Objective: Our objective is to determine the best implementation of spherical harmonic synthesis algorithms on a massively parallel architecture. Past experience on conventional supercomputers have indicated that this aspect of the geodetic estimation problem can be a severe bottleneck in efficient parallelization of the estimator. Re-evaluation of the problem in terms of scalable, distributed-memory architecture machines such as the CRAY T3D is fundamental to the implementation of these computers in geodetic applications.
Approach: In the process of formulating an approach, we make an assumption that is inherent to the nature of the satellite gradiometry problem: the number of observations expected to be processed at any one time will greatly exceed the number of processors available. This assumption allows us to take a data parallel approach. In other words, a single processor will perform all required calculations for any given observation.
Accomplishments: The project has developed fast algorithms to calculate gravitational potential, gradient, force, and associated partials. One potential difficulty in the approach described above involves the storage of the gravitational coefficients associated with the spherical harmonic expansion used to represent the gravity field. To obtain the best performance, each processor must possess a copy of the coefficients in its local memory. The number of coefficients varies with the size of the gravity field according the formula,
number of coefficients = (degree of gravity field + 1)**2 - 4
For a gravity field of degree and order 70, the number of coefficients is 5,037, or 20 kilobytes of information. For a gravity field of degree and order 180, the number of coefficients is 32,757, or 131 kilobytes. The CRAY T3D has 64 megabytes of memory per processor; therefore, lack of sufficient memory should not be a problem.
A second and more critical difficulty arises from the I/O required to load the gravitational coefficients and the location information necessary to perform the calculations. In this analysis, tables of latitudes and longitudes ranging from 10-degree grid spacing (684 locations, or 2.7 kilobytes) to 1-degree grid spacing (65,160 locations, or 260 kilobytes) are used. The following results were from a case using a gravity field of degree and order 64 and 1-degree spacing. This case was run on the CRAY T3D located at the Jet Propulsion Laboratory.
This program consists of two distinct parts. The first is the inner loop over which the calculations are distributed. This loop displays almost perfect linear speed-up as the number of processors is increased. The second part of the program performs the necessary I/O required to read the data from disk and initialize the local data structures in each processor's memory. The deviation from linear speedup reflects the increased percentage of time required to perform the necessary I/0.
Status/Plans: The completion of this task allows us to move forward with the development of an application software package to estimate geopotential coefficients from gravity gradient observations. The next step is to combine the above work with least squares accumulation techniques. This process will involve the performance analysis of each module and the estimation of small gravity fields (degree and order 50) from simulated data in order to verify the software and perform preliminary studies of algorithm performance.
Point of Contact:
Dr. Bob E. Schutz
University of Texas at Austin
Center for Space Research
Campus Mail C0605
Austin, TX 78722