Mogi (1958): FEM:

Conclusions

Goodness-of-fit tests (Pearson & Spearman correlations) likely produce irrelevant results, so RMSE was used to quantify similarity between the Mogi analytical model and FEMs. This value was also expressed in percentage of maximum analytical displacement. The maximum absolute difference in displacement between analytical and numerical models was also given in terms of percentage of maximum analytical displacement. This allows researchers to evaluate errors on the basis of their results and to determine acceptable limits with respect to the precision of their data acquisition method.

Results of the FEMs were far more sensitive to limits on lateral domain extent than domain depth. Therefore, the boundary conditions of a 50-km3 domain, as described by Charco & del Sastre (2011) may produce an unsuitable amount of error for some applications.

The FEM systematically underestimates the displacement due to the circular chamber being approximated by secant lines, resulting in a smaller effective radius. The underestimation can be reduced by increasing the number of nodes along the chamber. Alternately, it can be compensated by increasing the size of the chamber such that its effective radius is equal to the desired radius. If the latter method is chosen, note that the effective radius is a function of the number, size, and orientation of element faces that form the spherical chamber in three dimensions. In our models, with 16 elements on the magma chamber half-circle, there is little difference in the results between radius and effective radius. With 8 elements, the difference is still not great, but the model is already invalid.

r

r

P-z

A

(f2 + r2)3/2-z

uz

ur

Models – Analytical & Finite Element

Results

free surfacefree surface

axis

of s

ymm

etry

axis

of s

ymm

etry

uu rr==uu zz=

0=0

uurr==uuzz=

0=0

PP

zz

rr

50 km50 km

Abstract

Finite Element Models (FEMs) can accurately simulate ground deformation due to migration of subsurface magma. FEMs are becoming an increasingly important tool for volcano modelers. However, FEMs are computationally expensive. Thus, it is highly beneficial for modelers to minimize computation costs, especially in iterative model processing or in models for which restrictions of number of elements becomes a limiting factor. To the authors’ knowledge, there has not been a systematic study of model domain parameters, such as vertical and lateral model extent and mesh fineness, that can serve as a guide for producing the most efficient FEMs. As a result, modelers often choose domain parameters arbitrarily, by trial-and-error, or based on earlier researchers’ models, which may also be inefficient. Therefore, we present a systematic study of model parameters, including how mesh fineness, geometry, and extent interact. We apply statistical analyses to our results in order to determine ranges of acceptable parameters. It is our hope that this study can serve as a valuable guide for other modelers in determining optimum volcanic model parameters.

Quantitative analysis of volcano deformation FEM domain parameters – A guide for modelers (V11B-2756)

Jonathan Stone1 & Timothy Masterlark2

1jstone@crimson.ua.edu, Department of Geological Sciences, The University of Alabama, Tuscaloosa, AL 35487, USA2timothy.masterlark@sdsmt.edu, Department of Geology and Geological Engineering, South Dakota School of Mines & Technology, Rapid City, SD 57701, USA

References

Charco, M. & del Sastre, P. G. (2011), Finite element numerical solution for modelling ground deformation in volcanic areas, In: Pardo, L., Balakrishnan, N., & Gil, M. A., (Eds.) Modern Mathematical Tools and Teqhniques in Capturing Complexity, Springer, Berlin, pp. 223-238.

Masterlark, T. (2007), Magma intrusion and deformation predictions: Sensitivities to the Mogi assumptions. Journal of Geophysical Research 112, B06419, 1-17. doi:10.1029/2006JB004860.

Masterlark, T. & Stone, J. (2011), Simulating volcano deformation with Abaqus FEM software, University of Iceland short course, Reykavik, Iceland.

Masterlark, T., Feigl, K. L., Haney, M., Stone, J., Thurber, C., & Ronchin, E. (2012), Nonlinear estimation of geometric parameters in FEMs of volcano deformation: Integrating tomography models and geodetic data for Okmok volcano, Alaska, Journal of Geophysical Research, 117, B02407, doi:10.1029/2011JB008811.

Mogi, K. (1958). Relations between the Eruptions of Various Volcanoes and the Deformations of the Ground surface around them. Bulletin of the Earthquake Research Institute 36, 99-134.

