Problem 9SE

The article “Insights into Present-Day Crustal Motion in the Central Mediterranean Area from GPS Surveys” (M. Anzidei, P. Baldi, et al., Geophysical Journal International, 2001:98-100) reports that the components of velocity of the earth’s crust in Zimmerwald, Switzerland, are 22.10±0.34 mm/year in a northerly direction and 14.3 ± 0.32 mm/year in an easterly direction.

a. Estimate the velocity of the earth’s crust, and find the uncertainty in the estimate.

b. Using your answer to part (a), estimate the number of years it will take for the crust to move 100 mm, and find the uncertainty in the estimate.

Solution 9SE

Step1 of 3:

We have article that Insights into Present Day Crustal Motion in the Central Mediterranean Area from GPS Surveys reports that the components of velocity of the earth’s crust in Zimmerwald, Switzerland, are 22.100.34 mm/year in a northerly direction and 14.30.32 mm/year in an easterly direction.

That is and

Here our goal is:

a).We need to Estimate the velocity of the earth’s crust, and find the uncertainty in the estimate.

b).We need to estimate the number of years it will take for the crust to move 100 mm, and find the uncertainty in the estimate by using our answer to part (a).

Step2 of 3:

a).

Let us consider the measured northerly component of velocity is X and the measured easterly component of velocity is Y.

Now, the velocity of Earth’s crust is estimated by using the formula

From the given information we have X = 22.10 and Y = 14.3

Then

=

=

=

= 26.3229

Hence, 26.3229.

1).Differentiate with respect X then we get

Let us take u =

Apply chain rule to differentiate above equation we get

=

Hence, =

Substitute “u” value in above equation we get

=

=

=

Therefore, =

Consider

=

=

= 0.8395.

Therefore, = 0.8359.

2).Differentiate with respect Y then we get

Let us take u =

Apply chain rule to differentiate above equation we get

=

Hence, =

Substitute “u” value in above equation we get

=

=

=

Therefore, =

Consider

=

=

= 0.5432.

Therefore, = 0.5432.

Consider,

=

=

=

= 0.3330

Hence, =0.3330.

Therefore, The velocity of Earth’s crust is 26.320.33 mm/years.

Step3 of 3:

b).

Let us consider S be the estimated number of years it will take for the earth’s crust to move

100 mm. And from part (a), we have

Now

S =

=

= 3.7989

Hence, S = 3.7989.

Consider, S =

Differentiate above equation with respect to V then we get

=

=

= -0.1443

Hence, = -0.1443.

=

= 0.0480

Hence, =0.0480.

Therefore, S = 3.79890.0480 years.

From part (a), we have The velocity of Earth’s crust is 26.320.33 mm/years.

The uncertainty in the above estimate is 0.3330

Conclusion:

a).The velocity of Earth’s crust is 26.320.33 mm/years.

b).Therefore, S = 3.79890.0480 years and The uncertainty is 0.3330.