Least-squares full-waveform inversion (FWI) is considered in the frequency domain for a set of noise-free observations of time length $T$ at the surface obeying the 1D wave equation, with a known source. The initial model is of constant velocity. The first iteration, which equals the constant-velocity migration inversion (CVMI), is thoroughly analyzed. In CVMI, for the unit source power spectrum, it is within reach to analytically derive and interpret the mathematical formulas of the first-order partial derivatives of the modeled observations (Jacobian), and the gradient and Gauss-Newton Hessian of the objective function, and learn what information the calculation requires to obtain a successful physical result (i.e., velocity update). We recognize the gradient elements, except the last one, to be sums of reflection-amplitude weighted band-limited sign functions and the Hessian elements, except along the last column and row, to be band-limited, diagonal-centered triangle functions, which for infinite bandwidth reduces to the Kronecker delta function. When the fundamental frequency $1/T$ is lacking in the observations, the gradient loses information of the low-wavenumber trend of the velocity update. The Hessian becomes close to singular, and any stabilized inverse has no chance to repair the deficiencies of the gradient caused by any missing low frequency in the observations. FWI is started by applying CVMI. First, Jacobians are modeled by classic reflectivity modeling. Second, the diagonal Hessians can be used for estimating discrete velocity updates. Third, the Jacobian can be modeled in the first-order Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) approximation and by neglecting transmission effects. Finally, single-frequency and low-frequency seismograms can be inverted by using broadband Hessians. The main mathematical findings are developed by simple numerical models and data.

中文翻译：

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对反射地震图一维反问题的理论贡献

最小二乘全波形反演 (FWI) 在频域中被考虑用于一组时间长度的无噪声观测 $吨$在表面服从一维波动方程，具有已知的来源。初始模型是恒定速度的。第一次迭代，等于等速偏移反演 (CVMI)，被彻底分析。在 CVMI 中，对于单位源功率谱，可以解析推导出和解释建模观测值的一阶偏导数（雅可比）的数学公式，以及目标函数的梯度和 Gauss-Newton Hessian，并了解计算需要哪些信息才能获得成功的物理结果（即速度更新）。我们认识到梯度元素，除了最后一个，是反射幅度加权带限符号函数的总和，而 Hessian 元素，除了最后一列和最后一行，是带限的、以对角线为中心的三角形函数，对于无限带宽，它减少到 Kronecker delta 函数。当基频$1/吨$缺乏观测，梯度丢失了速度更新的低波数趋势信息。Hessian 变得接近奇异，并且任何稳定的逆都没有机会修复由观测中任何缺失的低频引起的梯度缺陷。FWI 是通过应用 CVMI 启动的。首先，雅可比矩阵是通过经典的反射率建模来建模的。其次，对角 Hessians 可用于估计离散速度更新。第三，Jacobian 可以在一阶 Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) 近似中建模并忽略传输效应。最后，可以使用宽带 Hessian 反演单频和低频地震图。主要的数学发现是通过简单的数值模型和数据得出的。