Earthquakes and Seismology

Learning Objectives

  • Separate source initiation, rupture extent, hypocentral path length and epicentral distance.
  • Compare body- and surface-wave particle motions without ranking damage from wave names alone.
  • Derive an S-P range from stated speeds, account for depth and test location consistency.
  • Distinguish seismic moment, moment magnitude, calibrated local amplitude, approximate energy scaling and intensity.
  • Interpret PGA, transient response spectra, damping and period-specific site response.
  • Assess geological hazards conditionally using ground conditions, exposure and project performance requirements.

Sources, Ruptures and Distance

Source geometry and a stated propagation model are necessary before converting an arrival pick into distance.

Earthquake Sources

An earthquake produces seismic waves through a sudden release of stored energy, commonly during fault slip. Strain can accumulate where fault motion is resisted; rupture can then propagate over a finite fault surface. Tectonic earthquakes occur at plate boundaries and within plates. Volcanic processes and human activities such as fluid injection, extraction or mining can also generate seismicity. Not every recorded seismic signal is a tectonic earthquake.

Normal, reverse and strike-slip describe slip geometry. They do not prescribe a single magnitude, depth or damage outcome. Large subduction-interface thrust ruptures can be megathrust earthquakes; not every reverse fault is a subduction interface.

Hypocenter and Epicenter

The hypocenter, or focus, is the estimated three-dimensional initiation point of an earthquake rupture. It can lie in the crust or deeper in a subducting slab. The epicenter is its projection onto the reference surface. A large rupture radiates from an extended region after initiation; the epicenter is neither the whole source nor necessarily the site of greatest damage.

Source Path and Surface Distance

Original synthetic flat-section geometry: horizontal distance 40 km, depth 30 km and straight source-to-station path 50 km. The initiation point does not represent the full extent of a real rupture. Curved rays and unequal station elevations are outside this model.

A surface station is 40 kilometres horizontally from the epicenter above a 30-kilometre-deep source, making a 50-kilometre straight source path

Wave Propagation and Particle Motion

The disturbance travels through material while individual particles move about reference positions.

Body and Surface Waves

  • P waves produce compression and dilation with particle displacement parallel to propagation. Mechanical compressional waves can propagate through solids and fluids.
  • S waves have particle displacement perpendicular to propagation. A fluid does not sustain the shear rigidity needed for a propagating bulk S wave. P and S speeds depend on material properties; fixed global speed ranges are not site measurements.
  • Love waves have horizontal transverse motion in a surface-guided wavefield.
  • Rayleigh waves combine horizontal and vertical motion, often represented by near-surface elliptical particle paths in a vertical propagation plane.

Real records combine direct, reflected, refracted, converted and surface phases. Both body and surface waves can contribute damaging motion. Amplitude, frequency content, duration, source/path effects and structural response matter; a wave name alone does not predict damage.

Two Surface-Wave Motion Planes

Qualitative original schematics, not measured displacement paths or a universal orbit-rotation rule. The Love panel is a plan and the Rayleigh panel a vertical section. Layering and depth affect the actual wavefield.

Love motion is horizontal and transverse in plan, while Rayleigh motion has horizontal and vertical components in a propagation section

Controlled Body-Wave Snapshot

A kinematic displacement field for comparing orientation, not a prediction of ground motion or damage.

u(x0,ϕ)=Asin⁡ ⁣[2π(x0λ−ϕ)].u(x_0,\phi)=A\sin\!\left[2\pi\left(\frac{x_0}{\lambda}-\phi\right)\right].P:(x,y)=(x0+u,y0),S:(x,y)=(x0,y0+u).\begin{aligned} \text{P:}\quad &(x,y)=(x_0+u,y_0),\\ \text{S:}\quad &(x,y)=(x_0,y_0+u). \end{aligned}

Variables

SymbolDescriptionUnit
x0,y0x_0,y_0Particle reference coordinatesm
A,λA,\lambdaDisplacement amplitude and reference-coordinate wavelengthm
ϕ\phiControlled phase in cycles; for a specified period it could represent time divided by perioddimensionless

Interactive Exploration

Advance phase and follow the marked material particle. Switch between P and S at the same phase, then set amplitude to zero. The reference particles and spatial scales remain visible.

