## 1 INTRODUCTION

Numerical simulations of glacial isostatic adjustment (GIA) have an important historical role in global geodynamics since they are a key to constrain the rheological profile of the mantle (Cathles 1975) and help in the interpretation of present-day sea level variations and geodetic observations (King *et al.* 2010). Although various benchmark studies of mantle convection have been successfully completed since the late 1980s (Busse *et al.* 1994; Blankenbach *et al.* 1989; Muhlhaus & Regenauer-Lieb 2005; Zhong *et al.* 2008), to date no extensive GIA benchmark has been published on an international journal of Geophysics, in spite of two remarkable initiatives in the last two decades. In the mid-1990s, a benchmark project for GIA codes was launched by Georg Kaufmann and Paul Johnston. Some results were collated until 1997, when the flow of contribution from the participants ceased (the initial proposal and some results are still available from the web page http://rses.anu.edu.au/geodynamics/GIA_benchmark/). Since the importance of a GIA benchmark was further stated at the eighth European Workshop on Numerical Modeling of Mantle Convection and Lithospheric Dynamics, held in Hruba Skala (Czech Republic) in 2003, another call for an international collaboration was opened. This new initiative, leaded by Jan M. Hagedoorn, was not limited to GIA only but also included the study of the sea level equation (SLE, Farrell & Clark 1976). While some GIA results are still available from the dedicated web page http://geo.mff.cuni.cz/gia-benchmark/, as far as we know the SLE benchmark has never been initiated. Our aim here is to present some results from a new benchmark initiative at a stage in which a significant number of submission has been reached, addressing all of the main aspects of GIA modelling. To strengthen our collaboration (and to increase the probability of success) these activities have been placed in the framework of the European COST Action ES0701 ‘Improved constraints on models of Glacial Isostatic Adjustment' (see http://www.cost-es0701.gcparks.com/). Our aims are (i) testing the codes currently in use by the various teams, (ii) establish a sufficiently large set of agreed results, (iii) correct possible systematic errors embedded in the various physical formulations or computer implementations and (iv) facilitate the dissemination of numerical tools for surface loading studies to the community and to early career scientists. Though the benchmark activities described here have been initially limited to members of the Action, they will be open to the whole GIA community through the COST Action ES0701 web pages (see http://www.cost-es0701.geoenvi.org/activities/publications). The collaboration will continue with an SLE benchmark whose details are now under discussion.

There are several motivations for a benchmark study of GIA codes. First, a number of methods and computer packages are now in use from different groups, which include an increasingly sophisticated description of the physics of GIA. Benchmarking the codes, in this context, is useful to strengthen confidence in the results and to validate the methods. A second motivation is the progressively increasing role played by GIA in the framework of global climate change. Fundamental issues such as future projections of vertical crustal movements and sea level variations on a regional and global scale critically rely upon correct modelling of GIA. Predictions of the geophysical quantities involved in this process often depend on several model assumptions and simplifications, whose impact may be crucial for future projections, and that must be verified within a benchmark programme including a significant number of investigators. This would help to identify the most critical issues from a numerical standpoint, and, possibly to determine upper and lower bounds to the errors intrinsically associated with numerical modelling. Third, improvements in modelling techniques are needed to place tighter constraints on ongoing GIA in regions of current ice mass fluctuation. In particular, a benchmark study may be useful for the interpretation of future geodetic measurements in deglaciated areas and for ongoing satellite missions focused on the study of GIA gravity signatures such as GRACE (Paulson *et al.* 2007; Tamisiea *et al.* 2007; Barletta & Bordoni 2009; Riva *et al.* 2009) and GOCE (Schotman *et al.* 2009; Vermeersen & Schotman 2009). Last, since no GIA benchmark of this extent has ever been accomplished to date (see discussion earlier), it is our opinion that the community could take advantage of the presentation of a number of agreed results obtained from independent techniques which are the basis for future model development. Since some of the scientists working on this benchmark agree to release their numerical codes (and some are available already, see Table 1), we expect that scientists approaching the topic of GIA for the first time could benefit from this project.

