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There are many ways to integrate different tomography types for velocity inversion, such as sequential inversion, joint inversion, or recursive inversion. Compared to other integration schemes, joint inversion is a more complex process but also expected to be more efficient. The complexity of joint inversion comes from the difference between the data sets used for inversion, the difficulty of balancing the different tomography operations, and the challenge of software design. The advantage of joint inversion is that different data sets are used simultaneously to invert the velocity. Therefore, the inversion result will be more accurate with lower uncertainty.
In this paper, I realize a joint inversion scheme with data fitting goal (3) and regularization goal (4):

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(3) |

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(4) |

Here, and are the reflection and VSP tomography operators, respectively. is the traveltime change for surface reflection and is the traveltime change for VSP tomography. Usually, the traveltime is acquired in different ways in reflection tomography and VSP tomography. In reflection tomography, the reflection points are picked from the migration result. Semblance scans are used to obtain the residual moveout in the angle domain common image gathers. The residual moveout is then converted to the corresponding traveltime change. For VSP tomography, traveltime is usually picked directly from the VSP data manually or automatically.
In order to balance the surface reflection tomography and VSP tomography in the integrated inversion, I introduce an integration operator **N** into the data fitting goal (3). **N** is efficiently approximated by the inverse noise covariance Tarantola (1987) which is difficult to get in practice. A correct design of operator **N** is critical to balance the two tomography operators in the final integration result. In this paper, I simply use an identity operator as the integration operator.

The regularization operator can be best approximated by the inverse model covariance that lacks *a priori* information in reality. I simply use the Laplacian operator as a regularizer in this paper.

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Stanford Exploration Project

7/8/2003