图  1  密集台阵子台网示意图，以台站为波场空间梯度计算的参考点. 灰色影区内的三角形表示子台网，其中红色三角表示参考台站$ {S}_{0}({r}_{0},{\theta }_{0}) $，白色三角为辅助台站$ {S}_{i}\left({r}_{i},{\theta }_{i}\right),i=\mathrm{1,2}\dots $ ；子台网对应的波形为$ {u}_{i}\left(i=\mathrm{0,1},2,..\right) $ （修改自Langston and Liang, 2008）

Figure  1.  Subarray diagram using stations as reference location for wave gradient analysis. Triangles represent seismic stations and triangles in gray areas are stations of subarray; red and white stations represent reference station [S₀ (r₀, θ₀)] and supporting stations, respectively (modified from Langston and Liang, 2008)

图  2  二维规则台阵子台网示意图，以网格点为波形空间梯度计算的参考点. 红色圆为网格点，白色三角形为辅助台站

Figure  2.  Subarray diagram in a 2-D regular seismic array, where the grid point (i.e., red dot) is the reference location for wave gradient analysis and white triangles represent supporting stations

图  3  二维不规则台阵中的子台网，以台站为波形空间梯度计算的参考点（修改自Liang and Langston, 2009）

Figure  3.  Subarray diagram in a 2-D regular seismic array, where the red station is the reference location for wave gradient analysis and black triangles represent supporting stations (modified from Liang and Langston, 2009)

图  4  2007年4月1日M8.1所罗门地震在南加州地震台网不同震相的波场梯度结果，从上到下依次是地震波形、方位角和慢度（引自Langston and Liang, 2008）

Figure  4.  Results of wave gradiometry analyses in different phases of the M8.1 Solomon earthquake wavefield, as recorded by the Southern California Seismic Network. From top to bottom, sections of the figure represent the waveform, azimuth, and slowness for the same (cited from Langston and Liang, 2008)

图  5  利用TAIGER记录的爆炸源的水平分量加速度计算的应变和水平旋转. U₁和U₂分别对应于X和Y方向的水平加速度. U_1,1+U _2,2、U_1,1−U_2,2、U_1,2+U_2,1和U_1,2−U_2,1分别为面应变、法向应变的微分、两倍的剪切应变和两倍的负水平旋度（引自Langston et al., 2009）

Figure  5.  Strain and horizontal rotation calculated from the horizontal component acceleration of the explosion source, as recorded by TAIGER. U₁ and U₂ correspond to the horizontal acceleration in the X and Y directions, respectively. U_1,1+U _2,2, U_1,1− U_2,2, U_1,2+U_2,1 and U_1,2−U_2,1 are the coordinate-invariant area, normal differential, twice of the shear strains, respectively, in the x-y coordinate system; the negative value is twice the horizontal rotation about the vertical axis (cited from Langston et al., 2009)

图  6  基于波场梯度分析的S波的视相速度[（a）、（b）、（d）和（e）的彩色底图]、地面转动[（b）和（e）的黑色曲线]、S波到时[（a）、（b）、（d）和（e）的红色虚线]和传播方向[（c）和（f）的箭头]的结果；图（c）和（g）的虚线表示震源和台阵的连线；（a-c） 和（d-f）分别代表震源EP-3 和 EP-5的结果；图（g）和 （h）分别为阿波罗 17 号着陆点下方月壳的震波速和泊松比结构（修改自Sollberger et al., 2016）

Figure  6.  Gradient-based estimates of apparent phase velocity [i.e., color maps (a), (b), (e), and (f)], rotational ground motion [i.e., black curves in (b) and (f)], and propagation direction [i.e., arrows in (c) and (g)]. Shear wave arrivals are identified based on their distinct increase in the amount of rotational energy and are marked in red. Dashed lines underlying the propagation direction estimates mark the source-receiver azimuth, according to the survey geometry; (h) shows the seismic velocity and Poisson's ratio structure of the lunar crust below the Apollo 17 landing site (modified from Sollberger et al., 2016 )

