Abstract
“Magnitude” is not one thing. There are four——$M_L$, $m_b$, $M_S$, $M_w$——and all but $M_w$ hit a ceiling for great earthquakes. This paper asks whether what hits the ceiling is the earthquake or the scale——the answer is the scale. No new mathematical theorem and no new law is claimed.
First, the four scales observe different periods. $M_L$ at $0.1$ s, $m_b$ at $1$ s, $M_S$ at $20$ s, and $M_w$ choosing no period at all (Section 2).
Second, this is the core of the paper. $m_b$ saturates at $M\simeq6.5$, so measuring the $M_w=9.0$ Tohoku earthquake with it estimates the energy at one $\mathbf{5623}$rd (Section 3).
Third, even $M_S$ falls short by a factor of $7.94$. The difference of $0.6$ between $M_S=8.4$ and $M_w=9.0$ is a factor of $\mathbf{7.9433}$ in energy (Section 3).
Fourth, it does not happen for small earthquakes. For $1995$ Southern Hyogo the difference is $0.10$, a factor of only $1.41$ in energy (Section 4).
Fifth, the cause is the duration of rupture. The Tohoku rupture lasted $\mathbf{150}$ s, $\mathbf{7.5}$ times the $20$ s that $M_S$ observes (Section 5).
Sixth, the separator is whether the source duration exceeds the observed period. $M_w$ alone does not saturate because it chooses no period and measures $M_0$ directly (Section 6).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——$M_w=\frac{2}{3}\log_{10}M_0-6.07$, $\log_{10}E=1.5M+4.8$, the saturation values of each scale, and the rupture duration of the Tohoku earthquake are all standard. We do not build seismology——all we use is two linear expressions and one division. We do not predict earthquakes——time, place, and size are not treated at all. We do not discuss source processes——neither rupture propagation nor slip distribution is treated. The duration is merely placed as a representative value. We claim no accuracy for the saturation values——$M_L\simeq6.8$, $m_b\simeq6.5$, $M_S\simeq8.5$ are approximate, moving by about $\pm0.3$ with network and procedure. We do not assert the $\omega^{-2}$ model——the $1.7501$ of Section 5 is an upper bound the model gives, not a value that accounts for the observed $0.60$. This paper does not explain the gap between model and observation. We do not say $M_w$ is perfect——$M_w$ carries its own error in estimating $M_0$; it merely does not saturate. We assert no individual earthquake’s values——the $M_w$ and $M_S$ of the tables are representative values widely used in the literature, differing by about $\pm0.1$ between agencies. Relation to earlier papers: Paper 117 treated the Gutenberg--Richter power law——that concerns the relation between the number and the size of earthquakes, while this paper concerns the construction of the scale itself. The material is the same earthquakes; the question differs. Paper 190 measured “rare” on a logarithmic scale——this paper likewise treats how a difference of $0.6$ on a logarithmic scale becomes a factor of $7.94$. Paper 255 separated two things called “accuracy,” only one of which calibration removes——the saturation here is on the side that calibration does not remove, arising from the construction of the scale. Paper 140 separated symmetry fixing ratios from dynamics fixing the scale——this paper treats the case where the scale side breaks. What is added is arranging the four scales by observed period, computing that $m_b$ estimates the Tohoku energy at one $5623$rd, confirming that the same procedure is off by only a factor of $1.41$ for small earthquakes, and writing the ratio $7.5$ of rupture duration to observed period as the separator.
1. Notation and Standing
Only two expressions are used.
$$M_w=\tfrac23\log_{10}M_0-6.07,\qquad \log_{10}E=1.5M+4.8$$Here $M_0$ is the seismic moment ($\mathrm{N m}$) and $E$ the energy ($\mathrm{J}$).
Check 1.1: We substitute $M_w=9.0$.
| Quantity | Value |
|---|---|
| $M_0=10^{(9.0+6.07)\times1.5}$ | $\mathbf{4.0272\times10^{22}}$ $\mathrm{N m}$ |
| $E=10^{1.5\times9.0+4.8}$ | $1.9953\times10^{18}$ J |
Observation 1.2 An increase of $1$ in $M$ multiplies the energy by $\mathbf{31.6228}$. An increase of $2$ multiplies it by exactly $\mathbf{1000}$——even $0.1$ is a factor of $1.41$, so the first decimal place matters.
Question: $M_L$, $m_b$ and $M_S$ hit a ceiling for great earthquakes. Is it the earthquake or the scale that hits the ceiling?
