How Much Bigger Is an Earthquake When the Magnitude Is 1 Higher?
― Two steps up means 1,000 times the energy. An M9 equals a thousand M7 quakes
When the news says "magnitude 7.3" and "9.0", the two numbers differ by less than 2. Yet in the energy they released, the gap is more than 300 times. That is because magnitude is a scale that grows by multiplying, not adding, with every step up.
A news flash comes on in the early evening. "The focus is at a depth of 10 kilometres. The magnitude, which shows the size of the quake, is estimated at 6.8."
A few days earlier, you heard about a magnitude 7.8 quake in a faraway country. 6.8 and 7.8 differ by only 1. It seems like "a bit bigger", more or less.
In fact, the 7.8 quake released about 32 times the energy of the 6.8 one. It is rare, in everyday life, to meet a scale where the look of the number and its real meaning are so far apart.
Why does a small difference in the number make such a huge gap? Two reasons
Magnitude counts the "number of digits" of the energy. One step up gives about 32 times the energy, and two steps give about 1,000 times. Even a difference of 0.2 means roughly double.
An earthquake is an event in which underground rock breaks and slips. The bigger the magnitude, the wider the broken surface, both lengthwise and across, and the larger the slip.
These two are the front and back of the same thing. The wider the surface and the bigger the slip, the more energy leaves as shaking. Let us look at them in turn.
One step up is 32 times, two steps up is 1,000 times
A magnitude number is not the energy itself. It is closer to a number that tells you "how many digits" the energy has.
The rule is this. When magnitude goes up by 2, the energy becomes exactly 1,000 times larger. Split that 1,000 into two steps, and each step is about 32 times. Multiply 32 by itself and you get about 1,000.
By this rule, a magnitude 9 quake releases in one go the energy of 1,000 magnitude 7 quakes. Compared with magnitude 5, it is a million times as much.
The 2011 Tohoku earthquake was magnitude 9.0. The 1995 Kobe earthquake (Hyogo-ken Nanbu) was magnitude 7.3. The difference is 1.7, but in energy it is about 350 times.
As magnitude rises, the broken surface grows enormously
So why is the energy so different? The source of an earthquake is the slipping of a fault, a crack in the rock underground. The bigger the magnitude, the wider the surface that slips.
Look at Figure 1. When you change the magnitude with the slider, the rough size of the slipped fault is drawn at the same scale as the map. At magnitude 5 it is a small rectangle a few kilometres across. At magnitude 7 it is about 50 kilometres long, and at 9 about 500 kilometres.
It is not only the area. The wider the surface, the larger the slip as well. In the Tohoku earthquake, a surface about 500 kilometres long and about 200 kilometres wide moved. In places the slip is estimated to have reached around 50 metres.
Because both "area" and "amount of slip" grow together, the energy grows by multiplication. That is what lies behind a one-digit difference becoming a huge gap.
Magnitude is the size of the earthquake itself. One quake gets one value. Seismic intensity is how strongly each place shook. Even for the same quake, a town near the focus and a town far away have different intensities.
Think of a light bulb. Magnitude is the brightness of the bulb (its wattage), and intensity is how bright it is on your desk. As Figure 2 shows, even a small quake shakes hard if it is right beneath you. Even a big quake shakes only a little if it is far away.
For the Tohoku earthquake, the first magnitude announced by the Japan Meteorological Agency was 7.9. It was later revised to 8.8, then 9.0. With the usual method, the number becomes hard to raise once a quake passes a certain size. This is called "saturation".
On a scale of 32 times per step, a gap of 1.1 is about 45 times in energy. From this experience, work is under way to improve ways of quickly estimating the size of huge earthquakes.
Summary
Magnitude is a "multiplying scale": one step up gives about 32 times the energy, and two steps give 1,000 times. Behind this is the fact that the bigger the quake, the wider the slipping fault surface and the larger the slip. Next time you hear "6.8" and "7.8" on the news, try reading it as "32 times".
One step of magnitude is not a difference of 1.
It is a difference of 32 times, and two steps make it 1,000 times.
For how small the shaking looks on a planet-wide scale, see If everyone in the world jumped at once, would the ground shake? For the shaking that follows a big quake, see Why do aftershocks go on for days? For why the same quake shakes different places differently, see Why does the ground shake more than in the next town? For an energy comparison with the sky, try The energy of one lightning strike.
- Open the earthquake information page of the Japan Meteorological Agency, pick two recent quakes, and copy down their magnitudes.
- Find the difference between the two. Using "0.2 is about 2 times, 1 is about 32 times, 2 is 1,000 times" as a guide, estimate how many times the energy differs.
- Compare the "maximum intensity" in the same list too. A quake with a large magnitude but small intensity is most likely far away or deep.
If you enter the same number with the slider in Figure 1, you also see the rough size of the fault. With a calculator, you can check by multiplying by 32 several times.
For those who want more ― terms, formulas, and links to textbooksEach part is labelled by level, from junior high science to university specialist courses
- Junior highCovered in junior high school science
- High schoolCovered in high school Earth science and mathematics (logarithms)
- High school+Advanced high school material, or textbook sidebars
- UniversityNot taught in high school; specialist university material (seismology)
- ResearchNot taught even at university as settled fact; researchers are studying it now
Junior highTerms: this phenomenon has names
- Magnitude: a number showing the size of the earthquake itself. One quake gets one value.
- Seismic intensity: the strength of shaking at a place. In Japan it is shown on a 10-step scale from 0 to 7.
- Fault: the surface where underground rock broke and slipped. Earthquakes happen when a fault slips.
- Focus (hypocentre): the underground point where the fault slip began. The spot on the ground directly above it is called the epicentre.
