Context : The \Delta -variance analysis , introduced as a wavelet-based measure for the statistical scaling of structures in astronomical maps , has proven to be an efficient and accurate method of characterising the power spectrum of interstellar turbulence .
It has been applied to observed molecular cloud maps and corresponding simulated maps generated from turbulent cloud models .
The implementation presently in use , however , has several shortcomings .
It does not take into account the different degree of uncertainty of map values for different points in the map , its computation by convolution in spatial coordinates is very time-consuming , and the selection of the wavelet is somewhat arbitrary and does not provide an exact value for the scales traced .

Aims : We propose and test an improved \Delta -variance algorithm for two-dimensional data sets , which is applicable to maps with variable error bars and which can be quickly computed in Fourier space .
We calibrate the spatial resolution of the \Delta -variance spectra .

Methods : The new \Delta -variance algorithm is based on an appropriate filtering of the data in Fourier space .
It uses a supplementary significance function by which each data point is weighted .
This allows us to distinguish the influence of variable noise from the actual small-scale structure in the maps and it helps for dealing with the boundary problem in non-periodic and/or irregularly bounded maps .
Applying the method to artificial maps with variable noise shows that we can extend the dynamic range for a reliable determination of the spectral index considerably .
We try several wavelets and test their spatial sensitivity using artificial maps with well known structure sizes .
Performing the convolution in Fourier space provides a major speed-up of the analysis .

Results : It turns out that different wavelets show different strengths with respect to detecting characteristic structures and spectral indices , i.e .
different aspects of map structures .
As a reasonable universal compromise for the optimum \Delta -variance filter , we propose the Mexican-hat filter with a ratio between the diameters of the core and the annulus of 1.5 .
When the main focus lies on measuring the spectral index , the French-hat filter with a diameter ratio of about 2.3 is also suitable .
In paper II we exploit the strength of the new method by applying it to different astronomical data .

Conclusions :