# Enhanced Upward Translations for Systems with Clusters
## 3 Enhanced Translations
Hence the far field (2) may be evaluated as the sum of far fields of clusters  
\[\mathbf{f}(\mathbf{x})=\sum_{p=1}^{N_{cl}}\mathbf{f}_{p}(\mathbf{x})\]  
where \(\mathbf{f}_{p}(\mathbf{x})\) is the far field of the cluster \(p\). It is  
\[\mathbf{f}_{p}(\mathbf{x})=\sum_{j=1}^{M_{clp}}\mathbf{K}\Big{(}\mathbf{x},\mathbf{y}_{j}\Big{)} \mathbf{q}_{j} \tag{5}\]  
Consider a particular cluster \(p\). Its representative source has the intensity \(\mathbf{q}_{clp}\), location \(\mathbf{y}_{clp}\), and normal \(\mathbf{n}_{clp}\), defined by (1). As emphasized in the previous section, with growing distance \(|\mathbf{x}-\mathbf{y}|\) from a cluster to a field point \(\mathbf{x}\), the field \(\mathbf{f}_{p}(\mathbf{x})\) tends to the field of its representative source. Thus the difference between the fields of the cluster and its representative source goes to zero. Clearly, it decreases faster than the field of the cluster. Of essence is also that the far field of the representative source is exactly evaluated as the product \(\mathbf{K}(\mathbf{x},\mathbf{y}_{clp})\mathbf{q}_{clp}\). This suggests using the \"subtract-and-add\" approach by subtracting from and adding to the right-hand side of (5) the known field \(\mathbf{K}(\mathbf{x},\mathbf{y}_{clp})\mathbf{q}_{clp}\) of the representative source. Then (5) is written as  
\[\mathbf{f}_{p}(\mathbf{x})=\mathbf{f}_{mp}(\mathbf{x})+\mathbf{K}\Big{(}\mathbf{x},\mathbf{y}_{clp}\Big{)} \mathbf{q}_{clp}\]  
where the field \(\mathbf{f}_{mp}(\mathbf{x})\) is the difference of the fields:  
\[\mathbf{f}_{mp}(\mathbf{x})=\mathbf{f}_{p}(\mathbf{x})-\mathbf{K}\Big{(}\mathbf{x},\mathbf{y}_{clp}\Big{)} \mathbf{q}_{clp} \tag{6}\]  
The difference will be called the residual (_modified_) field. As mentioned, the modified field decreases faster than the field of the cluster itself. Consequently, it is reasonable to apply the approximation of (3) type to the residual field, rather than to the field of the cluster. Thus, we approximate the modified field similar to (3)  
\[\mathbf{f}_{mp}(\mathbf{x})\approx\sum_{s=1}^{N_{e}}\mathbf{K}_{mp}(\mathbf{x},\mathbf{y}_{cs})\bm {q}_{msp} \tag{7}\]  
where \(\mathbf{q}_{msp}\) are modified equivalent intensities of sources located on the ES; \(\mathbf{K}_{mp}(\mathbf{x},\mathbf{y})\) is the _modified kernel_. It is defined as the difference  
\[\mathbf{K}_{mp}(\mathbf{x},\mathbf{y})=\mathbf{K}(\mathbf{x},\mathbf{y})-\mathbf{K}\Big{(}\mathbf{x},\mathbf{y}_{ clp}\Big{)} \tag{8}\]  
Definition (8) guarantees that with growing distance \(|\mathbf{x}-\mathbf{y}|\), the right-hand side of (7) goes to the exact modified field \(\mathbf{f}_{pm}(\mathbf{x})\). Similar to \(\mathbf{q}_{cs}\) in (3), the _modified_ equivalent intensities \(\mathbf{q}_{msp}\) of cluster \(p\) are found by equating the right-hand side of (7) to the exact modified field (6) at \(N_{e}\) check points \(\mathbf{x}_{i}\) at the ChS  
\[\sum_{s=1}^{N_{e}}\mathbf{K}_{mp}(\mathbf{x}_{i},\mathbf{y}_{cs})\mathbf{q}_{msp}=\sum_{j=1}^{ M_{clp}}\mathbf{K}\Big{(}\mathbf{x}_{i},\mathbf{y}_{j}\Big{)}\mathbf{q}_{j}-\mathbf{K}\Big{(}\mathbf{x}_{i},\mathbf{y}_{clp}\Big{)}\mathbf{q}_{clp},\,\,\,i=1,\ldots,N_{e} \tag{9}\]  
The ES and ChS are taken the same as those used in (3) and (4) for conventional translations. The system (9) is solved by using its inversed matrix. As for the conventional matrix in (4), the latter is calculated in advance and stays the same in all iterations. The found modified equivalent intensities \(\mathbf{q}_{msp}\) are used in Equation (7), which approximates the modified far field \(\mathbf{f}_{mp}(\mathbf{x})\) defined by (6). Then the sum \(\mathbf{f}_{p}(\mathbf{x})=\mathbf{f}_{mp}(\mathbf{x})+\mathbf{K}(\mathbf{x},\mathbf{y}_{clp})\mathbf{q}_{clp}\) gives the far field of the cluster. In this way, far fields \(\mathbf{f}_{p}(\mathbf{x})\) of all the clusters \((p=1,\ldots,N_{cl})\) become known. Finally, their sum \(\mathbf{f}(\mathbf{x})\approx\sum_{p=1}^{N_{cl}}\mathbf{f}_{p}(\mathbf{x})\) gives the needed approximation of the far field (2) of all leaf sources.