversion, not
merely resampling. Which means that the authors need to state what
data are being fit and how well by their models. Slide 58
Efficiency and Accuracy Accuracy includes how well model fits earth
features, not just how well quantities are computed in that model.
Efficiency trades of with generality. Computation efficiency can be
enhanced by clever choices of pre-computed information (e.g. hash
tables) but trades off with storage efficiency.
