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A new straightforward method was proposed to measure the local roughness exponent of a scanning tunnelling microscopy (STM) image. Its reliability was tested against some ideal self-affine surfaces. Taking the mean value of a number of local roughness exponents for the STM images, which were randomly recorded on a fracture surface, as the roughness exponent of the fracture surface, we quantitatively assessed the experimental error of the measured roughness exponent by statistical analysis. The experimental results for two different fracture surfaces showed that the difference between the two roughness exponents was not produced by their experimental errors to a 90% probability.