Junior highHigh schoolCheck with a formula: the energy gap between magnitude 7 and 9
The relation between the energy released as shaking and the magnitude is given by the formula below. The number of digits of the energy grows by 1.5 for each step of magnitude.
| In symbols | log₁₀ E ＝ 4.8 ＋ 1.5 × M |
| In words | Number of digits of the energy (joules) ＝ 4.8 ＋ 1.5 × magnitude |
| Where it comes from | An empirical formula (by Gutenberg and Richter) found by estimating the energy carried by seismic waves from the shaking records of many earthquakes and setting it beside magnitude. So when M differs by 2, the digits differ by 3 and the energy is 1,000 times larger |
| Symbol | Meaning and unit |
| E | Energy released as shaking (seismic waves). The unit is the joule |
| M | Magnitude (a number without a unit) |
| log₁₀ | A symbol meaning "how many times 10 is multiplied" (common logarithm) |
| The two quakes compared | Magnitude 7.0 and 9.0 |
| 10 to the power 1.5 | About 31.6 |
| Electricity one Japanese household uses in a year | About 4,000 kilowatt-hours, roughly 14 billion joules |
| Difference in magnitude | 9.0 − 7.0 ＝ 2.0 |
| Difference in digits | 1.5 × 2.0 ＝ 3.0 |
| Energy ratio (multiply 10 three times) | 10 × 10 × 10 ＝ 1000 |
| Ratio for one step of magnitude | 10 to the power 1.5 ≒ 31.6 (about 32 times) |
| Digits for magnitude 7 (first part) | 1.5 × 7 ＝ 10.5 |
| Digits for magnitude 7 (second part) | 4.8 + 10.5 ＝ 15.3 |
| Energy of magnitude 7 | 10 to the power 15.3 is about 2 quadrillion joules (2,000 trillion joules) |
| Years of household electricity (in units of 100 million joules) | 20000000 ÷ 140 ≒ 142857 |
| Gap between Kobe (7.3) and Tohoku (9.0) | 9.0 − 7.3 ＝ 1.7 |
| Difference in digits | 1.5 × 1.7 ＝ 2.55 |
| Energy ratio | 10 to the power 2.55 is about 355 (about 350 times) |
The energy one magnitude 7 quake gives off as shaking is equal to the electricity used by about 140,000 households in a year. For magnitude 9, it is 1,000 times that. Note, though, that only the part that left as shaking is counted here. The part spent on crushing and heating rock is not included.
High schoolHigh school+Fault length and magnitude
High schoolIn Earth science you learn that one more unit of magnitude means about 32 times the energy, and two more means 1,000 times. This is the formula above restated: the digits grow by 1.5 per step. Using logarithms from mathematics, this relation can be drawn as one straight line.
High school+Fault length also has an empirical formula. One commonly used rule of thumb is: digits of length ＝ 0.5 × M − 1.8 (length in kilometres). For magnitude 7, 0.5 × 7 ＝ 3.5, minus 1.8 gives 1.7, and 10 to the power 1.7 is about 50 kilometres. Figure 1 is drawn with this rule. Real earthquakes can differ from it by a factor of several.
UniversityMoment magnitude and seismic moment
What is used today to express the size of huge earthquakes is the moment magnitude (symbol Mw). Its basis is the seismic moment, given by the rigidity of the rock × the fault area × the amount of slip. It is exactly the product of "area" and "slip" in the main text.
The Japan Meteorological Agency magnitude, found from the amplitude of shaking, is hard to raise for huge earthquakes. This is called magnitude saturation. The seismic moment is fixed directly by the size of the fault, so it does not saturate. As for the relation between size and frequency, the Gutenberg-Richter law is known: quakes one magnitude smaller occur about 10 times more often.
📖 Derivations and further reading: Magnitude (Wikipedia, Japanese) ／ Moment magnitude (Wikipedia, Japanese)
ResearchWhat is still not clear
- How big can one get? The largest ever observed is the 1960 Chile earthquake, given as moment magnitude 9.5. Fault length is limited, so there seems to be an upper limit, but its value is unknown.
- Is the final size already set when the slip begins? Whether the first few seconds can tell if a small rupture will grow into a huge earthquake is still debated.
- The share that becomes shaking. Of the energy released by the slip, only a part leaves as seismic waves. Most of the rest goes into heat and breaking rock, but the share differs from quake to quake and is poorly understood.
So the content of this article too is "an explanation as far as we know now". Please note that both the energy formula and the fault length formula are rules of thumb averaged over many earthquakes.
Links to textbooks (by level)
| Level | Subject and unit | Where in this article |
|---|---|---|
| Junior high | Grade 7 science, "Changes of the Earth (earthquakes)" | Magnitude vs. intensity, faults |
| High school | Basic Earth Science / Earth Science "Earthquakes", Mathematics II "Logarithmic functions" | 32 times for 1, 1,000 times for 2 |
| High school+ | Advanced Earth Science | Empirical formula for fault length and magnitude |
| University | Seismology | Seismic moment, moment magnitude, saturation |
| Research | Studies of how earthquakes begin | Maximum size, prediction from the first few seconds |
| ― | Link to daily life | Reading news numbers as "times" |
- Magnitude (Wikipedia, Japanese)
- Moment magnitude (Wikipedia, Japanese)
- National Research Institute for Earth Science and Disaster Resilience (防災科学技術研究所), J-SHIS Japan Seismic Hazard Information Station
- Utsu, T., Jishingaku (地震学, Seismology), 3rd edition, Kyoritsu Shuppan (共立出版)
- Gutenberg, B. and Richter, C. F., Magnitude and energy of earthquakes, Annali di Geofisica (1956)
* This article is a popular-science explanation for general readers. The figures given are rough guides and estimates to help you understand how things work. For the latest information on earthquake size and shaking, check official announcements such as those of the Japan Meteorological Agency.