Design of Seismic Wavelets and Reflection Coefficient Sequences

The design of seismic wavelets and construction of reflection coefficient sequences are fundamental steps in seismic exploration data processing. By synthesizing single-trace seismic records and performing inversion to solve for wavelets and reflection coefficients, the effectiveness of algorithms can be validated while gaining deeper insights into the physical processes of seismic wave propagation.

Seismic Wavelet Design Seismic wavelets typically employ Ricker wavelets or other band-limited wavelet models, where the dominant frequency and phase characteristics directly influence the morphology of seismic records. Wavelet design must consider the actual propagation attenuation characteristics of seismic waves and instrument responses. In code implementation, this involves defining parameters like peak frequency and time duration, then generating the wavelet using mathematical formulas (e.g., Ricker wavelet formula: (1-2π²f²t²)e^(-π²f²t²)).

Reflection Coefficient Sequence Construction Reflection coefficient sequences represent the impedance contrasts in subsurface media, which can be generated randomly or based on geological models. The sparsity and amplitude distribution of the sequence should conform to actual formation reflection characteristics. Implementation typically involves creating sparse vectors with non-zero values at layer boundaries, often following statistical distributions similar to real geological data.

Seismic Record Synthesis Synthetic seismic records are generated by convolving the wavelet with the reflection coefficient sequence. This process simulates the propagation and reflection response of seismic waves in subsurface media. In programming terms, this is implemented using convolution operations (e.g., MATLAB's conv function or Python's numpy.convolve) between the wavelet vector and reflection coefficient array.

Logarithmic Decomposition for Wavelet Extraction By taking the logarithm of the seismic record's spectrum, the multiplicative relationship between wavelet and reflection coefficients is transformed into an additive relationship, allowing separation of the wavelet's spectral characteristics. This method relies on the randomness assumption of reflection coefficient sequences. Algorithm implementation involves Fourier transforms, logarithmic operations, and spectral averaging techniques.

Deconvolution for Reflection Coefficient Recovery Using the extracted wavelet, deconvolution processing of the seismic record can reconstruct the reflection coefficient sequence. The choice of deconvolution algorithm (such as least squares deconvolution) directly affects the resolution and stability of the solution. Code implementation typically involves designing inverse filters or using optimization methods to minimize the difference between convolved results and actual seismic data.

This workflow completely demonstrates the closed-loop verification approach for seismic data forward and inverse modeling, providing an effective practical path for understanding core algorithms in seismic signal processing.