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‎docs/lectures/01_source_and_wave.md‎

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@@ -639,12 +639,6 @@ We have $\tau_{\text{min}} \simeq 400$s, $\tau_{\text{max}} \simeq 600$; and ass
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* $\tau_d = \frac{1}{2} (\tau_\text{max} + \tau_\text{min}) = 500$ s
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* $v_r = L/\tau_d = 1260/500 = 2.52$ km/s
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---
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### The Haskell fault model
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![20250401002831](https://raw.githubusercontent.com/zhuwq0/images/main/20250401002831.png)
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How to quantify the size of an earthquake?
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* For historical reasons the most well-known measure of earthquake size is the earthquake magnitude.
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* Derived from the largest amplitude that is recorded on seismograms.
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* There are now many different types of magnitude scales, but all are connected in some way to the earliest definitions of magnitude.
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---
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### Richter Magnitude (Local magnitude $M_L$)
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The original magnitude scale is based on the maximum amplitude recorded on a standard Wood-Anderson torsion seismograph.
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$$
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M_L = \log_{10} A(X) - \log_{10} A_0(X)
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$$
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$A_0$: the amplitude of the reference event
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$X$: the epicentral distance
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![20250402232519 bg right:50% fit](https://raw.githubusercontent.com/zhuwq0/images/main/20250402232519.png)
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----
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### Richter Magnitude (Local magnitude $M_L$)
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An approximate empirical formula has been derived for $\log_{10} A_0(X)$ at different ranges.
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The local magnitude can be calculated by
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$$
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M_L = \log_{10} A(X) + 2.56 \log_{10} X - 1.67
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$$
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where $A(X)$ is the displacement amplitude in microns (10$^{-6}$ m) and X is in kilometers.
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* Events below about $M_L 3$ are generally not felt
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* Significant damage to structures in California begins to occur at about $M_L 5.5$
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* A $M_L 6.0$ earthquake implies amplitude 100 times greater than a $M_L 4.0$ event.
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### Global earthquakes: body wave magnitude $m_b$
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$$
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m_b = \log_{10} (A/T) + Q(h, \Delta)
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$$
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where A is the ground displacement in microns, T is the dominant period of the measured waves, $\Delta$ is the epicentral distance in degrees, and Q is an empirical function of range and event depth h.
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* Why $A/T$?
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* h?
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### Global earthquakes: surface wave magnitude $M_s$
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For Rayleigh waves on vertical instruments:
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$$
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M_s = \log_{10} (A/T) + 1.66 \log_{10} \Delta + 3.30
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$$
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Since the strongest Rayleigh wave arrivals are generally at a period of 20 s, this expression is often written as
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$$
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M_s = \log_{10} A_{20} + 2.46 \log_{10} \Delta + 2.0
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$$
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* Note that this equation is applicable only to shallow events
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* surface wave amplitudes are greatly reduced for deep events.
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---
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### Magnitude saturation
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![bg right:70% fit](https://raw.githubusercontent.com/zhuwq0/images/main/20250402233819.png)
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---
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### The Haskell fault model
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![20250331235242 height:250px](https://raw.githubusercontent.com/zhuwq0/images/main/20250331235242.png)
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![20250401002831 height:300px](https://raw.githubusercontent.com/zhuwq0/images/main/20250401002831.png)
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---
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### Source spectra
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A boxcar pulse in the time domain produces a sinc function in the frequency domain
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A boxcar pulse in the time domain produces a sinc function in the frequency domain: $\text{sinc}(x) = \frac{\sin(x)}{x}$
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![20250401092155 height:400px](https://raw.githubusercontent.com/zhuwq0/images/main/20250401092155.png)
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<div style="display: flex; justify-content: center;">
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<img src="https://raw.githubusercontent.com/zhuwq0/images/main/20250401092155.png" width="68%">
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<img src="https://raw.githubusercontent.com/zhuwq0/images/main/20250403000646.png" width="30%">
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</div>
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We can approximate $\left|\operatorname{sinc} x\right| \simeq 1$ for $x<1$ and $1/x$ for $x>1$
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$$
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\log |A(\omega)|-G = \log \left(M_0\right)+\log \left|\operatorname{sinc}\left(\omega \tau_r / 2\right)\right|+\log \left|\operatorname{sinc}\left(\omega \tau_d / 2\right)\right|
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$$
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$$
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\begin{aligned} \log |A(\omega)|-G & =\log M_0, & & \omega<2 / \tau_d \\ & =\log M_0-\log \frac{\tau_d}{2}-\log \omega, & & 2 / \tau_d<\omega<2 / \tau_r \\ & =\log M_0-\log \frac{\tau_d \tau_r}{4}-2 \log \omega, & & 2 / \tau_r<\omega\end{aligned}
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$$
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### The amplitude spectrum for the Haskell fault model.
