  We develop a geometrical interpretation of ternary probabilistic forecasts in
which forecasts and observations are regarded as points inside a triangle.
Within the triangle, we define a continuous colour palette in which hue and
colour saturation are defined with reference to the observed climatology. In
contrast to current methods, forecast maps created with this colour scheme
convey all of the information present in each ternary forecast. The geometrical
interpretation is then extended to verification under quadratic scoring rules
(of which the Brier Score and the Ranked Probability Score are well--known
examples). Each scoring rule defines an associated triangle in which the square
roots of the score, the reliability, the uncertainty and the resolution all
have natural interpretations as root--mean--square distances. This leads to our
proposal for a Ternary Reliability Diagram in which data relating to
verification and calibration can be summarised. We illustrate these ideas with
data relating to seasonal forecasting of precipitation in South America,
including an example of nonlinear forecast calibration. Codes implementing
these ideas have been produced using the statistical software package R and are
available from the authors.
