# On Some Properties of the Glacial Isostatic Adjustment Fingerprints

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## Abstract

**:**

## 1. Introduction

## 2. Theory

## 3. Methods

## 4. Some Properties of the GIA Fingerprints

#### 4.1. Relative Sea-Level Change

#### 4.2. Vertical Displacement

#### 4.3. Geoid Height and Absolute Sea-Level Change

#### 4.4. Surface Load

## 5. Observing the Global GIA Fingerprint by Vertical GPS Rates

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Sketches of the reference state for time $t={t}_{0}$ (

**a**) and of the general configuration for $t\ge {t}_{0}$ (

**b**) showing the three Earth’s portions that are interacting in the SLE: the solid Earth, the oceans and the ice sheets. With r we denote the radius of a given point relative to the center of mass of the whole Earth system. Changes in sea level relative to the solid Earth are observed by the red stick meter. The sea surface is equipotential in (

**a**) but also in (

**b**), after that the ice sheets have shrunk and the mass of the oceans has consequently varied to compensate exactly the ice mass loss. The vertical arrows in (

**b**) indicate that the sea surface and the solid Earth have moved relative to the origin of the reference frame. The spring-dashpot system is an exemplification of Earth’s rheology.

**Figure 2.**The top part of the triangle shows the elements that are perpetually interacting in the SLE (the solid Earth, the ice sheets and the oceans) through surface loading and mutual gravitational attraction. The bottom part qualitatively shows how Earth rotational effects are coming into play. The figure is inspired to that originally published by Clark et al. [2].

**Figure 3.**GIA fingerprint for $\dot{\mathcal{S}}$, the present-day rate of relative sea-level change, obtained by implementing model ICE-5G (VM2) in SELEN${}^{4}$. To better visualize the regional variations, the palette is limited to the range of $\pm 1$ mm year${}^{-1}$. The largest rates, marked by white dots, are associated with the isostatic disequilibrium still caused by the disintegration of the Laurentide ice sheet complex, with $\dot{\mathcal{S}}\sim -17.9$ and $\dot{\mathcal{S}}\sim +5.3$ mm year${}^{-1}$, respectively. The most significant regional variability, measured as the density of local maxima and minima of $\dot{\mathcal{S}}$ in this map, is found to within ∼1500 km from the continental margins.

**Figure 4.**GIA fingerprint for the current rate of crustal uplift $\dot{\mathcal{U}}$, according to our implementation of GIA model ICE-5G (VM2). The rates with largest absolute values, marked by white dots, are associated with the melting of the Laurentide ice sheet in north America and Canada, and are found in the same locations of Figure 3, with values of $\dot{\mathcal{U}}\sim +19.2$ and $\dot{\mathcal{U}}\sim -5.7$ mm year${}^{-1}$, respectively. The regional variability of the $\dot{\mathcal{U}}$ fingerprint appears to be comparable to that of $\dot{\mathcal{S}}$ in Figure 3 but the rotational lobes are much more developed.

**Figure 5.**GIA fingerprint for $\dot{\mathcal{G}}$ i.e., the current rate of geoid height variation, according to our GIA simulation based upon model ICE-5G (VM2). The white dots show where the largest rates are predicted, with values of $\dot{\mathcal{G}}\sim -0.6$ and $\dot{\mathcal{G}}\sim +1.7$ mm year${}^{-1}$, respectively. The regional variability in this map, i.e., the alternation of local minima and maxima, is drastically reduced in comparison with $\dot{\mathcal{S}}$ and $\dot{\mathcal{U}}$, giving to $\dot{\mathcal{G}}$ a very smooth semblance.

**Figure 6.**Fingerprint for $\dot{\mathcal{N}}$, which represents the current rate of sea surface variation or absolute sea-level change according to our implementation of GIA model ICE-5G (VM2). White dots mark the places where the largest rates are expected, with $\dot{\mathcal{N}}\sim -0.9$ and $\dot{\mathcal{N}}\sim +1.4$ mm year${}^{-1}$, respectively. The spatial variability of $\dot{\mathcal{N}}$ matches that of $\dot{\mathcal{G}}$ in Figure 5, since the two fingerprints only differ by the spatially invariant term $\dot{c}$, where c is the FC76 constant.

