Convolution is a form of superposition that efficiently deals with input varying arbitrarily in time or space. It works whenever superposition is applicable, that is, for linear systems. Even though convolution is well-known since the 19th century, this valuable method is still missing in most textbooks on ground water hydrology. This limits widespread application in this field. Perhaps most papers are too complex mathematically as they tend to focus on the derivation of analytical expressions rather than solving practical problems. However, convolution is straightforward with standard mathematical software or even a spreadsheet, as is demonstrated in the paper. The necessary system responses are not limited to analytic solutions; they may also be obtained by running an already existing ground water model for a single stress period until equilibrium is reached. With these responses, high-resolution time series of head or discharge may then be computed by convolution for arbitrary points and arbitrarily varying input, without further use of the model. There are probably thousands of applications in the field of ground water hydrology that may benefit from convolution. Therefore, its inclusion in ground water textbooks and courses is strongly needed.

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A heat pump combined with Aquifer Thermal Energy Storage (ATES) is proven technology to economically and sustainably provide space heating and cooling. The two most important preconditions for the applicability of ATES are favorable climatic conditions and the availability of a suitable aquifer. This paper shows how these two preconditions can be combined to identify where in the world ATES potential is present, or will become present as a consequence of climate change. Countries and regions are identified where regulation and stimulation measures may increase application of ATES technologies and thus help reduce CO2-emissions. Two types of data determine ATES suitability, and their combination with a 3rd identifies potential hot-spots in the world: 1) geo-hydrological conditions, 2) current and projected climate classification and 3) urbanization. Our method combines the data into an ATES-suitability score as explained in this paper. On the one hand the results confirm the suitability for ATES where it is already applied and on the other they identify places where the technology is or will become suitable. About 15% of urban population lived in areas with high potential for ATES at the start of the 21st century, but this figure will decrease to about 5% during the 21st century as a consequence of expected climate change. Around 50% of urban population currently lives in areas of medium ATES suitability, a percentage that will remain constant. Demand for ATES is likely to exceed available subsurface space in a significant part of the urban areas.

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