Acronym | Code | Author or user | Short description of method and notes |
---|---|---|---|

Bl | ABAQUS | B. Lund | Finite elements (FE) (Abaqus 2007); Flat-Earth approximation |

Gs | TABOO | G. Spada | Viscolastic Normal Modes (VNM). Fortran code and User Guide available from Gs or from the Samizdat Press (http://samizdat.mines.edu/taboo/) |

Gs | PMTF | G. Spada | A Fortran program that computes the polar motion transfer function using the VNM method (available from Gs) |

Gsa | ALMA | G. Spada | Post-Widder method (Spada & Boschi 2006; Spada 2008). Fortran code available from http://www.fis.uniurb.it/spada/ALMA_minipage.html |

Pg^{★} | ABAQUS | P. Gasperini | Finite elements (FE) (Abaqus 2007) |

Rr | FastLove-HiDeg | R. E. M. Riva | VNM (Riva & Vermeersen 2002) |

Vb | MHPLove | V. R. Barletta | A Mathematica™Mathematica 4.1 2001 program for computing high precision Love numbers |

Vk | VILMA | V. Klemann | Spectral-finite elements (SFE) Martinec (2000) |

Zm | VEENT | Z. Martinec | VNM (Martinec & Wolf 1998), Fortran code available from Zm |

Test | Bl | Gs | Gsa | Pg^{★} | Rr | Vb | Vk | Zm | Meaning of test |
---|---|---|---|---|---|---|---|---|---|

T1/1 | x | x | x | x | Isostatic relaxation times | ||||

T2/1 | x | x | x | x | Loading Love numbers | ||||

T3/1 | x | x | x | x | Tidal Love numbers | ||||

T4/1 | x | x | Polar motion transfer function | ||||||

T5/1 | x | x | x | Time-domain Love numbers | |||||

T6/1 | x | x | x | x | Love numbers of multistratified model | ||||

T7/1 | x | x | x | x | x | Degree-1 Love numbers | |||

T1/2 | x | x | x | x | x | Geodetic quantities | |||

T2/2 | x | x | x | x | Rates of geodetic quantities | ||||

T3/2 | x | x | Polar motion and LOD |

Owing to space limitations, a complete review of the GIA theory is not possible here. In the body of the manuscript a basic (but certainly not exhaustive) outline is given to facilitate the reader. A complete summary of state-of-the-art GIA theory is presented in the recent report of Whitehouse (2009). The viscoelastic normal mode (VNM) method for a spherical Earth, introduced in the seminal work of Peltier (1974) and later refined by Wu & Peltier (1982), Sabadini *et al.* (1982) and Peltier (1985), is at the basis of several numerical contributions presented in this manuscript. An ancillary presentation of mathematical details for the VNM is given by the booklet of Spada (2003), while for a broad geophysical view of the topic the reader is referred to the treatise of Sabadini & Vermeersen (2004). Possible caveats of the VNM approach, particularly regarding the implementation of compressibility and multilayered models in GIA investigations, have been discussed by James (1991) and Han & Wahr (1995), and later reconciled by Vermeersen *et al.* (1996a) and Vermeersen & Sabadini (1997). In this study, some GIA results obtained by the VNM method are compared to finite elements (FEs) or spectral-finite element (SFE) computations. The applications of these techniques to GIA are briefly summarized in Sections 3.3 and 3.6, respectively.

An important aspect of GIA concerns the rotational variations of the Earth in response to the melting of the continental ice sheets, which is in fact one of the topics of this benchmark. The problem has been stated by Nakiboglu & Lambeck (1980) and analysed in depth by Sabadini & Peltier (1981), who set the theoretical framework which is used in our polar motion benchmark. Then, it was further developed by Yuen *et al.* (1982), Wu & Peltier (1984) and Yuen & Sabadini (1985). Since the observed secular drift of the rotation axis is currently small (somewhat less than 1 degree Myr^{−1}, see, e.g. Lambeck 1980; Argus & Gross 2004) linearized Euler equations (Ricard *et al.* 1993) can be employed on the GIA timescales, as done here (for a review of the True Polar Wander problem, which entails the fully non-linear Liouville equations, the reader is referred to Sabadini & Vermeersen 2004, and references therein). The study of polar motion excited by deglaciation has continued through the 1990s (Spada *et al.* 1992; Peltier & Jiang 1996; Vermeersen & Sabadini 1996; Vermeersen *et al.* 1997), accompanied by a number of contributions addressing the more general problem of rotational feedbacks, in which sea level fluctuations are driven by the changing position of the Earth's rotation axis responding to unloading (Han & Wahr 1989; Sabadini *et al.* 1990; Milne & Mitrovica 1996; Sabadini & Vermeersen 1997; Milne & Mitrovica 1998; Peltier 2001; Mitrovica *et al.* 2005). A further aspect studied is the harmonic degree one displacement, which describes the geocentre motion. Here, GIA contributes a significant secular trend (Greff-Lefftz 2000; Klemann & Martinec 2009).

The paper is organized as follows: in Section 2 the two Test Classes considered in this study are defined and described and their background theory is presented; they pertain to the spectral (2.1) and to the spatial domain (2.2), respectively; numerical methods employed by the contributors are presented in Section 3; results (Section 4) are presented separately for the spectral (4.1) and the time domain analyses (4.2) and discussed in Section 5.