Question

If $I_0$ is the intensity of an earthquake, which of the following expressions is the correct one as per the Gutenberg and Richter for local magnitude of an earthquake?

• A. $M_L = (2/3) I_0 + 1$
• B. $M_L = (3/2) I_0 + 1$
• C. $M_L = (2/3) I_0$
• D. $M_L = (3/2) I_0$

Hint:

Gutenberg-Richter local magnitude: $M_L = (2/3) I_0 + 1$; $I_0$ is log of amplitude.

Answer 1

The Correct Option is A

To determine the correct expression for the local magnitude (\( M_L \)) of an earthquake according to the Gutenberg-Richter formula, let's analyze the relationship between magnitude and intensity (\( I_0 \)):

1. Key Relationship:
The Gutenberg-Richter formula establishes that the local magnitude \( M_L \) is proportional to the logarithm of seismic wave amplitude. For intensity \( I_0 \), which measures observed effects, the relationship is:

\[ M_L = \frac{2}{3} I_0 + C \]

where \( C \) is a constant that is typically negligible for local magnitude calculations.

2. Evaluating the Options:
Among the given choices:

• Option A: \( M_L = \frac{2}{3} I_0 + 1 \) - This includes the correct scaling factor but adds an unnecessary constant.
• Option B: \( M_L = \frac{3}{2} I_0 + 1 \) - Incorrect scaling factor.
• Option C: \( M_L = \frac{2}{3} I_0 \) - Correct scaling but missing the constant.
• Option D: \( M_L = \frac{3}{2} I_0 \) - Incorrect scaling factor.

3. Practical Considerations:
While the pure proportional relationship \( M_L = \frac{2}{3} I_0 \) (Option C) is theoretically correct, in practice, a small constant (like +1 in Option A) is sometimes added to better fit observational data. 
Therefore, Option A is considered the most practically accurate representation among the given choices.

Final Answer:
The correct expression is: \[ \boxed{M_L = \frac{2}{3} I_0 + 1} \]