Calculate moment ratios from logarithmic magnitude

Compare the quantities behind the scale

Select Tohoku 2011 as Event A and Ridgecrest 2019 as Reference event B. Their stored moment magnitudes differ by 9.1 − 7.1 = 2.0. That difference is not a claim that one earthquake is “two times bigger.” A logarithmic scale compresses large ratios into manageable numerical differences.

Seismic moment describes a source property related to rigidity, rupture area and slip. Moment magnitude converts that quantity to a logarithmic scale. It is different from radiated seismic energy and from ground-motion amplitude at a particular site. The browser derives ratios from the stored moment magnitudes; it has not measured either event’s energy or a site’s shaking.

Recover a ratio from a logarithmic difference

Let M_A and M_B be the two moment magnitudes, and let ΔM = M_A − M_B. The moment-magnitude relation gives a seismic-moment ratio of 10^(1.5ΔM). The ratio compares the moment implied for A with that implied for B, using the same relation and units for both. USGS explanation of moment magnitude and energy scaling.

For ΔM = 2.0, first multiply 1.5 × 2.0 = 3.0. Then evaluate 10³ = 1,000. Ten cubed means 10 × 10 × 10. This is a dimensionless ratio; the corresponding moment units cancel between numerator and denominator.

The algebra can be read without calculating absolute moments. From Mw = (2/3)(log₁₀(moment) − constant), subtract the two magnitude equations. The common constant cancels, leaving ΔM = (2/3)log₁₀(moment_A/moment_B). Multiplying by 3/2 and reversing the base-ten logarithm yields the ratio formula. The absolute-moment convention affects the constant, but using a common convention makes that constant disappear from this comparison.

Check direction and equality

Swap A and B. The difference becomes −2.0, so the ratio becomes 10⁻³ = 0.001. A negative exponent gives a reciprocal, not a negative physical moment. The forward and reversed ratios multiply to one, apart from display rounding.

Now choose the same event in both controls. ΔM is zero, so 10⁰ = 1. This compares a stored record with itself. It does not demonstrate that two distinct physical earthquakes have identical rupture properties. A control experiment should be interpreted according to what was actually selected.

For a paper check, use hypothetical moment magnitudes 6 and 4. The difference is again two and the implied moment ratio is again 1,000. Equal magnitude differences give equal ratios in this relation even when the absolute values differ. These hypothetical numbers are a transfer exercise, not additional catalog entries.

Keep the energy approximation separate

The second displayed comparison uses an approximate radiated-energy scaling relation: log₁₀E = 5.24 + 1.44Mw. Subtracting two applications cancels the intercept 5.24 and gives the estimated ratio 10^(1.44ΔM). This scaling is an estimate from magnitude, not a measurement of the energy radiated by these particular events. The empirical energy relation and the moment definition describe different quantities.

At ΔM = 2, the energy exponent is 2.88 and the estimated ratio is about 758.6. It is neither the moment ratio of 1,000 nor a site-amplitude ratio of 100. Keep the quantity name beside the number whenever you quote it. Different expressions of source size should not share an unlabeled “times stronger” heading.

Audit the precision of a comparison

The arithmetic is deterministic, but the input magnitudes remain catalog estimates. Reporting many ratio digits does not establish matching scientific accuracy. The energy scaling adds another approximation. In your reflection, distinguish numerical rounding in the displayed calculation from uncertainty in the input estimates and limitations of the energy relation.

Capture a forward pair, the reverse pair and the same-record comparison. Recalculate the exponent for each and label the numerator and denominator. This makes a mistaken direction or an inappropriate linear interpretation easier to diagnose.

Transfer to a half-unit difference

For a hypothetical ΔM of 0.5, calculate the moment exponent and the energy-scaling exponent. Use a calculator for the powers of ten, then predict the reversed ratios before revealing the result.

Check the magnitude-ratio transfer

The moment exponent is 0.75, giving about 5.623. The approximate energy-scaling exponent is 0.72, giving about 5.248. Reversing the comparison gives their reciprocals, about 0.1778 and 0.1905 respectively. Neither result specifies shaking at a site or damage.