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The forward and adjoint operators for shot-profile leastsquares migration are derived. The forward operator is demigration, and the adjoint operator is migration. The demigration operator is derived from the Born approximation. The process begins with a Green's function that allows for a laterally varying migration velocity model using the split-step approximation. Next, the earth is divided into horizontal layers, and within each layer the migration velocity model is made to be constant with respect to depth. For a given layer, ͑1͒ the source-side wavefield is propagated down to its top using the background wavefield. This gives a background wavefield incident at the layer's upper boundary. ͑2͒ The layer's contribution to the scattered wavefield is computed using the Born approximation to the scattered wavefield and the background wavefield. ͑3͒ Next, its scattered wavefield is propagated back up to the measurement surface using, again, the background wavefield. The measured wavefield is approximated by the sum of scattered wavefields from each layer. In the derivation of the measured wavefield, the shot-profile migration geometry is used. For each shot, the resulting wavefield modeling operator takes the form of a Fredholm integral equation of the first kind, and this is used to write down its adjoint, the shot-profile migration operator. This forward/adjoint pair is used for shot-profile least-squares migration. Shot-profile least-squares migration is illustrated with two synthetic examples. The first uses data collected over a four-layer acoustic model, and the second uses data from the Sigsbee 2a model.

We present a three-stage algorithm for adaptive separation of free-surface multiples. The free-surface multiple elimination (FSME) method requires, as deterministic prerequisites, knowledge of the source wavelet and deghosted data. In their absence, FSME provides an estimate of free-surface multiples that must be subtracted adaptively from the data. First we construct several orders from the free-surface multiple prediction formula. Next we use the full recording duration of any given data trace to construct filters that attempt to match the data and the multiple predictions. This kind of filter produces adequate phase results, but the order-by-order nature of the free-surface algorithm brings results that remain insufficient for straightforward subtraction. Then we construct, trace by trace, a mixing model in which the mixtures are the data trace and its orders of multiple predictions. We separate the mixtures through a blind source separation technique, in particular by employing independent component analysis. One of the recovered signals is a data trace without free-surface multiples. This technique sidesteps the subtraction inherent in most adaptive subtraction methods by separating the desired signal from the free-surface multiples. The method was applied to synthetic and field data. We compared the field data to a published method and found comparable results.