图  7  （a）区域构造与（b）台站分布. 图（a）中白线是块体边界；左下方的子图为地震事件分布图，红色圆点表示地震，绿色五角星表示鞑靼海峡M6.2地震，黄色小方块表示研究区域. ANHF：安宁河断层；LMSF：龙门山断裂；XSHF：鲜水河断裂； BFZ：板块边界断裂带；LXF：丽江—小金断裂；LRBF：龙日坝断裂. 图（b）中浅蓝色三角形是川西流动台阵（TWSA）（修改自Cao et al., 2020）

Figure  7.  Geological setting and seismic array. (a) White lines denote the boundaries of tectonic blocks; red dots represent earthquakes; green pentagram indicates the M6.2 earthquake in the Tatar Strait. SGB is the Songpan-Ganzi block; SCB is the South China Block; SCDsB is the south Chuandian subblock; NCDsB is the north Chuandian subblock; ANHF is the Anning He Fault; LMSF is the Longmenshan fault; XSHF is the Xianshuihe fault; BFZ is the boundary fault zone; LXF is the Lijiang-Xiaojin fault. (b) Light blue triangles denote stations of the Temporary West Sichuan Array (modified from Cao et al., 2020)

图  8  单个参考位置的波场梯度分析结果. （a）子台网空间分布，红色三角形为参考位置的台站，蓝色三角形为辅助台站，红色直线为大圆路径 （图7a 绿色五角星）；（b）子台网波形，红色为参考位置台站记录的波形，蓝色为辅助台站的记录的波形；（c）参考位置的波场梯度分析在事件序列上的结果，从上到下依次为参考位置的波形、相速度、方位角变化、几何扩散和辐射花样，蓝色直线标记了大圆路径对应的方位角，紫色短棒标记了瑞利面波最大振幅所在相位的结果，绿色短棒标记了一个周期的时间序列上的结果，该时间序列上的结果用于计算单个地震事件波场梯度分析的标准偏差

Figure  8.  Wave gradiometry analyses for a single reference location. (a) Triangles represent stations and the red triangle represents the reference location; blue triangles represent supporting stations; the red straight line shows the ray direction (i.e., green pentacle in Figure 7a); (b) Waveforms of subarray with period of T = 40 s; the red trace is the waveform of the reference location, and blue traces are waveforms of supporting stations; (c) Subfigures, from top to bottom, are waveform of the reference location, phase velocity, azimuth, geometrical spreading and radiation pattern, respectively; the blue straight-line marks the great circle azimuth; the vertical bar in the middle marks the timing of the waveform peak; another two green bars are about half of the period apart from the bar in the middle. The standard deviation between these two times is given as errors for corresponding parameters

图  9  基于波场梯度分析的2007年8月2日鞑靼海峡M6.2地震在川西地区的瑞利面波传播参数（a-d）及对应的标准差（e-h），其中T = 40 s，白色粗线为块体边界，白色细线为断层线，青色曲线为河流（修改自Cao et al., 2020）

Figure  9.  Rayleigh wave propagation parameters (a–d) and corresponding standard deviations (e–h) of the M6.2 Tatar Strait earthquake at T = 40 s in the western Sichuan region based on wave gradiometry analysis. Thick and thin white curves represent block boundaries and faults, respectively, and cyan lines represent rivers (modified from Cao et al., 2020)

图  10  青藏高原东南缘T=40 s瑞利面波的方位各向异性拟合（修改自Cao et al., 2020）. （a）基于后方位角变化的相速度，黑色圆圈表示被删除的相速度，青色圆点为保留的相速度值，棕色虚线标记了各向同性相速度；（b）各向异性拟合，黑色圆点为每10°后方位角相速度的中值，青色曲线为方位各向异性最终的拟合结果；（c）不同区域的方位各向异性拟合的结果

