next up previous [pdf]

Next: Interval parameter estimation Up: Sripanich & Fomel: Interval Previous: Traveltime expansion

General formulas for traveltime derivative tensors

Using equations 9 and 10 and applying the chain rule, we can differentiate the one-way traveltime $ t$ with respect to half offset $ h_i$ to derive the following equations:

$\displaystyle t_i$ $\displaystyle =$ $\displaystyle g_{\hat{i}i}t_{,\hat{i}} = p_i~,$ (11)
$\displaystyle t_{ij}$ $\displaystyle =$ $\displaystyle g_{\hat{j}j} p_{i,\hat{j}} = g_{\hat{j}j} \delta_{i\hat{j}} = g_{ij}~,$ (12)
$\displaystyle t_{ijk}$ $\displaystyle =$ $\displaystyle g_{\hat{k}k}t_{ij,\hat{k}} = g_{\hat{k}k} g_{ij,\hat{k}} ~,$ (13)
$\displaystyle t_{ijkl}$ $\displaystyle =$ $\displaystyle g_{\hat{l}l}t_{ijk,\hat{l}} = g_{\hat{l}l}g_{\hat{k}k,\hat{l}}g_{ij,\hat{k}} + g_{\hat{l}l}g_{\hat{k}k} g_{ij,\hat{k}\hat{l}}~,$ (14)

where the derivatives with respect to $ p_1$ and $ p_2$ are represented by comma (e.g, $ \frac{\partial t}{\partial p_i}$ corresponds to $ t_{,i}$ ), $ \delta_{ij}$ denotes the Kronecker delta, $ g_{ij}$ denotes $ \frac{\partial p_i}{\partial h_j}$ , and $ \hat{i}$ , $ \hat{j}$ , $ \hat{k}$ , $ \hat{l}$ represent dummy indices. Equations 11-14 can be used to compute $ t_{ij}$ and $ t_{ijkl}$ terms needed by equations 5 and 6 using explicit relationships for $ h(p_1,p_2)$ and $ t(p_1,p_2)$ .

According to the chain rule and the symmetry of the time derivative tensors, the second-derivative tensor $ g_{ij}$ and its derivatives in equations 13 and 14 can be related to the derivatives of half offset $ h$ as follows:

$\displaystyle g_{ij,\hat{k}}$ $\displaystyle =$ $\displaystyle - g_{im}h_{m \hat{m},\hat{k}}g_{\hat{m}j}~,$ (15)
$\displaystyle g_{ij,\hat{k}\hat{l}}$ $\displaystyle =$ $\displaystyle 2g_{im}h_{m \hat{m},\hat{k}}g_{\hat{m} n} h_{n \hat{n},\hat{l}} g_{\hat{n} j} - g_{i m}h_{m \hat{m},\hat{k}\hat{l}}g_{\hat{m}j}~,$ (16)

where $ m$ , $ \hat{m}$ , $ n$ , $ \hat{n}$ are dummy indices. The matrix $ h_{ji}=\frac{\partial h_j}{\partial p_i}$ is the inverse of the matrix $ g_{ij}$ (Grechka and Tsvankin, 1998). Substituting equations 15 and 16 into equations 13 and 14, we subsequently arrive at expressions
$\displaystyle t_i$ $\displaystyle =$ $\displaystyle p_i~,$ (17)
$\displaystyle t_{ij}$ $\displaystyle =$ $\displaystyle g_{ij}~,$ (18)
$\displaystyle t_{ijk}$ $\displaystyle =$ $\displaystyle -g_{\hat{k}k}g_{im}h_{m \hat{m},\hat{k}}g_{\hat{m}j} ~,$ (19)
$\displaystyle t_{ijkl}$ $\displaystyle =$ $\displaystyle 3 g_{\hat{l}l}(g_{\hat{k} m}h_{m \hat{m},\hat{l}}g_{\hat{m}k}) (g...
...n}j})- g_{\hat{l}l}g_{\hat{k}k}g_{im}h_{m \hat{m},\hat{k}\hat{l}}g_{\hat{m}j}~,$ (20)

which only involve derivatives of explicitly defined functions. Subsequently, we have at zero offset ($ p_i$ =0):
$\displaystyle t_i\vert _{h=0}$ $\displaystyle =$ $\displaystyle 0 ~,$ (21)
$\displaystyle t_{ij}\vert _{h=0}$ $\displaystyle =$ $\displaystyle g_{ij}~,$ (22)
$\displaystyle t_{ijk}\vert _{h=0}$ $\displaystyle =$ $\displaystyle 0 ~,$ (23)
$\displaystyle t_{ijkl}\vert _{h=0}$ $\displaystyle =$ $\displaystyle - g_{\hat{l}l}g_{\hat{k}k}g_{im}h_{m \hat{m},\hat{k}\hat{l}}g_{\hat{m}j}~.$ (24)



Subsections
next up previous [pdf]

Next: Interval parameter estimation Up: Sripanich & Fomel: Interval Previous: Traveltime expansion

2017-04-14