Wave propagation and particle motion

Concept and model scope

Advance phase to move the disturbance to the right. Compare one marked particle's motion with the propagation direction, then change between P and S waves without changing the displacement profile.

Model and limits: A controlled kinematic snapshot in a solid: u=A sin[2π(x₀/λ−φ)]. Phase φ is a fraction of a cycle, not elapsed seconds; no wave speed, frequency, attenuation, stress or damage is calculated. P displacement is parallel to +x; the illustrated S polarization is transverse. Coordinates and displacement use metres on equal spatial scales, with no geometric exaggeration. Controls: amplitude 0–1 m (0.05 m steps), wavelength 10–40 m (1 m steps), phase 0–1 cycle (0.025-cycle steps). Reference rows are synthetic particles, not mapped rock layers.

Controls

Wave type

In a P wave particles move parallel to propagation. In the illustrated S wave they move perpendicular to propagation. This solid-medium model does not transmit an S wave through a fluid or compare P/S speeds.

1.00 m

Displacement amplitude

Maximum magnitude of particle displacement, not travel distance or measured site shaking. Zero leaves every current particle at its equilibrium point.

Range: 0–1 m. Step: 0.05 m.

20 m

Wavelength

Reference-coordinate spacing between equal phases. The model changes the spatial pattern without assigning a frequency or propagation speed.

Range: 10–40 m. Step: 1 m.

0.250 cycles

Phase

A controlled cycle fraction in x₀/λ−φ. Increasing phase translates the pattern in +x; each material particle oscillates about its own reference point. One full cycle returns the same snapshot.

Range: 0–1 cycles. Step: 0.025 cycles.

Increasing phase advances the disturbance →P: particle motion parallel to propagation-4-2024010203040Along-propagation coordinate x (m); transverse ticks in m○ reference● currentMarked: (10,0) m

The marked particle moves parallel to propagation.

Marked x displacement: 1.000 m

Marked transverse displacement: 0.000 m

Material motion is oscillation about reference positions. The travelling pattern is not transport of these particles across the scene; neither amplitude nor wave type determines building damage.

Arrival Times and a Conditional Range

An S-P delay removes the unknown origin time from this simple two-speed calculation.

Straight-Path S-P Range

Assume one homogeneous medium, the same straight source-to-station path for P and S, and speeds satisfying 0 < Vs < Vp.

tP=LVP,tS=LVS,Δt=TS−TP=L(1VS−1VP).t_P=\frac{L}{V_P},\qquad t_S=\frac{L}{V_S},\qquad \Delta t=T_S-T_P =L\left(\frac{1}{V_S}-\frac{1}{V_P}\right).L=kΔt,k=VPVSVP−VS.L=k\Delta t,\qquad k=\frac{V_PV_S}{V_P-V_S}.

Variables

SymbolDescriptionUnit
LLStraight hypocentral source-to-station path lengthkm when speeds use km/s
tP,tSt_P,t_STravel durations from the common origins
TP,TST_P,T_SObserved arrival clock times at one stationtime on the same clock
VP,VS,kV_P,V_S,kAssumed P/S speeds and their derived multiplierkm/s

There Is No Universal Multiplier

For VP=6V_P=6 and VS=3 km/sV_S=3\ \text{km/s}, the multiplier is k=6 km/sk=6\ \text{km/s}. For VP=6V_P=6 and VS=3.5 km/sV_S=3.5\ \text{km/s}, it is 8.4 km/s8.4\ \text{km/s}. The multiplier is not one wave's speed, a universal constant or a unique indicator of density.

