solution, i.e., the optimal covariance matrix ${\mathbf{R}}$, are functions of the steering vectors $\a(\r)$. In practice, however, the assumed steering vector model used to optimize ${\mathbf{R}}$ is inaccurate. Hence using *nominal* steering vectors $\ahat(\r)$ based on an ideal model, in lieu of the unknown *true* steering vectors $\a(\r)$ in , may result in undesired beampatterns with low power at the tumor region and damaging power deposition in the healthy tissue region. Such health considerations in medical applications motivate an approach that is robust with respect to the worst-case model uncertainties.

In order to formulate the robust design problem mathematically, we parameterize the steering vector uncertainties as follows. Let the true steering vector for the transducer array be $\mathbf{a(\r)} = \mathbf{\hat{a}(\r)} + \mathbf{\tilde{a}(\r)}$ where $\mathbf{\tilde{a}(\r)}$ is an unknown perturbation from the nominal steering vector. The deterministic perturbation at any generic point $\r$ belongs to uncertainty set $\mathcal{E}_{\r}$ that is bounded $$\begin{split}
\mathcal{E}_{\r} \triangleq \{ \mathbf{\tilde{a}}(\r) \hspace{.2cm} | \hspace{.2cm} \| \mathbf{\tilde{a}}(\r) \|_{\mathbf{W}}^2 \leq \epsilon_{\r} \},
\end{split}$$ where $\mathbf{W}$ is an $M \times M$ diagonal weight matrix with positive elements. The weight matrix $\mathbf{W}$ can be derived based on the type of uncertainty. Using $\mathbf{W}$, the set $\mathcal{E}_{\r}$ indicates an ellipsoidal region. The 