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General formulas for traveltime derivative tensors

Using equations 9 and 10 and applying the chain rule, we can differentiate the one-way traveltime with respect to half offset 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

and

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

and

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 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

, $\hat{m}$ ,

, $\hat{n}$ 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 (

=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

Interval parameter estimation

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

2017-04-14