Use consistent dates when arrival times cross midnight. Adding the same clock offset to both picks leaves their difference unchanged, but an incorrect phase pick or time gap changes the inferred range. A zero delay gives zero path length in this model; it does not demonstrate a real event at the station.

Depth Correction and Timing Sensitivity

Assume a flat common station surface and a known or explicitly assumed depth h.

D=L2−h2,L≥h≥0.D=\sqrt{L^2-h^2},\qquad L\ge h\ge0.Lmin⁡=kmax⁡(0,Δt−ε),Lmax⁡=k(Δt+ε).L_{\min}=k\max(0,\Delta t-\varepsilon),\qquad L_{\max}=k(\Delta t+\varepsilon).

Variables

SymbolDescriptionUnit
D,hD,hHorizontal epicentral distance and source depth below the station reference surfacekm
ε\varepsilonSupplied bound on the S-P timing error; speeds/depth held fixed for this calculations

Impossible Inputs and Uncertainty

If LL is shorter than hh, there is no real horizontal radius under these assumptions. Report the inconsistency instead of clamping the radius to zero. Equality gives a zero-radius surface point.

For a fully valid interval with Lmin⁡≥hL_{\min}\ge h, transform both path bounds through the square root to obtain horizontal-distance bounds. They need not be symmetric about the central value. If a bound falls below the assumed depth, part of the timing interval is incompatible with that depth; do not silently interpret it as a valid distance interval.

Real location also requires appropriate velocity structure, station coordinates/elevations, phase identification and uncertainties. Straight local paths are not a global Earth travel-time model.

Circle Geometry and Location Consistency

Three distances only support a location if the original distance equations agree.

A Candidate from Three Surface Ranges

For known horizontal radii and noncollinear stations, subtract the first squared equation from the other two, solve for a candidate, then check all original ranges.

(x−xi)2+(y−yi)2=ri2,i=1,2,3.(x-x_i)^2+(y-y_i)^2=r_i^2,\qquad i=1,2,3.2(xj−x1)x+2(yj−y1)y=r12−rj2+xj2+yj2−x12−y12,j=2,3.2(x_j-x_1)x+2(y_j-y_1)y =r_1^2-r_j^2+x_j^2+y_j^2-x_1^2-y_1^2, \qquad j=2,3.qi=(x−xi)2+(y−yi)2−ri,qRMS=q12+q22+q323.q_i=\sqrt{(x-x_i)^2+(y-y_i)^2}-r_i,\qquad q_{\mathrm{RMS}}=\sqrt{\frac{q_1^2+q_2^2+q_3^2}{3}}.

Variables

SymbolDescriptionUnit
xi,yi,rix_i,y_i,r_iKnown station coordinates and inferred horizontal radius, in one local coordinate systemkm
x,y,qix,y,q_iAlgebraic candidate coordinates and signed distance residualskm

Trilateration Is a Simplified Location Model

One circle leaves many possible positions. Two circles can leave two intersections, one or none. A third noncollinear station can resolve an ideal consistent case; it does not guarantee a common intersection for arbitrary data. Collinear or nearly collinear geometry can leave ambiguity or make the solution sensitive to errors.

Solving the subtracted equations gives an algebraic candidate even when the circles disagree. That point is not automatically a least-squares best fit or an acceptable location. Compare residuals with justified data/model uncertainties. Distance-circle location is geometrically trilateration, though introductory earthquake lessons often call it triangulation. Operational earthquake location uses a network and fits source position, depth and origin time using an Earth model.

Check Arrival Data before Reporting a Location

A simplified horizontal-circle workflow with an explicitly supplied depth and velocity model. Real network location fits source position, depth and origin time and reports uncertainty.

Review phase picks, dates, station reference coordinates, assumed straight paths and speeds. If path length is shorter than depth, station geometry is degenerate or original range residuals exceed justified uncertainty, revise the inputs/model rather than labelling an algebraic candidate an exact location. The interactive model's 0.01 km tolerance is only a teaching convention.