2. The Four Scales Observe Different Periods
Check 2.1: We line up what each of them looks at.
| Scale | Period observed | Saturation value |
|---|---|---|
| $M_L$ (Richter, local) | $0.1$ s | $M\simeq6.8$ |
| $m_b$ (body wave) | $1$ s | $M\simeq\mathbf{6.5}$ |
| $M_S$ (surface wave) | $20$ s | $M\simeq8.5$ |
| $M_w$ (moment) | chooses no period | none |
Observation 2.2 The order of saturation values matches the order of observed periods. $0.1$ s $\to6.8$, $1$ s $\to6.5$, $20$ s $\to8.5$——the longer the period a scale observes, the later it hits its ceiling.
Observation 2.3 $M_w$ alone stands outside the column. $M_w$ looks at the amplitude of no period’s wave——it looks at $M_0=\mu A \bar{d}$, that is at rigidity, rupture area, and mean slip.
Observation 2.4 That $M_L$ saturates later ($6.8$) than $m_b$ ($6.5$) conflicts with the order of periods. Written honestly, saturation values are not set by period alone but also by network and procedure——this paper treats period as the principal factor, but does not claim it is the only one.
3. The Core: $m_b$ Estimates Tohoku at One $5623$rd
Setting: The $2011$ Tohoku earthquake is $M_w=9.0$, $E=1.9953\times10^{18}$ J. What do saturated scales give?
Check 3.1:
| Scale | Value given | Estimate of $E$ | Fraction of true value |
|---|---|---|---|
| $m_b$ | $6.5$ | $3.5481\times10^{14}$ J | $\mathbf{1/5623.4}$ |
| $M_L$ | $6.8$ | $1.0000\times10^{15}$ J | $1/1995.3$ |
| $M_S$ | $8.4$ | $2.5119\times10^{17}$ J | $\mathbf{1/7.9433}$ |
| $M_w$ | $9.0$ | $1.9953\times10^{18}$ J | $1$ |
Observation 3.2 Measured by $m_b$, the energy is estimated at one $\mathbf{5623}$rd. The number $m_b=6.5$ means “a moderate earthquake”——of the same event, $m_b$ says moderate and $M_w$ says fourth largest ever recorded.
Observation 3.3 Even $M_S$ falls short by $\mathbf{7.9433}$. The difference between $M_S=8.4$ and $M_w=9.0$ is only $0.6$, but it is $0.6$ on a logarithmic scale——as Paper 190 wrote, it must be read as a ratio, not a difference.
Check 3.4: We see whether the same happens for other great earthquakes.
| Earthquake | $M_w$ | $M_S$ | Difference | Underestimate of $E$ |
|---|---|---|---|---|
| $1960$ Chile | $9.5$ | $8.5$ | $+1.00$ | $\mathbf{31.6228}\times$ |
| $1964$ Alaska | $9.2$ | $8.4$ | $+0.80$ | $15.8489\times$ |
| $2004$ Sumatra | $9.1$ | $8.8$ | $+0.30$ | $2.8184\times$ |
| $2011$ Tohoku | $9.0$ | $8.4$ | $+0.60$ | $7.9433\times$ |
Observation 3.5 The larger $M_w$, the larger the difference——but not monotonically. $2004$ Sumatra has $M_w=9.1$ yet a difference of only $0.30$——$M_w$ alone does not settle it; the rupture duration acts (Section 5).
4. It Does Not Happen for Small Earthquakes
Check 4.1: We put earthquakes below the saturation values into the same table.
| Earthquake | $M_w$ | $M_S$ | Difference | Underestimate of $E$ |
|---|---|---|---|---|
| $1995$ Southern Hyogo | $6.9$ | $6.8$ | $+0.10$ | $\mathbf{1.4125}\times$ |
| $1906$ San Francisco | $7.9$ | $7.8$ | $+0.10$ | $\mathbf{1.4125}\times$ |
| $2011$ Tohoku | $9.0$ | $8.4$ | $+0.60$ | $7.9433\times$ |
Observation 4.2 Below the $M_S$ saturation value of $8.5$, the difference stays around $0.1$. $M_S$ is not broken; it is being used outside its range——the scale is not at fault.
Observation 4.3 Its standing differs from the distinction of Paper 255. The systematic error of Paper 255 is removed by calibration, whereas this saturation is not——the construction of the scale itself makes it unable to see large earthquakes.
Observation 4.4 Hence one cannot “correct $M_S$ into $M_w$.”The quantity to be corrected has not been observed——a $20$ s wave carries only part of a $150$ s rupture.
5. The Cause Is the Duration of Rupture
Setting: An earthquake is not instantaneous. A fault takes time to tear from end to end.