Figure  10.  Azimuthal anisotropy fitting of Rayleigh surface waves at T=40 s (modified from Cao et al., 2020). (a) Azimuth-dependent velocity; black circles represent the discarded data points, cyan dots are the phase velocity used for anisotropy fitting, and the brown dotted line marks the isotropic phase velocity; (b) Anisotropy fitting; black dots are the median of the phase velocity in every ten-degree window of the back azimuth and the cyan curve is the final fitting result of azimuth anisotropy; (c) Fitting results of azimuthal anisotropy in different regions

图  11  青藏高原东南缘T=40 s瑞利面波的各向同性相速度与方位各向异性结果. （a）瑞利面波各向同性相速度; （b）Rayleigh波方位各向异性，短棒的方向代表快波方向，颜色代表各向异性强度，蓝色曲线为敏感核曲线；（c）各向异性标准偏差，底图表示快波方向标准偏差，黄色条的长度表示各向异性强度的标准偏差（修改自Cao et al., 2020）

Figure  11.  Isotropic phase velocity and azimuthal anisotropy of Rayleigh wave at T=40 s. (a) Rayleigh wave isotropic phase velocity. (b) Azimuthal anisotropy of Rayleigh waves, where the short bar direction represents the fast orientation, the color represents the anisotropic magnitude, and the blue curve represents the sensitivity kernel. (c) Anisotropic standard deviation, where the base represents the standard deviation of the fast orientation, and the length of the yellow bar represents the standard deviation anisotropic magnitude (modified from Cao et al., 2020)

图  12  美国大陆背景噪声成像和波场梯度法计算的瑞利面波相速度对比（修改自Porter et al., 2016）. ANT：背景噪声成像；WG：波场梯度法

Figure  12.  Absolute value images of differences between phase velocities calculated using ambient noise tomography and wave gradiometry (modified from Porter et al., 2016)

图  13  青藏高原东南缘波场梯度法和程函方程面波成像获取的瑞利面波相速度和方位各向异性结果对比. （a-c） 波场梯度法基于川西台阵获得的20 s、40 s和60 s周期的结果（修改自Cao et al., 2020）；（d-f） 程函方程面波成像获得的18 s、40 s 和60 s的结果（修改自王怀富等，2020）

Figure  13.  Comparisons of Rayleigh wave phase velocity and azimuthal anisotropy, as calculated by wave gradiometry and Eikonal tomography in the southeastern margin Tibetan Plateau. (a-c) Results at periods of 20, 40, and 60 s calculated using wave gradiometry (modified from Cao et al., 2020); (d-f) Results at periods of 20, 40, and 60 s calculated using Eikonal tomography (modified from Wang et al., 2020)

图  14  美国大陆程函方程成像（a-c）（修改自Jin and Gaherty, 2015）和波场梯度法（d-f）计算的瑞利面波相速度对比 （修改自Porter et al., 2016）

Figure  14.  Comparisons of Rayleigh wave phase velocity calculated by Eikonal tomography (a-c) (modified from Jin and Gaherty, 2015) wave gradiometry method and (d-f) in the North American continent (modified from Porter et al., 2016)

图  15  青藏高原东南缘波场梯度法和接收函数Pms震相计算的各向异性比较（修改自Cao et al., 2020）. 每个子图的玫瑰图统计了两种方法获得的各向异性在快波方向上的差异；右下方的蓝色曲线为敏感核曲线；黑色短棒为波场梯度法获得的方位各向异性结果（来自Cao et al., 2020）; 图（a）和图（b）绿色短棒为接收函数Pms震相获得的方位各向异性（来自Zheng et al., 2018）, 黄色阴影区标记了方位各向异性差异较大的区域

Figure  15.  Comparisons of anisotropy calculated using wave gradiometry and Pms phase of receiver function in the southeastern margin Tibetan Plateau (modified from Cao et al., 2020). Rose diagrams indicate the statistics of fast-orientation differences in two methods; blue curves denote the sensitivity kernel, black bars represent the results from wave gradiometry (from Cao et al., 2020), green bars are the results obtained by the Pms phase (from Zheng et al., 2018), and the yellow shadow marks the area with relatively large difference in fast propagation direction between the two methods