Check Arrival Data before Reporting a LocationA simplified horizontal-circle workflow with an explicitly supplied depth and velocity model. Real network location fits source position, depth and origin time and reports uncertainty.. Review phase picks, dates, station reference coordinates, assumed straight paths and speeds. If path length is shorter than depth, station geometry is degenerate or original range residuals exceed justified uncertainty, revise the inputs/model rather than labelling an algebraic candidate an exact location. The interactive model's 0.01 km tolerance is only a teaching convention.. Start: locate a synthetic event → P/S picks, dates and station coordinates; P/S picks, dates and station coordinates → State velocity paths and depth assumption; State velocity paths and depth assumption → Picks and model basis adequate?; Picks and model basis adequate? — Yes → Compute S-P paths and horizontal radii; Picks and model basis adequate? — No → Hold: obtain or revise needed data; Compute S-P paths and horizontal radii → L ≥ h and station geometry usable?; L ≥ h and station geometry usable? — Yes → Solve candidate; check original range residuals; L ≥ h and station geometry usable? — No → Review picks, depth, velocities and geometry; Solve candidate; check original range residuals → Residuals consistent with justified uncertainty?; Residuals consistent with justified uncertainty? — Yes → Report conditional location and uncertainty; Residuals consistent with justified uncertainty? — No → Review picks, depth, velocities and geometry; Review picks, depth, velocities and geometry → State velocity paths and depth assumption; Report conditional location and uncertainty → End: retain model limits

Start: locate a synthetic event → P/S picks, dates and station coordinates; P/S picks, dates and station coordinates → State velocity paths and depth assumption; State velocity paths and depth assumption → Picks and model basis adequate?; Picks and model basis adequate? — Yes → Compute S-P paths and horizontal radii; Picks and model basis adequate? — No → Hold: obtain or revise needed data; Compute S-P paths and horizontal radii → L ≥ h and station geometry usable?; L ≥ h and station geometry usable? — Yes → Solve candidate; check original range residuals; L ≥ h and station geometry usable? — No → Review picks, depth, velocities and geometry; Solve candidate; check original range residuals → Residuals consistent with justified uncertainty?; Residuals consistent with justified uncertainty? — Yes → Report conditional location and uncertainty; Residuals consistent with justified uncertainty? — No → Review picks, depth, velocities and geometry; Review picks, depth, velocities and geometry → State velocity paths and depth assumption; Report conditional location and uncertainty → End: retain model limits

  • Start: locate a synthetic event: terminator
  • P/S picks, dates and station coordinates: input output
  • State velocity paths and depth assumption: process
  • Picks and model basis adequate?: decision
  • Hold: obtain or revise needed data: terminator
  • Compute S-P paths and horizontal radii: process
  • L ≥ h and station geometry usable?: decision
  • Review picks, depth, velocities and geometry: process
  • Solve candidate; check original range residuals: process
  • Residuals consistent with justified uncertainty?: decision
  • Report conditional location and uncertainty: document
  • End: retain model limits: terminator

Interactive Exploration

Compare the default consistent circles with zero delays, a changed single arrival and an impossible assumed depth. With all delays at 12.5 s and depth 45 km, the horizontal circles return to the default geometry.

Arrival circles and location consistency

Concept and model scope

Convert three S-P delays to straight source ranges, account for the assumed depth, and inspect whether the resulting surface circles actually support their algebraic candidate.

Model and limits: Synthetic local flat coordinates in km, surface stations and homogeneous straight paths with fixed Vp=6 and Vs=3 km/s (k=6 km/s). Three noncollinear stations are A(0,0), B(72,0), C(0,96). The assumed source depth is common to all. Controls: each S-P delay 0–20 s in 0.5 s steps, depth 0–60 km in 1 km steps. This solves differences of squared circle equations, not a least-squares location or a real Earth velocity model. A maximum absolute range residual ≤0.01 km is only a teaching consistency tolerance, not measurement accuracy. All allowed circles fit the fixed equal-scale map; there is no hidden target.

