WEIGHTED MEDIANS

The simple median is too simple a concept. We need the weighted median concept to properly allow for the fact that a stacking hyperbola is large near its top and tapers off with distance and angle. A simple median is like a sum over the hyperbola without allowing for amplitude as a function of offset. We need to allow for amplitude with medians too. Weighted medians are explained in FGDP, but I give a short summary here. These two equations are equivalent:

$$\begin{eqnarray} 0 &=& \mbox{median}_i \left( {r_i\over\Delta r_i}+\alpha \right... ...\alpha} \sum_i \left\vert {r_i\over\Delta r_i}+\alpha \right\vert\end{eqnarray}$$ (14) (15)

The weighted median with weights $\Delta r_i$is defined instead by the equilibrium:

$$\begin{eqnarray} 0 &=& {\partial\over\partial\alpha} \sum_i \vert\Delta r_i \ve... ...l\over\partial\alpha} \sum_i \; \vert r_i+\alpha \Delta r_i \vert\end{eqnarray}$$ (16) (17)

The weighted median minimizes the so-called L₁ norm of the residual thus the weighted median seems a more standard optimization concept. The simple median, not the weighted median however, is more appropriate in the autoregression problem where noise in field data finds its way into the operator. (The data is in the convolution matrix (operator) that multiplies the filter vector.)