Bissection method

Without adding a new parameter, the bissection method may bring a better prior distribution than the basic interpolation method. The velocities are defined by a minimum value (first parameter, $V$ ). The other parameters are between 0 and $1$ . A total velocity variation is calculated from the second parameter $p_1$ , $\delta V=p_1*(V_{max}-V)$ . The velocity of the first layer and last layer are set to $V$ and $V+\delta V$ , respectively. The calculation of velocities starts from the layer at the middle of the stack, $V_i=V+p_i\delta V$ . The stack is then cut in two sub-stacks limited by velocities $V$ , $V_i$ and $V_i$ , $V+\delta V$ . The calculation is the same in each sub-stack until every layer has been affected a velocity. The resulting prior distribution is shown in figure B.5. Comparing it with figures B.2 and B.3, a more uniform distribution is achieved without adding a new parameter. Contrary to the preceding method, each basic random parameter is directly linked to the velocity at a fixed depth, which tends to simplify the parameter space. Profiles with a low velocity at depth are rarely generated. If the velocity of the last and first layer are set to $V$ , and $V-\delta V$ , respectively, a symmetric image is obtained.

Figure B.5: Prior information carried by parameterization: bissection. The black lines are the minimum and maximum admissible velocity profiles.

Figure B.6: Prior information carried by parameterization: diagonal. The black lines are the minimum and maximum admissible velocity profiles.