Controls
10.0 s

Station A S-P delay

S arrival minus P arrival on this station's clock. At the fixed model speeds the straight hypocentral path is 6 times this delay in km; it is not automatically the horizontal circle radius.

Range: 0–20 s. Step: 0.5 s.

10.0 s

Station B S-P delay

S arrival minus P arrival on this station's clock. At the fixed model speeds the straight hypocentral path is 6 times this delay in km; it is not automatically the horizontal circle radius.

Range: 0–20 s. Step: 0.5 s.

10.0 s

Station C S-P delay

S arrival minus P arrival on this station's clock. At the fixed model speeds the straight hypocentral path is 6 times this delay in km; it is not automatically the horizontal circle radius.

Range: 0–20 s. Step: 0.5 s.

0 km

Assumed source depth

Known or assumed common vertical depth below the station reference surface. Horizontal radius is √(L²−h²). If the inferred path L is shorter than h, the input combination is impossible in this model and no candidate is drawn.

Range: 0–60 km. Step: 1 km.

Model speeds: Vp = 6 km/s; Vs = 3 km/s. Each circle uses horizontal radius √(L² − h²), with L = 6 × S-P delay.

Synthetic local plan; north is +yA solid · B dashed · C dotted50 kmX: algebraic candidate; check residuals-100-10000100100200200A (0, 0)B (72, 0)C (0, 96)XEast x (km)

All three distances agree within the 0.01 km teaching tolerance.

Station A: path 60.00 km; horizontal radius 60.00 km; range residual 0.000 km

Station B: path 60.00 km; horizontal radius 60.00 km; range residual 0.000 km

Station C: path 60.00 km; horizontal radius 60.00 km; range residual 0.000 km

Candidate east: 36.000 km

Candidate north: 48.000 km

RMS range residual: 0.000 km

Maximum absolute residual: 0.000 km

Residual = candidate-to-station distance minus the inferred horizontal radius. An algebraic intersection of subtracted equations need not satisfy any original circle. Real location uses additional data, velocity/depth estimation and uncertainty analysis.

Moment, Magnitude, Amplitude and Intensity

These quantities describe different aspects of an event and its effects.

Seismic Moment and Moment Magnitude

Use shear modulus in pascals, slipping fault area in square metres, mean slip in metres and seismic moment in N·m. The magnitude constant depends on these units.

M0=μAfδ‾,Mw=23[log⁡10 ⁣(M0 in N⋅m)−9.1].M_0=\mu A_f\overline{\delta},\qquad M_w=\frac{2}{3}\left[\log_{10}\!\left(M_0\text{ in N·m}\right)-9.1\right].M0,2M0,1=101.5(Mw,2−Mw,1).\frac{M_{0,2}}{M_{0,1}}=10^{1.5(M_{w,2}-M_{w,1})}.

Variables

SymbolDescriptionUnit
μ\muShear modulus, not an intact failure strengthPa
Af,δ‾A_f,\overline{\delta}Fault area that slipped and area-averaged slipm² and m
M0,MwM_0,M_wScalar seismic moment and moment magnitude estimateN·m and dimensionless magnitude

What the Logarithmic Comparisons Mean

Moment measures source size through rigidity, area and slip; it is not radiated energy even though both can be expressed with dimensions of force times length. Magnitude estimates can be revised and have uncertainty. Local and other magnitude types still have appropriate measurement uses; they are not simply obsolete.

For introductory comparisons, an explicitly approximate energy scaling is E2/E1≈101.5ΔME_2/E_1\approx10^{1.5\Delta M}, about 31.631.6 per unit and 10001000 for two units. Repeatedly squaring a rounded factor 3232 gives 10241024, an avoidable rounding artifact. Actual radiated energy is not determined exactly by moment magnitude.

A calibrated local amplitude-magnitude comparison at the same distance and instrument correction has ΔML=log⁡10(A2/A1)\Delta M_L=\log_{10}(A_2/A_1). Its tenfold amplitude rule does not mean that a unit increase in MwM_w gives ten times the PGA or a particular recorded amplitude at a site.

