Presentation on theme: "Representation of Global Earth Models"— Presentation transcript:
1Representation of Global Earth Models Interoperability, efficiency and prediction accuracyBill MenkeLamont-Doherty Earth Observatory
2Being able to use someone else’s model in a quantitative way InteroperabilityBeing able to use someone else’s model in a quantitative way
3Goal providing the right information to make your model interoperable
4even simple, well known issues can cause substantial interoperability problems here’s an example
5the world is not a sphere …. to centerdownnorth pole, ZqfX(X,Y,Z) Earth-Centered, Earth Fixed Cartesian coordinatesGeographical latitude f not equal to geocentric latitude qDown not the same as direction to center
6Recommendation – 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 equatorand prime meridianAnd you should describe how to do it for any models that you publish or make widely available
7Example: formula for converting geodetic coordinates to EDEF
8ellipticity correction ~0.5s over continental distances
9issues associated with ellipticity Broad consensus to use geodetic lat, lon’sBut what datum (model for earth’s shape)?GPS satellite system: WGS-84hundreds of others in useMight effect:station locationsevent locationsregistration of regional tomographic results withrest of world
10positioning errors of a few hundred meters are possible from using wrong datum significant in earthquake location; somewhat significant in global tomography
11issues 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
12Oops! … depth defined this way misses the center of the earth to centerdowncenterYet if you define depth as straight-line distance to the earth’s center, then geographic (lat, lon) varies along that line.
13issues 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?
14issues associated with ellipticity 4. What is meant by a traveltime observation?A. TraveltimeB. Traveltime corrected for somebody’smodel of ellipticityI suspect that the reference to iasp91 in the AK135 Tables means that traveltime data were de-ellipsified using before being used
15recommendation Good Model prediction + correction = observed data Bad Model prediction = observed data – correctionbad because it tempts you to publish “data” that have been “corrected” in some not-very-obvious way.corrections: ellipticity, station elevation, anelasticity …
16Case Study #1 of Interoperabilty comparing continental scale models of P and S velocity Interoperabilty includes having enough information to understand the comparison
17P velocity: Phillips (2007). isotropic model. database of traveltimes P velocity: Phillips (2007) isotropic model database of traveltimes 0.5x0.5 grid, registered on 0, ½ 50 km depth S velocity: Nettles & Dziewonski (2007) Voigt average of anisotropic model Love & Rayleigh waveforms 0.5x0.5 grid, registered on ¼, ¾ 50 km depthHere’s the biggest problem …
18My 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.
20Clearly some systematic disagreement in location western edge of craton what would we need to know to understand it ?
21there’s a rather lot of scatter around dVs/dVp3 expected from effect of temperature what would we need to know to understand it ?
22Case Study #2 of Interoperabilty how well does surface wave derived shear velocity model predict S-wave traveltimes?
23Shear 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 0.5x0.5 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?
24Map of ray exit pointsWesternNAcratonArrivals from Transition zone
25(but for a different reason) Another Oops …(but for a different reason)The surface wave model hasnegative lithospheric velocity gradient over most of North AmericaSo there are no S-wave rays turning in the lithosphere
26This 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.
27The Nettles and Dziewonski (2007) model and body wave travel times are inherently interoperable Why, we don’t know … maybeEarth looks different at very long periodsorS-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
28Example 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 lithosphereTravetime residuals predicted by modelTraveltime residuals observed for Gnome explosion
29many different model representations spherical harmonicsvoxelssplines, with varyingarrangement of nodesinterpolantetc.
30How should models be converted between representations? Guiding principlea model is a way of summarizing the datathereforethe conversion should try toreproduce an “idealized” version of the dataused to create the model“idealized in the sense that one might not know exactly what data were used to create the model
31ExampleThe 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).crustmantle
32crust calculated from simple formula mantle Layer Cake Model Traveltime Prediction
33crust grey – tt’s from linear spline model calculated from inversion mantlegrey – tt’s from linear spline modelcalculated from inversionsolid – tt’s from layer-cake modelTraveltime “observations” and predictionsComparison of layer-cake and linear spline modelsThe 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.
34recommendationModel representations always be converted to preserve “idealized data”,Even though this requires solving an inverse problemHowever, the in hard cases the sense of idealization can be broadened to include preserving quantities that are merely “data-like”, rather than exactly data.
35Example – traveltime data traveltime-like data: integrals of slowness along a suite of ray paths, whose shape is itself determined by the modeltraveltime-like data: integrals of slowness along a suite of prescribed curves that look something like ray paths
36efficiency and prediction accuracy Some of my experiences with aray-based traveltime tomography - earthquake location algorithmthat uses a 3D model representation based upon linear tetrahedral splines
37Choice of linear splines and tetrahedra motivated by efficiency
38Advantages of tetrahedral models Ray paths known function (arc of circle) within tetrahedronImportant ray integrals can be performed analytically, e.g. traveltime, TWhere v is velocity, g is its gradient and
39Disadvantages 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
43Map view of ray exit points for source at Fan array in next sliderays loop in LVZMoho triplicationupper crustal triplicationMap view of ray exit points for source at
44Traveltime plot for fan array geometry traveltime, sLVZ reverberationsdelayed by LVZFan array Y-distance
45Efficiency issues Given an irregular tetrahedral grid .. 1. How do you figure out whethera given (x,y,z) point is in a given tetrahedron ?2. How do you figure out whichtetrahedron a given (x,y,z) point is in ?
461. Is point in a given tetrahedron? Yes or No Its is if for each of the 4 faces of the tetrahedronit is on the same side of the faceas the excluded vertex* of the faceThis one’s notOne of the 4 facesIts 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
47The 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.
482. 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 localizedAn if not, then its probably in a tetrahedron that is close-by
49Finding the tetrahedron by walking toward it current pointtest pointNew pointAlways move to tetrahedron adjacent to face with outward pointing normal is most parallel to the line connecting test point and new pointvector connecting old and new points
50Deciding which tetrahedron is adjacent to a given face of a given tetrahedron needs to be very efficient, because it is used so oftenThus adjacency (nearest 4 neighbors) information needed needs to be pre-computed and stored.
51ECEF Cartesian Hash Table can be used to facilitate finding tetrahedron that encloses a point 7204621051621398191524113262729171228141825223023393236505137544740315355343833354146444248454352Point is in Hash Cell (5,5,0) which overlaps 5 tetrahedra 16, 19, 20, 21, 26
52Station-specific 3D Traveltime Tables that allow multiple arrivals Rays shot at suite of angles from station
53Ray tubes with traveltime increments tabulated And hashed onto underlying tetrahedral model
54Multiple ray tubes allowed Point tetrahedra ray tubes overlapping this tetrahedron walk each ray tube to find segment inclosing point
56InteroperabilityGrowing need to use each others models in quantitative waysOther people to be able to understand your model representation in considerable detailMany subtle issues related to model representation make interoperability difficultModels are not as interoperable as they might seem, for reasons that go beyond mere differences in representation
57Conversion 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.
58Efficiency 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.