Estimating the potential predictability of Australian surface maximum and minimum temperature

Abstract

A study is made of the potential predictability of seasonal means in Australian surface maximum and minimum temperature using monthly data from December 1950 to November 2000. Because the usual assumption of stationarity cannot be applied to the observations at all stations and for all seasons, a modification to an existing methodology is proposed. Here, we show that, to a first order, monthly mean variances within a season can be modeled by a linear relationship, and inter-monthly correlations can be assumed to be stationary. The intraseasonal component of variability can then be estimated using monthly data. Removing the intraseasonal variance from the total interannual variance allows an estimate of the potential predictability to be made. Surface maximum and minimum temperature has high potential predictability over most of northern Australia in the four main seasons. However, there is high potential predictability only in some of the four seasons for the centre and south of Australia. Surface minimum temperature is generally more potentially predictable than surface maximum temperature. The spatial and temporal patterns of potential predictability are generally consistent with published patterns of hindcast skill from a statistical forecast scheme. A comparison between the intraseasonal variance of Australian surface maximum and minimum temperature estimated using the stationary variance assumption and the linear assumptions showed qualitatively and quantitatively similar patterns of distribution.

Appendix

Applying the constraints Eqs. 15, 16, 19–21, Eqs. 9–11) become

$$ (\sigma_{2} - \beta )^{2} - 2\sigma_{2} \left( {\sigma_{2} - \beta } \right)\varphi_{12} + \sigma_{2}^{2} \approx a,$$

(25)

$$ \sigma_{2}^{2} (2\varphi_{12} - 1) \approx b,$$

(26)

$$ (\sigma_{2} + \beta )^{2} - 2\sigma_{2} \left( {\sigma_{2} + \beta } \right)\varphi_{12} + \sigma_{2}^{2} \approx c. $$

(27)

A cubic equation for φ ₁₂ can be derived from Eqs. 25–27 and takes the form

$$ A\varphi_{12}^{3} + B\varphi_{12}^{2} + C\varphi_{12} + D = 0, $$

(28)

where

$$ A = a + 2b + c, $$

(29)

$$ B = - \frac{5}{2}A - b - \frac{{\left( {c - a} \right)^{2} }}{4b}, $$

(30)

$$ C = 2A + 2b + \frac{{\left( {c - a} \right)^{2} }}{4b}, $$

(31)

$$ D = - \frac{1}{2}A - b - \frac{{\left( {c - a} \right)^{2} }}{16b}. $$

(32)

The real root of Eq. 28 has the standard solution (e.g. NBS 1964)

$$ \varphi_{12} = - \frac{{B^{*} }}{3} + \sqrt[3]{{E + \sqrt {E^{2} + F^{3} } }} + \sqrt[3]{{E - \sqrt {E^{2} + F^{3} } }}, $$

(33)

where

$$ E = \frac{{\left( {9B^{*} C^{*} - 27D^{*} - 2B^{{*^{3} }} } \right)}}{54}, $$

(34)

$$ F = \frac{{\left( {3C^{*} - B^{{*^{2} }} } \right)}}{9}, $$

(35)

$$ B^{*} = \frac{B}{A}, $$

(36)

$$ C^{*} = \frac{C}{A}, $$

(37)

$$ D^{*} = \frac{D}{A}. $$

(38)

As in Zheng et al. (2000), φ ₁₂ is further constrained to lie in [0, 0.1] to reduce the estimation error. Given an estimate for φ ₁₂, estimates for σ ₂ and β can then be found by back substitution of φ ₁₂ into Eqs. 25 and 27.