Specify the Scale Before Comparing Ratios

Analytical log-ratio relationships over a stipulated magnitude difference of 0–2. The moment relationship follows Mw; the energy curve is an introductory approximation, and the amplitude line applies to a calibrated local-amplitude scale. These are not site-PGA predictions.

Logarithm of moment ratio grows by 1.5 per unit moment-magnitude difference, while a separately calibrated local-amplitude ratio grows by one per local-magnitude difference

Intensity

Intensity describes location-specific observed shaking effects on people, objects, structures and the environment using a named scale. Modified Mercalli Intensity and the PHIVOLCS Earthquake Intensity Scale are different scales; Philippine intensity reports commonly use PEIS. Do not convert an intensity label directly into a unique magnitude, PGA or design force.

PGA and Transient Structural Response

A ground-motion peak is not the response of every structure exposed to that motion.

Ground Peak and a Rigid-Mass Limit

Specify the acceleration component. The rigid-mass calculation assumes the mass follows ground acceleration without flexible dynamic amplification.

PGA=max⁡t∣ag(t)∣,Frigid,max=m PGA,W=mg.\mathrm{PGA}=\max_t|a_g(t)|,\qquad F_{\mathrm{rigid,max}}=m\,\mathrm{PGA},\qquad W=mg.

Variables

SymbolDescriptionUnit
ag,m,ga_g,m,gChosen ground-acceleration component, ideal rigidly attached mass and gravitym/s², kg and 9.81 m/s² in examples

A Response Spectrum Is Not Steady-State Motion

A response spectrum records peak transient responses of a family of oscillators to a specified motion, over a stated period range and damping ratio. Displacement, relative velocity, absolute acceleration and pseudo-spectral acceleration are different response measures.

For an elastic oscillator, displacement spectrum SdS_d is the maximum magnitude of relative displacement. Define pseudo-spectral acceleration PSA=ωn2Sd\mathrm{PSA}=\omega_n^2S_d, where ωn=2π/Tn\omega_n=2\pi/T_n. PSA is not generally identical to the peak absolute acceleration spectrum, especially at other damping levels or long periods.

Pseudo-Acceleration and Relative Displacement

When the supplied PSA ordinate is expressed as a multiple s of gravity, use g to obtain SI acceleration.

Sd=PSA(Tn2π)2=s g(Tn2π)2.S_d=\mathrm{PSA}\left(\frac{T_n}{2\pi}\right)^2 =s\,g\left(\frac{T_n}{2\pi}\right)^2.

Variables

SymbolDescriptionUnit
Tn,Sd,sT_n,S_d,sNatural period, peak relative displacement and supplied PSA/g ratios, m and dimensionless
Read a Stipulated Spectral Ordinate

Synthetic reading exercise, not a code spectrum or the computed spectrum of a real accelerogram. Listed (period s, PSA/g) pairs are (0.1,0.4), (0.2,0.8), (0.5,0.8), (1,0.4), (2,0.2), (3,0.12). Lines interpolate these supplied values only; no extrapolation or compliance is implied.

Stipulated five-percent-damped pseudo-spectral acceleration rises to 0.8g at 0.2–0.5 seconds and declines to 0.4g at one second

Ground Motion Is an Input, Not a Completed Design

The synthetic ordinates above support calculation practice only. A flexible structure's demand depends on motion, damping, period, mode shapes, nonlinearity and performance criteria. A short period does not universally prescribe one acceleration, and a tall building does not universally have low demand.

Site-specific hazard assessment and applicable current project/code criteria govern real design spectra and structural/geotechnical checks. PGA times total mass is not a general building base-shear design method.

Site Response and Geological Hazards

Interpret the site, the motion and the exposed system together.