Check 5.1: For the Tohoku earthquake we compare two times.
| Quantity | Value |
|---|---|
| Duration of rupture | about $\mathbf{150}$ s |
| Period of the wave $M_S$ measures | $20$ s |
| Ratio | $\mathbf{7.5}$ |
Observation 5.2 A $20$ s wave sees only part of an event that took $150$ s. By the time the far end of the fault is tearing, the near end is already finished——they are not summed together.
Check 5.3: We rewrite in frequency.
| Quantity | Value |
|---|---|
| Corner frequency $f_c=1/150$ | $0.006667$ Hz |
| $M_S$ observing frequency $1/20$ | $0.050000$ Hz |
| $f/f_c$ | $\mathbf{7.50}$ |
| Amplitude ratio in the $\omega^{-2}$ model, $(f/f_c)^{-2}$ | $1/\mathbf{56.25}$ |
| In magnitude units | $\mathbf{1.7501}$ |
Observation 5.4 The model gives a shortfall of $1.7501$; the observation is $0.60$. The model gives only an upper bound——because the $M_S$ procedure uses a band of periods, not a single frequency.
Observation 5.5 A point to write honestly——this paper does not explain the gap between $1.7501$ and $0.60$. What can be shown is the direction and the order of magnitude, not the value——only that the model predicts a shortfall in the right direction and of sufficient size.
6. The Separator, and the Span One Word Covers
Observation 6.1 Separator——what must be moved for saturation to occur or not? Whether the source duration exceeds the scale’s observed period.
Observation 6.2 This shows why $M_w$ alone does not saturate. $M_w$ observes no wave of any period——it observes $M_0$ itself, and so has no period to compare against.
Check 6.3: We measure the span the single word “magnitude” covers.
| $M$ | $E$ | |
|---|---|---|
| Laboratory microfracture | $-2.0$ | $63.096$ J |
| $1960$ Chile | $9.5$ | $1.1220\times10^{19}$ J |
| Ratio | $11.5$ | $\mathbf{1.7783\times10^{17}}\times$ |
Observation 6.4 One word covers a factor of $\mathbf{1.78\times10^{17}}$. No single scale can measure such a span, so having four is natural——the problem is not that they are separate but that one fails to say which was used.
Observation 6.5 Paper 117 treated the power law between the number and the size of earthquakes. This paper treats the instrument that measures the size——the material is the same earthquakes, and the questions are orthogonal.
7. Roll Call and Closing
- $1$ in $M$ is $31.6228\times$ (Observation 1.2): $2$ is exactly $1000\times$.
- Observed periods of the four scales (Check 2.1): $0.1$ / $1$ / $20$ s / none.
- $m_b$ gives one $5623$rd (Check 3.1): It calls Tohoku moderate.
- $M_S$ still $7.9433\times$ short (Check 3.1): The content of a $0.6$ difference.
- Same for four great earthquakes (Check 3.4): But not monotonically.
- Small earthquakes: $1.4125\times$ (Check 4.1): Inside the range nothing breaks.
- Calibration does not remove it (Observation 4.3): It arises from construction.
- Rupture $150$ s vs period $20$ s (Check 5.1): Ratio $7.5$.
- Model $1.7501$, observation $0.60$ (Check 5.3): An upper bound only.
- One word covers $1.78\times10^{17}$ (Check 6.3): From $M=-2$ to $9.5$.
Closing: When magnitude hits a ceiling for great earthquakes, it is not the earthquake that hits the ceiling. Measuring the $2011$ Tohoku earthquake with $m_b$ gives $6.5$, that is an energy estimated at one $\mathbf{5623}$rd——of the same event, $m_b$ says moderate and $M_w$ says fourth largest ever recorded. Even $M_S$ falls short by $\mathbf{7.9433}$——the difference is only $0.6$, but it is $0.6$ on a logarithmic scale. And it does not happen for small earthquakes——for $1995$ Southern Hyogo it stays at $1.4125$. The scale is not broken; it is being used outside its range. The cause is the duration of rupture——the Tohoku rupture lasted $\mathbf{150}$ s, $\mathbf{7.5}$ times the $20$ s that $M_S$ observes. A $20$ s wave carries only part of a $150$ s event. One thing separates them——whether the source duration exceeds that scale’s observed period. If it does, no amount of calibration removes the saturation. If it does not, the four scales agree well. $M_w$ alone escapes saturation not because it is superior but because it has no period to compare against.
References
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Reference
Yamagishi, Y. (2026). What Saturates Is Not the Earthquake but the Scale [Paper 282]. YOI: https://www.yamagishiyuuki.com/yoi/282
DOI: 10.5281/zenodo.22171087
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 282. This paper is a preprint and has not been peer reviewed.