## 2 Methodology
### Influence of Attenuation Media on Depth-Domain Source Wavelets
Therefore, the complex wavenumber for the depth domain is written as follows:  
\[\widetilde{k}=\frac{1}{v}\left[k-i\tan\left(\frac{\pi\gamma}{2}\right)\left| k\right|\right]. \tag{9}\]  
Substituting Equations (8) and (9) into Equation (5) yields the 3D Green function in the wavenumber domain, which can be expressed as follows:  
\[g_{0,3D}(x|x_{0},k)=\frac{1}{4\pi|x-x_{0}|}e^{-i(\frac{1}{v_{0}}\frac{1}{2}| \gamma^{\prime})\left|k-i\tan\left(\frac{\pi\gamma}{2}\right)\left|k\right| \right|\left|x-x_{0}\right|}. \tag{10}\]  
Then, the wavenumber-domain analytical solution \(s(k)\) is defined as follows:  
\[s(k)=F(k)\cdot g_{0,3D}(x|x_{0},k), \tag{11}\]  
where \(F(k)\) represents the wavenumber spectrum of the source wavelet. Furthermore, the depth-domain analytic solution of wavelets can be obtained by the inverse Fourier transform of \(s(k)\). It can be found from Equation (10) that the effect of the wavenumber spectrum of the source wavelet by the wavenumber-domain 3D Green function can be divided into three parts: spherical divergence caused by \(1/(4\pi|x-x_{0}|)\), attenuation of different wavenumber components caused by \([k-i\tan(\pi\gamma/2)\left|k\right|]\), and dispersion caused by \(|k_{0}|^{\gamma}/\left(v_{0}|k|^{\gamma}\right)\). This results in the attenuation of amplitude and the waveform change for the depth-domain source wavelet. The attenuation of amplitude is easy to compensate, so this paper only considers the effect of attenuation media on the source-wavelet waveform, and the change of amplitude is temporarily ignored. The classical Ricker wavelet is commonly used as a source wavelet in seismic exploration. A standard time-domain Ricker wavelet can be written as follows [40]:  
\[w(t)=\left(1-\frac{1}{2}\omega_{p}^{2}(t-\tau_{0})^{2}\right)\exp\left(-\frac {1}{4}\omega_{p}^{2}(t-\tau_{0})^{2}\right), \tag{12}\]  
where \(\omega_{p}\) is the most energetic or dominant angular frequency (in radians per second) and \(\tau_{0}\) is the time delay. The time-depth relation links the depth and time domains, and is written as follows:  
where \(k_{p}\) is the dominant angular wavenumber (in radians per meter), which is equal to \(2\omega_{p}/v\), and \(l_{0}\) is the depth delay. The Fourier transform of Equation (15) is expressed as follows:  
\[w(k)=4\sqrt{\pi}\frac{k^{2}}{k_{p}^{3}}\exp\left(-\frac{k^{2}}{k_{p}^{2}}\right). \tag{16}\]  
For a homogeneous viscoacoustic medium with a velocity of 2 km/s and a \(Q\) value of 20, the wavenumber-domain 3D Green function is shown as the red curve in Figure 1a, indicating that the higher the wavenumber, the greater the attenuation. When a depth-domain Ricker wavelet with a dominant wavenumber of 50 km\({}^{-1}\) propagates 0.2 km in this medium, its wavenumber spectrum and depth-domain waveform are shown as the red curves in Figure 1b,c. As shown in Figure 1c, due to the attenuation, the wavelet is no longer symmetric, the waveform widens, and the amplitude of the right sidelobe decreases. The remaining black, green, and blue curves in Figure 1a-c, respectively, show the change in the depth-domain Ricker wavelet after it propagates 0.2 km in this medium with \(Q\) values of infinity (elastic), 100, and 50. It is evident that the waveform of the wavelet becomes wider and the right sidelobe becomes more attenuated with decreased \(Q\) values. Figure 1d-i show the changes of the depth-domain Ricker wavelet after it propagates 0.5 and 1.0 km in this medium with different \(Q\) values, respectively. It can be found that the larger the distance under the same \(Q\) value, the greater the distortion of the waveform. The red curve in Figure 1h gives the wavenumber spectrum of the source wavelet after a 1.0 km propagation in this medium with a \(Q\) value of 20. It can be seen that the peak wavenumber is about 28 km\({}^{-1}\), indicating that the degree of attenuation media is about 45%. The corresponding waveform of the depth-domain wavelet is represented by the red curve in Figure 1i, where the right sidelobe is completely absorbed.