Velocity estimation of spatial aliased coherent noises propagating along a plurality of sensors

A computing device for calculating a velocity vp(f, Topt) of a spatially aliased wave that propagates along plural sensors, the computing device comprising:

• an interface configured to receive data Dt, wherein the data Dt is associated with a parameter measured by the plural sensors distributed along a cable, and the parameter is indicative of vibrations that propagate along the cable;
• and
• a processor connected to the interface, and configured to,
• calculate a phase velocity vp(f) of the spatially aliased wave that propagates along the cable, as a function of a temporal frequency fin a spatial-temporal frequency domain FK;
• calculate a model-based velocity vp(f, T) of the spatially aliased wave as a function of the temporal frequency f and a tension T in the cable;
• and
• calculate the velocity vp(f, Topt) of the spatially aliased wave using a model-guided regression, which is based on an optimized tension Topt, the phase velocity vp(f) and the model-based velocity vp(f, T),
• wherein the velocity vp(f, Topt) is a function of the temporal frequency f, and
• wherein the optimized tension Topt is calculated by:
• estimating a first tension of the cable;
• measuring, in the time-space domain with the plural sensors, the parameter that is associated with the vibrations that propagate along the cable, wherein the parameter may be an acceleration, velocity, displacement, or pressure;
• transforming the data Dt associated with the parameter from the time-space domain to the FK domain to obtain FK data F(f, k), wherein f is the temporal frequency and k is the spatial frequency;
• periodizing the data F(f, k) according to the spatial frequency k, wherein a factor C is used to replicate the data F(f, k) C times to obtain periodized data Fp(f, k);
• defining a mask M(f, k) in the FK domain to isolate an upward or downward wave and applying the mask M(f, k) to the periodized data Fp(f, k) to obtain masked data Wm(f, k);
• generating a first list L1, for each of the spatial frequency k, that includes (1) a frequency fmax that maximizes the masked data |Wm(f, k)|, and (2) a first speed fmax/k;
• generating a second list L2, for each the temporal frequency f, that includes (i) a spatial frequency kmax that maximizes the masked data |Wm(f, k)|, and (2) a second speed f/kmax;
• optionally generating a third list L3, that includes pairs common to the first list L1 and the second list L2
• for each of the generated lists, calculating the phase velocity vp(f) of the spatially aliased wave that propagates along the cable;
• selecting a tension Tj, and calculating the model-based velocity vp(f, Tj) for the tension Tj;
• calculating a root mean square error (RMSE) between (a) the model-based velocity vp(f, Tj) for the tension Tj, and (b) the velocity associated with each of the lists;
• and
• selecting the optimized tension Topt associated with the minimum RMSE.