Coordinate system
This class of models uses a 3-D cartesian grid of points to represent central Asia. The spherical shape of the Earth is projected directly into cartesian space. The Earth is NOT "flattened". The coordinate transformation function takes latitiude, longitude, and depth in km from Earth's surface and returns X,Y,Z coordinates. The origin of the grid is at (48°00'N,75°00'E). At the origin, the positive X direction is west; the positive Y direction is north; and the positive Z direction is the normal to the surface of the Earth (up). This coordinate system is generated by:

1. applying a spherical polar to cartesian transform (polar angle = 90°-latitude / azimuth = longitude-75°)
2. a -48° rotation of the XYZ grid in the XZ plane
3. a 90° rotation of the XYZ grid in the XY plane
4. swapping the positive and negative X-axis
5. setting the Z coordinate to 6371-Z

Though unconventional to use an X-axis which is positive to the west, this makes the coordinate system right-handed for depth increasing into the Earth. The X and Y components are fairly intuitive. Z is defined as the depth beneath the the plane tangent to the surface at the origin.
(Lat/Lon to XYZ coordinate transformation script)
(XYZ to lat/lon coordinate transformation script)

Name Code lat. lon. X, km Y, km Z, km Ala-Archa AAK 42.6300 74.4800 42.54 70.18 0.53 Aktyubinsk AKTO 50.4340 58.0180 1185.26 1052.84 200.40 Borovoye BRVK 53.0581 70.2828 314.89 1230.66 127.93 Kurchatov KURK 50.7200 78.6200 -254.67 971.27 79.62 Makanchi MAK 46.8001 82.0001 -531.51 554.88 46.50 Pari NIL 33.6500 73.2512 161.85 -923.54 69.37 Zalesovo ZAL 53.9367 84.7981 -638.24 1354.33 178.42

Left Model grid shown with regionalization and seven proposed IMS stations in central Asia. Center Same information displayed in cartesian coordinate system. Right A slice through the model in the XZ plane. This view is essentially an east-west cross-section. Note how Z coordinate is defined below a tangent plane, as opposed to depth below the surface. (~250 km x 250 km node spacing shown). Table lists locations of central Asia IMS stations in this coordeinate system

Model Grid
Gridding of the A class models is performed in the latitude, longitude reference frame. These points are then transformed into the cartesian system. The four corners of grid are:

Vertical nodes lie along radii passing through the center of the Earth and the surface grid points discussed above. The vertical node spacing is defined to approximate the IASPEI 1991 1-D Earth model. The 1-D velocity profile is "suspended" from each of the grid points to. The raytracer used in this project assumes linear gradients between node points - an assumption well suited to the IASPEI model. The 2 layer 35 km crust defined in the IASPEI model is represented here as a single 35 km layer with an average slowness that is the same as IASPEI. Boundaries at the Moho, 410 km and 660 km are smoothed by a linear velocity gradient typically over a 10-20 km thickness centered on the actual discontinuity.

Model validation
Replacing the spherical shape of the Earth with a multi-faceted cartesian representation can introduce error into ray calculation. Tests of traveltime as a function of range, azimuth, and model node spacing were carried out to assess such errors. (The tests shown in the figures below are based on a model which differs slightly from IASPEI. While the tests are entirely valid, the traveltimes differ slightly from IASPEI predictions.)

The total error in traveltime prediction due to model coarseness is as follows:
~250 km x 250 km grid: < 0.4 seconds
~125 km x 125 km grid: < 0.1 seconds
~ 62 km x 62 km grid: < 0.02 seconds

Node density comparison: The model space shown above was gridded with horizontal node spacings of ~60-500 km. The results were encouraging. At grid spacings of ~500 km, the effects of the faceted Earth model are clear. Each flat block in the model is observed in the traveltime curve. Evenso, traveltime errors are generally less than 0.2 s. The traveltime surface smooths considerably for finer node spacings. Error introduced for a node spacing of ~125 km is < 0.05 s and is negligible for a spacing of ~62 km. For the purposes of this investigation, a grid spacing of 125 km or better is more than adequate. The other tests displayed here were performed on a grid with 125 km node spacing (a 40-by-50 horizontal grid of nodes with 14 "vertical" layers).

Fixed receiver azimuth test: In this test, the receiver positions are fixed to lie along lines of constant azimuth. The minimum traveltime path to each receiver is then determined. The results show error < 0.1 s associated with different paths in a ~125 km x 125 km gridded model. The oscillatory nature of the traveltime surface is a function of model coarseness discussed in the node density test. The 0.05 s constant variation between different azimuths is also a function of the faceted model, and depends on the receiver location on the gridded surface.

Revised node gridding
Model tests were performed on the grid described above were rows and columns on nodes lie on great circle paths. For mundane tasks such as plotting and model examination, it is easier to manipulate and work with models in which rows and columns have constant X and Y values, respectively.

Using the same coordinate system and corner points, a model of this type was created. The average node spacing do not differ significantly from the "A" models above. This new set of models, the "B" class, is warped in a slightly different way to get acheive rows and colums of constant X and Y. It is this model scheme that is actually used in the 1D and 3D velocity modeling throughout this project. Useless otherwise stated, all models in this project are of this type.
LeftViews of model series "B". View descriptions are identical to the descriptions pf "A" model images above. (~125 km x ~125 km node spacing shown)

Model Files

"A" class models	"B" class models	Description (files are Z compressed)
A: 10x12 nodes	B: 10x12 nodes	Very fast. Too coarse to be useful for much expect prelimenary code testing
A: 20x25 nodes	B: 20x25 nodes	Fast. Good for prelimenary modeling
A: 40x50 nodes	B: 40x50 nodes	Suitable when errors of 0.05 s are acceptible
A: 80x100 nodes	B: 80x100 nodes	Negligible model error but combersome and slow