**Figure 7.**GIA fingerprint for the present-day rate of variation of the surface load $\dot{\mathcal{L}}\left(\omega \right)$, in units of mm year${}^{-1}$ of water equivalent. The whole-Earth average is $<\dot{\mathcal{L}}{>}^{e}=0.00$ mm year${}^{-1}$ to a very high precision. In oceanic areas, $\dot{\mathcal{L}}$ is strongly correlated with $\dot{\mathcal{S}}$ in Figure 3. In continental areas, $\dot{\mathcal{L}}$ only takes contributions in regions where, according to ICE-5G (VM2), ice thickness variations are still occurring or where the OF is still varying. These conditions are met in Greenland, where $\dot{\mathcal{L}}$ shows the extreme values (white dots), and in West Antarctica, respectively.

**Table 1.**Values of density, rigidity and viscosity adopted in our realization of the rheological model VM2, with abbreviations LT, UM, TZ and LM denoting the lithosphere, the upper mantle, the transition zone and the lower mantle, respectively. With ${r}_{-}$ and ${r}_{+}$ we denote the radii of the base and of the top of each layer. A few spectral properties of this model are shown in Table 2.

Radius, ${\mathit{r}}_{-}$ (km) | Radius, ${\mathit{r}}_{+}$ (km) | Density, $\mathit{\rho}$ (kg m ^{−3}) | Rigidity, $\mathit{\mu}$ (Pa × 10 ^{11}) | Viscosity, $\mathit{\eta}$ (Pa·s × 10 ^{21}) | Layer |
---|---|---|---|---|---|

6281.000 | 6371.000 | 3192.800 | 0.596 | ∞ | LT |

6151.000 | 6281.000 | 3369.058 | 0.667 | 0.5 | UM1 |

5971.000 | 6151.000 | 3475.581 | 0.764 | 0.5 | UM2 |

5701.000 | 5971.000 | 3857.754 | 1.064 | 0.5 | TZ1 |

5401.000 | 5701.000 | 4446.251 | 1.702 | 2.7 | LM1 |

5072.933 | 5401.000 | 4615.829 | 1.912 | 2.7 | LM2 |

4716.800 | 5072.933 | 4813.845 | 2.124 | 2.7 | LM3 |

4332.600 | 4716.800 | 4997.859 | 2.325 | 2.7 | LM4 |

3920.333 | 4332.600 | 5202.004 | 2.554 | 2.7 | LM5 |

3480.000 | 3920.333 | 5408.573 | 2.794 | 2.7 | LM6 |

0 | 3480.000 | 10931.731 | 0 | 0 | Core |

**Table 2.**Numerical values of the elastic (with superscript e) and fluid (with superscript f) loading Love numbers ${k}_{l}^{L}$ and ${h}_{l}^{L}$ for the rheological model VM2 (see Table 1), for some harmonic degrees l in the range $1\le l\le 1024$. We use the compact notation ${v}_{exp}=v\times {10}^{-exp}$ and ${v}^{exp}=v\times {10}^{exp}$, where v is any value in the table and $exp$ is an exponent. Note that, for this model, the elastic tidal Love numbers of degree $l=2$ are $({k}_{2}^{Te},{h}_{2}^{Te})$ = $(0.{289}^{0},0.{524}^{0})$ while the fluid values are $({k}_{2}^{Tf},{h}_{2}^{Tf})$ = $(0.{931}^{0},0.{191}^{1})$.