2/322

3

2/322

3

)(4

3

)(4

3

rf

f

G

Pau

rf

r

G

Pau

z

r

where

ur is the displacement in the radial direction on the surface

uz is the vertical displacement on the surface

a is the radius of the spherical magma source

ΔP is the change in magma source pressure

G is the shear modulus

r is the radial distance on the surface from a point A

f is the depth of the center of the spherical magma source (z = -f)

Domain depth

Domain radius

ur (500-m radius)

RMSE

RMSE % of

max ur

max error % of

max ur

ur (effective radius)

RMSE

RMSE % of

max ur

max error % of

max ur

uz (500-m radius)

RMSE

RMSE % of

max uz

max error % of max

uz

uz (effective radius)

RMSE

RMSE % of

max uz

max error % of max

uz

50 km 50 km 0.00004726 1.572 2.560 0.00004723 1.571 2.560 0.00003178 0.4067 0.8949 0.00003166 0.4053 0.8865

25 km 50 km 0.00006002 1.996 2.560 0.00005998 1.995 2.560 0.00007617 0.9749 2.1606 0.00007603 0.9733 2.1533

15 km 50 km 0.00007959 2.647 3.266 0.00007954 2.646 3.265 0.00008826 1.1297 2.3271 0.00008813 1.1281 2.3182

10 km 50 km 0.00011665 3.880 5.864 0.00011663 3.879 5.863 0.00009984 1.2780 3.7517 0.00009995 1.2795 3.7617

5 km 50 km 0.00018565 6.174 15.283 0.00018555 6.171 15.277 0.00069189 8.8561 25.5484 0.00069171 8.8547 25.5414

50 km 40 km 0.00006950 2.311 3.966 0.00006946 2.310 3.966 0.00003696 0.4732 1.1119 0.00003682 0.4713 1.1030

50 km 35 km 0.00008786 2.922 5.144 0.00008781 2.921 5.144 0.00004226 0.5409 1.2036 0.00004210 0.5389 1.1953

50 km 30 km 0.00011595 3.856 6.927 0.00011589 3.854 6.927 0.00005134 0.6571 1.2729 0.00005116 0.6549 1.2645

50 km 25 km 0.00016159 5.374 9.799 0.00016152 5.372 9.799 0.00007000 0.8960 1.5025 0.00006980 0.8936 1.4942

50 km 10 km 0.00071093 23.644 46.479 0.00071076 23.641 46.479 0.00042514 5.4417 8.9443 0.00042484 5.4384 8.9443

Elements along

chamber (half- circle)

Effective chamber

radius

ur (500-m radius)

RMSE

RMSE % of

max ur

max error % of

max ur

ur (effective radius)

RMSE

RMSE % of

max ur

max error % of

max ur

uz (500-m radius)

RMSE

RMSE % of

max uz

max error % of max

uz

uz (effective radius)

RMSE

RMSE % of

max uz

max error % of max

uz

160 499.9839 m 0.00004726 1.572 2.560 0.00004723 1.571 2.560 0.00003178 0.4067 0.8949 0.00003166 0.4053 0.8865

80 499.9357 m 0.00004749 1.579 2.560 0.00004734 1.575 2.560 0.00003369 0.4312 1.0866 0.00003317 0.4247 1.05615

32 499.5984 m 0.00004780 1.590 2.560 0.00004688 1.563 2.560 0.00003639 0.4658 1.2953 0.00003291 0.4223 1.08635

16 498.3941 m 0.00005030 1.673 2.560 0.00004604 1.546 2.560 0.00004952 0.6338 2.0097 0.00003384 0.4373 1.15864

8 493.5827 m 0.00017020 5.661 16.749 0.00013032 4.505 13.526 0.00035553 4.5507 18.4840 0.00028107 3.7398 15.26294

4 474.4250 m 0.00031702 10.543 29.378 0.00015993 6.226 17.330 0.00057799 7.3983 28.4306 0.00029188 4.3735 16.34651

2 398.9423 m 0.00068619 22.821 61.573 0.00015462 10.124 25.634 0.00115178 14.7427 54.9503 0.00019280 4.8585 15.90264

Domain depth: 50 km

Domain radius: 50 km

Chamber elements: 160

Dis

plac

emen

t (m

)

Distance from axis of symmetry (m)

Analytical solutions for Mogi:

z = 3000 m a = 500 m ΔP = 50 MPa G (shear modulus) = 2.4 × 1010 ν (Poisson’s ratio) = 0.25 E (Young’s modulus) = 6.0 × 1010

Circles show corresponding FEM solutions

Domain depth: 10 km

Domain radius: 50 km

Chamber elements: 160

Distance from axis of symmetry (m)

For all the above models, chamber depth = 5000 m, chamber radius = 500 m, effective chamber radius = 499.9839 m, the number of elements on the half-circle representing the chamber wall = 160 (with the exception of the 5000-deep model, wherein the chamber is cut in half and, thus, contains 80 elements).

For all the above models, model domain depth = 50 km, model domain radius = 50 km, chamber depth = 5000 m, chamber radius = 500 m.

The triangular elements of the model result in a reduction of the chamber radius, which, in the axisymmetric case (assuming the formation of a regular n-gon), is given by:

where rE is the effective radius, rC is the modeled radius and n is the number of sides of the n-gon.

2 sin cosE C

nr r

n n

Dis

plac

emen

t (m

)

Domain depth: 50 km

Domain radius: 35 km

Chamber elements: 160

Domain depth: 50 km

Domain radius: 50 km

Chamber elements: 16

uz

ur

uz

ur

uz

ur

uz

ur