1. Representation of Global Earth Models
Interoperability, efficiency and prediction accuracy
Bill Menke
Lamont-Doherty Earth Observatory

2. Being able to use someone else’s model in a quantitative way
Interoperability
Being able to use someone else’s model in a quantitative way

3. Goal providing the right information to make your model interoperable

4. even simple, well known issues can cause substantial interoperability problems here’s an example

5. the world is not a sphere ….
to center
down
north pole, Z q f X
(X,Y,Z) Earth-Centered, Earth Fixed Cartesian coordinates
Geographical latitude f not equal to geocentric latitude q
Down not the same as direction to center

6. Recommendation – ECEF Coordinates
You should always specify know to translate from what ever coordinate system that you’re 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

7. Example: formula for converting geodetic coordinates to EDEF

8. ellipticity correction ~0.5s over continental distances

9. issues associated with ellipticity
Broad consensus to use geodetic lat, lon’s
But what datum (model for earth’s 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

10. positioning errors of a few hundred meters are possible from using wrong datum
significant in earthquake location; somewhat significant in global tomography

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

12. Oops! … depth defined this way misses the center of the earth
to center
down
center
Yet if you define depth as straight-line distance to the earth’s center, then geographic (lat, lon) varies along that line.

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 e(r)!
Why different?

14. issues associated with ellipticity
4. What is meant by a traveltime observation?
A. Traveltime
B. Traveltime corrected for somebody’s
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

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 …

16. Case Study #1 of Interoperabilty comparing continental scale models of P and S velocity Interoperabilty includes having enough information to understand the comparison

17. P velocity: Phillips (2007). isotropic model. database of traveltimes
P velocity: Phillips (2007) isotropic model database of traveltimes 0.5x0.5 grid, registered on 0, ½ 50 km depth S velocity: Nettles & Dziewonski (2007) Voigt average of anisotropic model Love & Rayleigh waveforms 0.5x0.5 grid, registered on ¼, ¾ 50 km depth
Here’s the biggest problem …

18. My solution: interpolate S to 0, ½ 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.

19. 1:3 slope

20. Clearly some systematic disagreement in location western edge of craton what would we need to know to understand it ?

21. there’s a rather lot of scatter around dVs/dVp3 expected from effect of temperature what would we need to know to understand it ?

22. Case Study #2 of Interoperabilty how well does surface wave derived shear velocity model predict S-wave traveltimes?

23. Shear velocity: Nettles & Dziewonski (2007)
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 ¼, ¾ Traveltime calculation 3D model with 0, ½  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?

24. Map of ray exit points
Western NA
craton
Arrivals from Transition zone

25. (but for a different reason)
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

26. This problem is not limited to Nettles and Dziewonski’s (2007) North America model. Kutowski’s (2007) Eurasian model has regions with strong negative velocity gradients, too.

27. The Nettles and Dziewonski (2007) model and body wave travel times are inherently interoperable
Why, we don’t 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

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

29. many different model representations
spherical harmonics
voxels
splines, with varying
arrangement of nodes
interpolant
etc.

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

31. Example
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

32. crust calculated from simple formula mantle Layer Cake Model
Traveltime Prediction

33. crust grey – tt’s from linear spline model calculated from inversion
mantle
grey – tt’s from linear spline model
calculated from inversion
solid – tt’s from layer-cake model
Traveltime “observations” and predictions
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.

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.

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

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

37. Choice of linear splines and tetrahedra motivated by efficiency

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

39. Disadvantages of linear tetrahedral models
Curved surface of approximated as surface of polyhedron (but ½  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

40. A few examples

41. Low Velocity Zone (LVZ)
Curved interface approximated as surface of polyhedron
Slice through a 3D model represented with linear tetrahedral splines

42. Some sample ray paths

43. Map view of ray exit points for source at
Fan array in next slide
rays loop in LVZ
Moho triplication
upper crustal triplication
Map view of ray exit points for source at

44. Traveltime plot for fan array geometry
traveltime, s
LVZ reverberations
delayed by LVZ
Fan array Y-distance

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 ?

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
This one’s not
One of the 4 faces
Its excluded vertex
* Three of the four vertices of the tetrahedron define a face. The fourth vertex is the “excluded” vertex.
This point is on the same side as the excluded vertex

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.

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

49. Finding the tetrahedron by walking toward it
current point
test point
New point
Always move to tetrahedron adjacent to face with outward pointing normal is most parallel to the line connecting test point and new point
vector connecting old and new points

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.

51. ECEF Cartesian Hash Table can be used to facilitate finding tetrahedron that encloses a point 7 20 4 6 2 10 5 16 21 3 9 8 19 15 24 1 13 26 27 29 17 12 28 14 18 25 22 30 23 39 32 36 50 51 37 54 47 40 31 53 55 34 38 33 35 41 46 44 42 48 45 43 52
Point is in Hash Cell (5,5,0) which overlaps 5 tetrahedra 16, 19, 20, 21, 26

52. Station-specific 3D Traveltime Tables that allow multiple arrivals
Rays shot at suite of angles from station

53. Ray tubes with traveltime increments tabulated
And hashed onto underlying tetrahedral model

54. Multiple ray tubes allowed
Point  tetrahedra  ray tubes overlapping this tetrahedron  walk each ray tube to find segment inclosing point

55. Some Closing Remarks

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

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.

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.