Site Effects Depend on Frequency and Shaking Level

Layering, stiffness, impedance contrasts, basin geometry, damping and nonlinear response can modify motion and duration. Soft deposits can amplify some frequency ranges and reduce others. An automatic three-to-five multiplier from “energy conservation,” equal epicentral distance, or a soil name is not a site-response analysis.

The ratio of two stipulated spectral ordinates must compare the same period, damping and response measure. For example, soil/rock PSA ratios of 0.24/0.30=0.80.24/0.30=0.8 at 0.2 s0.2\ \text{s} and 0.50/0.20=2.50.50/0.20=2.5 at 1.0 s1.0\ \text{s} demonstrate why one uniform factor is inadequate. These are synthetic supplied values, not measurements from a named basin.

A Site Ratio Depends on Period

Original synthetic comparison at 5% damping: rock/soil PSA values are 0.30/0.24g at 0.2 s and 0.20/0.50g at 1.0 s. No real site, transfer function, universal amplification factor or design spectrum is represented.

Two synthetic PSA comparisons show a soil-to-rock ratio below one at 0.2 seconds and above one at one second

Hazards and Conditional Responses

  • Shaking: assess frequency content, duration and system response; damping, ductility, isolation or other measures require project-specific design.
  • Surface displacement: locate and characterize active faults and deformation zones. Avoidance and applicable setbacks are important planning measures; do not invent one universal setback or assume that an isolation bearing solves ground offset.
  • Liquefaction-related deformation: assess susceptible saturated materials, cyclic demand, drainage, density/state, layering and consequences. Effects can include settlement, lateral spreading and buoyancy problems; none follows inevitably from the name “reclaimed sand.”
  • Slope movement: investigate geometry, strength, groundwater and shaking, including possible rockfall runout.
  • Tsunami: rapid displacement of water can result from suitable earthquakes, landslides or volcanic processes. Not every offshore earthquake generates a damaging tsunami. Use official current hazard information and site-specific assessment.

A Pore-Pressure Accounting Check

A simplified point-state calculation with fixed total normal stress, not a liquefaction-triggering or deformation model.

σv0′=σv−u0,σv′=σv−(u0+Δu),ru=Δuσv0′,σv0′>0.\sigma'_{v0}=\sigma_v-u_0,\qquad \sigma'_v=\sigma_v-(u_0+\Delta u),\qquad r_u=\frac{\Delta u}{\sigma'_{v0}},\quad \sigma'_{v0}>0.

Variables

SymbolDescriptionUnit
σv,u0,Δu\sigma_v,u_0,\Delta uAssumed total vertical stress, initial pore pressure and excess pore-pressure risekPa
σv′,ru\sigma'_v,r_uEffective vertical stress and excess-pressure ratio for that reference statekPa and dimensionless

Pore Water Already Contributes to Total Stress

Total stress is shared between the grain skeleton's effective stress and pore pressure. Water does not begin carrying stress only when shaking occurs. A pressure rise can reduce effective stress; this need not mean all strength instantly vanishes or that the soil becomes a literal liquid. The simplified ratio is not a universal acceptance threshold.

For reclaimed land or a basin project, combine hazard information with suitable ground investigation, groundwater data and system-specific performance requirements. Compare feasible treatment/foundation alternatives and verify their performance rather than prescribing compaction, piles or dampers from a short scenario alone.

Key Takeaways
  • A hypocenter is an initiation point; source path, source depth and surface distance are different quantities.
  • Body- and surface-wave motions contribute to complex site shaking; particle displacement differs from propagation.
  • Derive the S-P multiplier from stated speeds, account for depth, and report impossible or inconsistent data.
  • Check original distance residuals after solving for an algebraic circle candidate.
  • Mw derives from seismic moment; corrected local amplitude and approximate energy ratios require separate statements.
  • Name the intensity scale and keep it separate from measured motion and design demand.
  • Response spectra describe transient oscillator peaks at stated damping; PSA and peak absolute acceleration are distinct.
  • Evaluate frequency-dependent site effects and geological hazards with site/project evidence.

References