$\mathit{l}=1$ | 2 | 4 | 16 | 64 | 128 | 256 | 512 | 1024 | |
---|---|---|---|---|---|---|---|---|---|

${k}_{l}^{Le}$ | $-1.{000}^{0}$ | $-0.{235}^{0}$ | $-0.{117}^{0}$ | $-0.{558}_{1}$ | $-0.{231}_{1}$ | $-0.{123}_{1}$ | $-0.{642}_{2}$ | $-0.{323}_{2}$ | $-0.{162}_{2}$ |

${h}_{l}^{Le}$ | $-0.{102}^{1}$ | $-0.{442}^{0}$ | $-0.{463}^{0}$ | $-0.{975}^{0}$ | $-0.{165}^{1}$ | $-0.{181}^{1}$ | $-0.{188}^{1}$ | $-0.{190}^{1}$ | $-0.{190}^{1}$ |

${k}_{l}^{Lf}$ | $-1.{000}^{0}$ | $-0.{980}^{0}$ | $-0.{981}^{0}$ | $-0.{956}^{0}$ | $-0.{192}^{0}$ | $-0.{249}_{1}$ | $-0.{675}_{2}$ | $-0.{323}_{2}$ | $-0.{162}_{2}$ |

${h}_{l}^{Lf}$ | $-0.{163}^{1}$ | $-0.{267}^{1}$ | $-0.{481}^{1}$ | $-0.{173}^{2}$ | $-0.{139}^{2}$ | $-0.{363}^{1}$ | $-0.{198}^{1}$ | $-0.{190}^{1}$ | $-0.{190}^{1}$ |

**Table 3.**Ocean (top) and whole-Earth surface averages (bottom) of the present-day rate of change of GIA fingerprints considered in this study. In this table, the outputs of SELEN${}^{4}$ have been rounded to two significant figures. Although in the text we dwelt upon the new rotation theory (column (a)), results for the the traditional theory are also shown here in (b) while in (c) no rotational effects are taken into account. It is apparent that the spatial averages are only moderately affected by the choice of the rotation theory. The values of $<\dot{\mathcal{U}}{>}^{e}$, $<\dot{\mathcal{G}}{>}^{e}$ and $<\dot{\mathcal{L}}{>}^{e}$ are numerically found to be <${10}^{-5}$ mm year${}^{-1}$ in modulus. By virtue of mass conservation, their expected theoretical value should be exactly zero.

Average | (a) New Theory (mm year ^{−1}) | (b) Traditional Theory (mm year ^{−1}) | (c) No Rotation (mm year ^{−1}) |
---|---|---|---|

$<\dot{\mathcal{S}}{>}^{o}$ | −0.06 | −0.06 | −0.06 |

$<\dot{\mathcal{U}}{>}^{o}$ | −0.27 | −0.30 | −0.24 |

$<\dot{\mathcal{N}}{>}^{o}$ | −0.33 | −0.35 | −0.30 |

$<\dot{\mathcal{G}}{>}^{o}$ | −0.06 | −0.09 | −0.04 |

$<\dot{\mathcal{L}}{>}^{e}$ | +0.00 | +0.00 | +0.00 |

$<\dot{\mathcal{S}}{>}^{e}$ | −0.27 | −0.27 | −0.26 |

$<\dot{\mathcal{U}}{>}^{e}$ | +0.00 | +0.00 | +0.00 |

$<\dot{\mathcal{N}}{>}^{e}$ | −0.27 | −0.27 | −0.26 |

$<\dot{\mathcal{G}}{>}^{e}$ | +0.00 | +0.00 | +0.00 |

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Spada, G.; Melini, D.
On Some Properties of the Glacial Isostatic Adjustment Fingerprints. *Water* **2019**, *11*, 1844.
https://doi.org/10.3390/w11091844

**AMA Style**

Spada G, Melini D.
On Some Properties of the Glacial Isostatic Adjustment Fingerprints. *Water*. 2019; 11(9):1844.
https://doi.org/10.3390/w11091844

**Chicago/Turabian Style**

Spada, Giorgio, and Daniele Melini.
2019. "On Some Properties of the Glacial Isostatic Adjustment Fingerprints" *Water* 11, no. 9: 1844.
https://doi.org/10.3390/w11091844