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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:
| |
(14) |

| (15) |

The weighted median
with weights is defined instead by the
equilibrium:
| |
(16) |

| (17) |

The weighted median minimizes the so-called *L*_{1} 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.)

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** Up:** Claerbout: Medians in regression
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Stanford Exploration Project

11/12/1997