## 3 Computing and the Singular Value Decomposition  
For computational purposes it is necessary to form a discrete approximation to the [PERSON]'s function and Equation (14). We begin by partitioning the circular region into approximately equal-area _cells_. This is performed with a regular partition of radius \(r\) into \(m\) equal sub-intervals, followed by regular partitions in angle \(\theta\) for each sub-interval of \(r\), starting with \(n\) sub-intervals in \(\theta\) for the central disc, then a partition to \(2n\) sub-intervals in \(\theta\) for the next ring out, and so on. All but the inner-most sub-interval in \(r\) contains \(2n(k-1)\) cells, where \(k>1\) is the sub-interval number. The area of cell \(k\) is  
\[a_{k}=\frac{2k-1}{2k-2}a_{1}, \tag{16}\]  
which tends towards \(a_{1}\), the area of a centre division. A similar polar grid was used in [21] to define a resistor network discretization of the disk, equivalent to a finite element method discretization, that was used to for solve [PERSON]'s equation and reproduce [PERSON]'s weight function shown in Figure 1.  
Flow \(f\) is discretized as piece-wise constant in each cell occupying region \(a\). Using subscript \(ij\) to denote the value in cell with centre at \((r_{i},\theta_{j})\), each term in (15) has the discrete form  
\[u(\mathbf{x})=\sum f_{ij}\int\limits_{a_{ij}}\frac{\partial g}{\partial x}d \mathbf{\xi}. \tag{17}\]The map from flow to potential is therefore written as a matrix equation  
\[\mathbf{u}=\mathbf{A}\mathbf{f},\text{ where} \tag{18}\] \[\mathbf{A}=\left\{\int\limits_{d_{ij}}\frac{\partial g}{\partial x} d\xi\right\}\quad\text{and}\quad\mathbf{f}=\{f_{ij}\}.\]  
The elements of \(\mathbf{A}\) can be resolved individually by Gaussian quadrature, with  
\[\frac{\partial g}{\partial x}=\frac{1}{2\pi}\bigg{[}\frac{r\cos(\theta)-R\cos( \phi)}{R^{2}+r^{2}-2 rR\cos(\theta-\phi)}+\frac{r_{r}\cos(\theta)-R\cos(\phi)}{ R^{2}+r_{r}^{2}-2 Rr_{r}\cos(\theta-\phi)}\bigg{]}. \tag{19}\]  
It is worth noting that there is a point of difference between this work and much of existing literature in the field. Many EM flowmeter papers examine a specific shape of the magnetic field and focus on what that implies for measurements of the fluid's velocity. This approach fits with Bevir's weight function. We, however, are combining the velocity and magnetic field into a single entity. In practice, the magnetic field will be reasonably well known and the velocity is, thus, inferable. The singular value decomposition (SVD) provides an orthonormal basis for \(\mathbf{u}\) and \(\mathbf{f}\) that diagonalizes the map \(A\). Hence the SVD gives us a coordinate system for the measurement process such that individual components in flow are mapped to individual components in potential, with a scalar multiplier that is the singular value. Because these basis vectors are normalized, the singular values are exactly the sensitivity of measurements to that specific vector. Mathematically, for the \(i\)th singular vectors  
\[\mathbf{u}_{i}=\sigma_{i}\mathbf{f}_{i}, \tag{20}\]  
where \(\sigma_{i}\) is the corresponding singular value. By approximating the continuous system by a large number of measurement points and a fine discretization of functions, the number of singular vales above the signal to noise ratio (SNR) is an estimate of the number of pieces of information that can be extracted from the system through this measurement process. Figure 2 shows the singular values of this zero-mean measurement flowmeter setup. There are many pieces of information to be gained from this system, with over 100 singular values above \(10^{-1}\) of the maximum value. The first two singular vectors in both flow- and measurement-space are shown in Figures 3 and 4.