Geophysical Research Letters
Supporting Information for
Gravity increase before the 2015 Mw7.8 Nepal earthquake
Shi Chen1, Mian Liu2, Lelin Xing3, Weimin Xu1, Wuxing Wang4, Yiqing Zhu5, and Hui Li3
1. Institute of Geophysics, China Earthquake Administration, Beijing, 100081, China
2. University of Missouri, Columbia, MO, 65211, USA
3. Hubei Earthquake Administration, Wuhan, 430071, China
4. Institute of Earthquake Science, China Earthquake Administration, Beijing, 100036, China
5. Second Crust Monitoring and Application Center, China Earthquake Administration, Xi’An, 710054, China.
Contents of this file
Text S1
Figures S1 to S2
Introduction
This supporting information provides the details of the forward modeling using gravity
change and figures of the vertical displacement of the gravimetric stations from the time
series of continuous GPS measurements and GRACE data of gravity changes at the
four stations in southern Tibet.
1
Text S1.
Forward modeling of gravity anomaly of a disk body with a uniform density change.
(1) Equation of gravity anomaly ∆𝒈(𝒙, 𝒛)
We used the analytical solution to forward gravity change with a disk geometry
[Singh, 1977].
∆𝑔(𝑥, 𝑦) = 2𝐺𝜌 |
𝑅 2 −𝑥 2
√(𝑥+𝑅)2 +𝑦
𝑍
𝛫(𝑘) + √(𝑥 + 𝑅)2 + 𝑦 2 𝛦(𝑘) + 0.5𝜋𝑦𝛬(𝜑, 𝑘) − 𝜋𝑦|
2
𝑍+𝐻
where the ∆𝑔 is the gravity change with a uniform density 𝜌, 𝐺 is gravitation constant.
The x and y are the horizontal and vertical position of observed station, respectively. The
R, Z and H are the radius, top depth and thickness of disc model, respectively. As shown
in Figure 2, 𝑍 = 𝑍ℎ + 𝑍𝑡 and 𝑍ℎ is the known quantity and equal to the elevation, list in
Table 1. The K(k) is the first kind complete elliptic integral:
𝜋/2
Κ(k) = ∫
0
𝑑𝑡
√1 − 𝑘 2 𝑠𝑖𝑛2 𝑡
and E(k) is the second kind complete elliptic integral:
𝜋/2
Ε(k) = ∫
√1 − 𝑘 2 𝑠𝑖𝑛2 𝑡𝑑𝑡
0
and the Heuman Lambda function is:
𝛬(𝜑, 𝑘) = 𝛦(𝑘)𝐹(𝜑, 𝑘) + 𝐾(𝑘)𝛦(𝑘) − 𝐾(𝑘)𝐹(𝜑, 𝑘)
where k and φ as follows:
4𝑅𝑧
(𝑥 + 𝑅)2 + 𝑧 2
𝜋
x−R
φ = + tan−1
2
z
k2 =
(2) Bouguer layer equation for correcting the effects of elevation change:
∆𝑔 = −(3.086 − 0.419𝜌)∆ℎ 𝜇𝐺𝑎𝑙/𝑐𝑚
Assuming crustal density to be 𝜌 = 2.7 − 2.9𝑔/𝑐𝑚3 for the Tibetan crust [Chen, et al.,
2004], the gravity vertical gradient is from -1.87~1.95 𝜇𝐺𝑎𝑙/𝑐𝑚. We used the
−1.9𝜇𝐺𝑎𝑙/𝑐𝑚 in this paper.
(3) Strain rate estimations
According to the definition of dilatation Θ:
Θ = Δ𝑉/𝑉
where 𝑉 is a volume element, Δ𝑉 is the incremental change of the volume element.
After the deformation, volume element is
𝑉1 = 𝑉 + Δ𝑉 = (1 + Θ)𝑉
If the total mass is conserved, the density can be expressed as
1/𝜌1 = (1 + Θ)/𝜌0
where 𝜌0 is the density of element before deformation, and 𝜌1 is the density after
deformation.
Therefore, the density change Δ𝜌 is:
Δ 𝜌 = 𝜌1 − 𝜌0 = −
Θ
𝜌
1+Θ 0
The equation can be transformed as:
2
Θ=−
Δ𝜌
Δ𝜌 + 𝜌0
Using the plane stress equation [Turcotte et al., 2002], there is:
Θ=
2(1 − 2υ)
𝜀1
1−υ
where υ is the Poisson ration, and 𝜀1 is the horizontal strain.
Figure S1. Time series of daily solution of vertical displacements at the Naqu, Lhasa,
Shigatse, and Zhongba stations from continuous GPS recording. Red lines are the linear
fits. The GPS stations are collocated with the gravimetric stations. We fit the average
vertical motion rate at these stations and use the results to correct for the effects of
elevation changes on gravity at these stations.
3
Figure S2. Gravity change observed by the Gravity Recovery and Climate Experiment
(GRACE) satellite. We computed the series of average monthly gravity change at the
four gravimetric stations using a 300 km Gauss filter. The GRACE data show clear
seasonal variations and provide the long-wavelength gravity change. The average rate is
-0.048μGal/yr for the Lhasa station, -0.046μGal/yr for the Shigatse station, -0.042μGal/yr
for the Zhongba station, and -0.018μGal/yr for the Naqu stations. These trends of
decreasing gravity are consisted with the continuing uplift of the Tibetan Plateau. The
long-wavelength gravity decrease from the GRACE data are is opposite to the gravity
increase observed by absolute gravity measurements at these stations. Hence gravity
increases at these stations are not artifacts of background gravity variations; we suggest
that they reflect mass change in a broad source region related to the tectonic processes
leading to the 2015 Nepal earthquake.
References
Chen,W. P., and Z. Yang (2004), Earthquakes beneath the Himalayas and Tibet:
Evidence for strong lithospheric mantle, Science, 304, 1949-1952.
Singh, S.K. (1977), Gravitational attraction of a vertical right circular cylinder,
Geophysical Journal of the Royal Astronomical Society, 50, 243–246.
Turcotte, D. L., and G. Suchubert, G (2002), Geodynamics, 2nd ed. Cambridge, U.
K.:Cambridge Univ. Press.
4