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Keywords:

  • Global mean temperature;
  • surface temperature field;
  • spherical harmonics

Abstract

This paper considers the mean squared error (MSE) incurred in estimating an idealized earth's global average temperature with a finite number of point gauges located in a specified or stochastic way over the globe. For a class of model earths with rotationally invariant statistics, the MSE formula can be cast into the form of a summation over spherical harmonic indices. The summand factors into a part which depends on the design of the gauge network and a second part which is the degree variance spectrum of the surface temperature field. After presenting this formalism, we provide an example spectrum for the surface temperature field derived from a simple two parameter stochastic climate model defined on the sphere. An example calculation is given for the case of N gauges randomly arranged on the sphere. In addition, the sampling error is computed for some simple regular arrays of gauges as illustrative examples.