The theory behind TABOO - a posT glAcial reBOund calculatOr

This is an attempt to collect in a single document the basic traits of the the- ory describing the deformations of the Earth under peculiar surface loads: the ice sheets. The book has a mainly pedagogical purpose. It is written in a simple way, and an e®ort is made to avoid the sentence it can be shown that. Almost all of the propositions given here are demonstrated step{by{step, even when they may appear obvious a priori. This is mainly done to facili- tate the beginners in the 'art' of the postglacial rebound, but I also hope that this transparent style of writing could be useful for more experienced inves- tigators. The book is written according to an austere minimalism: we only give the statements which are strictly needed to understand the basic con- cepts. For this reason, the chapter devoted to the mathematical background is largely biased towards the main tools, such has di®erential operators and spherical harmonics. The theory illustrated here is implemented in the source code taboo.f90 which is freely distributed by the Samizdat Press along with this document and the accompanying user guide. At the core of TABOO there is the assump- tion that the Earth is spherically layered. In the common language this means that the problems which can be solved by TABOO are 1D problems. Nowadays several research groups have developed more advanced codes, which account for the 2D or even for the 3D structure of the lithosphere and the mantle. However, these codes are not publically available to date, mainly for two rea- sons. First, their are not totally developed, and some work is still to be done. Second, di®erently from TABOO they are often based on numerical techniques developed with the aid of software packages that are not publically available. In a sense, TABOO has the aim of closing the chapter of the 1D problems giving the chance of obtaining a portable source code and a full account of the theory behind. It is hoped that this will encourage the developers of 2D and 3D models to to the same with their procedures in the future. The reader should be warned that TABOO is not a sealevel equation [4] solver! The sealevel equation will be the subject of a separate review coming in the next months along with a freely available code (SELEN). The theory behind TABOO has only the purpose of collecting formulas and results in an ordered structure. By no means the results presented here are the product of my own research work. Rather, they constitute a theoretical framework which has been constructed by a number of Authors in the course of the last decades. It is not possible to mention all of the contributors to this enormous (but sparse) work, and for this reason I must apologize for the very poor bibliography that I have written at the end of this document. The full set of original papers where the basic ideas have been ¯rst developed canii be reconstructed on the basis of bibliographies of the manuscripts and books quoted here. While I have done my best to present a complete account of the theory behind TABOO (and consequently a complete source code), some work is still to be done. In particular, the present version of TABOO does not explicitely compute relevant physical quantities such as the gravity anomalies, the stress ¯eld in the lithosphere and the rotational variations of the Earth. It is my intention to include these topics in the next versions of the code (which will also include ¯gures). A ¯nal note concerning notation. I do not like to write vectors by bold face letters, so that I use arrows throughout. I have been very pedantic in the demonstration of the various propositions given in this document, certainly too much for an experienced reader. This is admittedly boring, but I hope it helps the novices, who are indeed the main target of this booklet. Since the source code TABOO is totally accessible, I have not described in detail how and where the single propositions are numerically implemented. I have been involved in the research on these topics for ¯fteen years, ¯rst as a student, and later as a teacher. Both need a place where a given formula can be easily found and demonstrated. After all, this is the main purpose of TABOO. The future releases of this document (if any) will bene¯t from the feedback of the readers of this ¯rst edition. Please feel free to write to spada@fis.uniurb.it for questions, comments, and suggestions. Urbino, October 10, 2003