Atmospheric Satellite Observations

(6.47)

The boundary terms disappear if the values of the adjoint variables are set to zero at the boundaries.

Once $L$ is expressed as an explicit function of the model variables ($u,v,\Phi$), that is, the last equation in (6.47), derivatives of L with respect to $u,v$, and $\Phi$ can be easily obtained:

$\{\begin{array}{l}\frac{\partial L}{\partial u}={\int }_{\mathrm{\Omega }}\left(-\frac{\partial \hat{u}}{\partial t}+\hat{v}\beta y-g{h}_e\frac{\partial \hat{\Phi }}{\partial x}\right)d\mathrm{\Omega }\hfill \\ \frac{\partial L}{\partial v}={\int }_{\mathrm{\Omega }}(-\frac{\partial \hat{v}}{\partial t}-\hat{u}\beta y-g{h}_{}\hfill \end{array}$