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![bg right:65% fit](https://raw.githubusercontent.com/zhuwq0/images/main/20250401092815.png)
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$$
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\begin{aligned} \log |A(\omega)|-G & =\log M_0, & & \omega<2 / \tau_d \\ & =\log M_0-\log \frac{\tau_d}{2}-\log \omega, & & 2 / \tau_d<\omega<2 / \tau_r \\ & =\log M_0-\log \frac{\tau_d \tau_r}{4}-2 \log \omega, & & 2 / \tau_r<\omega\end{aligned}
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$$
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![bg right:55% fit](https://raw.githubusercontent.com/zhuwq0/images/main/20250401092815.png)
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---
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### Magnitude saturation
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![bg right:70% fit](https://raw.githubusercontent.com/zhuwq0/images/main/20250402233819.png)
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---
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### Moment magnitude $M_w$
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The saturation of the and scales for large events helped motivate development of the moment magnitude $M_w$
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$$
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M_w = \frac{2}{3} (\log_{10} M_0 - 9.1)
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$$
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where is the moment measured in N-m.
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* The advantage of the scale is that it is clearly related to a physical property of the source and it does not saturate for even the largest earthquakes.
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* One unit increase in $M_w$ corresponds to a $10^{3/2} \approx 32$ times increase in the moment.
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* A $M_w 7$ earthquake releases about 1000 times more energy than a $M_w 5$ event.
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---
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### Magnitude as a function of moment
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![20250402234333 height:450px](https://raw.githubusercontent.com/zhuwq0/images/main/20250402234333.png)
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[USGS Magnitude Types](https://www.usgs.gov/programs/earthquake-hazards/magnitude-types); [Latest earthquake](https://earthquake.usgs.gov/earthquakes/eventpage/us7000pn9s/origin/magnitude)
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![20250402235133 bg fit](https://raw.githubusercontent.com/zhuwq0/images/main/20250402235133.png)
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![20250402235213 bg fit](https://raw.githubusercontent.com/zhuwq0/images/main/20250402235213.png)
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### The intensity scale
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The local strength of ground shaking as determined by damage to structures and the perceptions of people who experienced the earthquake.
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* One earthquake can have different intensities at different locations.
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[USGS Latest Earthquakes](https://earthquake.usgs.gov/earthquakes/map/?extent=26.07652,-136.80176&extent=49.92294,-98.48145&range=month&listOnlyShown=true&settings=true&search=%7B%22name%22:%22Search%20Results%22,%22params%22:%7B%22starttime%22:%222020-12-02%2000:00:00%22,%22endtime%22:%222023-12-09%2023:59:59%22,%22maxlatitude%22:37.642,%22minlatitude%22:37.588,%22maxlongitude%22:-122.344,%22minlongitude%22:-122.412,%22orderby%22:%22time%22%7D%7D)
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### The Mercalli intensity scale (MMI)
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![20250403001052 height:500px](https://raw.githubusercontent.com/zhuwq0/images/main/20250403001052.png)
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### JMA intensity scale
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![bg right:60% fit](https://raw.githubusercontent.com/zhuwq0/images/main/202504030012755.png)
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