I/O implications

I/O is an important consideration when dealing with industrial-size data sets. Contrary to the naive impression that the I/O cost of the proposed method in Kit iterations is Kit times that of the standard migration, here I show that the actual I/O cost of the former is only $2+\epsilon$ times^2.2 the latter, assuming the migrations are carried out in the frequency domain while the original data are stored in the time domain. If, however, the data have been transformed into the frequency domain already, then the I/O cost of the former method is only $\epsilon$ times the latter.

Let I/O cost be identified with the size^2.3 of data passing through I/O, and assume the data size is $M_{CSG}$ . The standard migration entails reading every shot gather, followed by the Fourier transform and then the migration. So the I/O cost is $C_0 = M_{CSG}$ . On the other hand, the work flow of the proposed method consists of two stages. (1) Preparation. All input data are read, transformed to the frequency domain and saved to disk. The I/O cost of this stage is $C_1 = 2 M_{CSG}$ . (2) Migration. The I/O cost per iteration is $M_{CSG_{enc}}$ . In $K_{it}$ iterations, the I/O cost is $C_2 = K_{it} M_{CSG_{enc}} = K_{it} \frac{M_{CSG}}{S} = \epsilon M_{CSG}$ , where $\epsilon = K_{it}/S \ll 1$ as $K_{it}$ is typically an order of magnitude smaller than $S$ , which is how speedup can be gained by iterative multisource methods. Therefore the I/O cost $C_1 + C_2 = (2 + \epsilon) C_0$ of the proposed method is a little more than twice that of the standard approach for any $K_{it} \ll S$ . If the data is in the frequency domain already, then the work pertaining to stage (1) is unnecessary. In this case, the I/O cost of the proposed method is only $C_2 = \epsilon C_0$ .

If ${\textrm{CSG}_\textrm{enc}}$ can fit in a computer's memory, $C_2$ can be further reduced as follows. Read the CSGs from disk to form a ${\textrm{CSG}_\textrm{enc}}$ , which is kept in the memory, then make $K_{CGit}$ iterative updates^2.4 to the trial model. In this scheme, $C_2$ is reduced by a factor^2.5 of $K_{CGit}$ .