• from Kramer (1996:67-68)

Peak Ground Acceleration (PGA) is the maximum absolute acceleration recorded during an earthquake ground-motion time history. It is the most commonly used measure of ground-motion amplitude and is typically reported in units of g or cm/sec². Because acceleration records contain a large proportion of high-frequency energy, PGA is particularly sensitive to short-period ground motion and often correlates with the response of stiff, low-rise structures. Although PGA is a useful measure of shaking intensity, it provides little information about the frequency content or duration of motion. Consequently, two earthquakes with identical PGA values may produce very different levels of damage. Kramer notes that very high peak accelerations can occur for only a small fraction of a second and may therefore be less destructive than lower accelerations sustained over a longer duration. PGA is therefore most useful when supplemented by other measures of ground motion. In historical and archaeoseismic studies, PGA is often linked indirectly to intensity scales through empirical relationships.

Peak Ground Velocity (PGV) is the maximum absolute velocity recorded during an earthquake ground-motion time history, typically expressed in cm/sec. Because integration of acceleration filters out some of the highest-frequency components, PGV is less sensitive to high-frequency shaking than PGA and is often regarded as a better indicator of structural response at intermediate periods. Kramer notes that PGV is particularly useful for evaluating the performance of flexible structures such as tall buildings, bridges, and lifelines. In many cases PGV has been found to correlate more closely with damage than PGA, especially for structures whose natural periods are long enough to respond strongly to velocity pulses. PGV has also been widely correlated with macroseismic intensity, making it particularly valuable for relating instrumental observations to observed damage patterns.

Peak Ground Displacement (PGD) is the maximum absolute displacement recorded or calculated from a ground-motion time history. It is generally associated with the lowest-frequency and longest-period components of earthquake shaking. Unlike PGA and PGV, PGD is often difficult to determine accurately because displacement records are usually obtained by integrating accelerograms twice, making them highly sensitive to baseline errors, filtering choices, and long-period noise. Nevertheless, PGD is of particular importance in the near-fault environment where large permanent or quasi-permanent displacements may occur. Because it reflects long-period ground motion, PGD is especially relevant to the response of long-span bridges, pipelines, dams, and large flexible structures. In archaeoseismology, PGD may be one of the parameters most closely related to large coherent movements, tilting, and displacement of architectural elements, although its estimation from historical evidence remains challenging.

Added Notes Kramer emphasizes that acceleration, velocity, and displacement are all derived from the same ground motion but emphasize different frequency bands. Acceleration is dominated by higher frequencies, velocity occupies an intermediate range, and displacement reflects the lowest-frequency content. No single parameter adequately characterizes an earthquake; a complete description of ground motion requires consideration of amplitude, frequency content, and duration together.

• from Kramer (1996:70-79)

Fourier Amplitude Spectrum is a frequency-domain representation of earthquake ground motion obtained by decomposing a time history into a series of sinusoidal components of different frequencies, amplitudes, and phases. The Fourier amplitude spectrum shows how the amplitude of ground motion is distributed among frequencies. Narrow spectra indicate motion dominated by a small range of frequencies, whereas broad spectra indicate energy spread across many frequencies. Smoothing of Fourier spectra is commonly required because raw spectra are often highly irregular and jagged. Kramer notes that large earthquakes generally contain proportionally more low-frequency energy than smaller earthquakes, and that the overall shape of the spectrum provides a fundamental description of earthquake frequency content. When Fourier amplitude spectra are smoothed and plotted on logarithmic scales, their characteristic shapes become easier to interpret. Fourier amplitudes tend to be largest over an intermediate frequency range bounded by the corner frequency (f_c) on the low-frequency side and the cutoff frequency (f_max) on the high-frequency side. The corner frequency is theoretically related to source size and seismic moment, being inversely proportional to the cube root of seismic moment (f_c ∝ 1/M₀^1/3). Consequently, large earthquakes tend to generate relatively more low-frequency energy than smaller earthquakes. The physical significance of the cutoff frequency remains less well understood; it has been interpreted both as a near-site effect and as a source effect, and is commonly assumed to be approximately constant within a given geographic region. The shape of the Fourier amplitude spectrum therefore contains information about both the earthquake source and the propagation and site effects that modify seismic waves before they reach a recording station.



Power Spectral Density (PSD) describes the distribution of ground-motion energy as a function of frequency. Unlike the Fourier amplitude spectrum, which shows the amplitudes of individual frequency components, the PSD describes how the energy or intensity of the motion is distributed across the frequency range. Conceptually, the PSD is closely related to the square of the Fourier amplitude spectrum, since energy is proportional to amplitude squared. Frequencies with large Fourier amplitudes therefore contribute disproportionately to the power spectrum. The PSD provides a statistical description of ground motion and is widely used in random vibration analyses and stochastic descriptions of earthquake shaking. Rather than focusing on the exact sequence of peaks and troughs in a time history, it describes how much motion intensity is associated with each frequency band. This makes it useful for estimating the statistical properties of earthquake motions and for constructing idealized models of ground shaking. Kramer notes that the total intensity of a ground motion can be expressed as the integral of squared acceleration through time,

I₀ = ∫ a(t)² dt

and that an equivalent expression can be written in the frequency domain using the power spectral density. Thus, the area beneath the PSD curve is directly related to the total intensity of the ground motion. A narrow PSD peak indicates that much of the motion energy is concentrated in a small range of frequencies, whereas a broad PSD indicates that energy is distributed across a wider frequency band. Power spectral density functions are particularly useful for estimating statistical properties of motion, determining central frequency and bandwidth, and modeling the frequency characteristics of earthquake records. They form the basis for several stochastic ground-motion models, including the Kanai–Tajimi model commonly used in earthquake engineering. Response Spectrum is one of the most important tools in earthquake engineering because it describes how structures respond to earthquake ground motion. Rather than focusing on the frequency content or energy of the earthquake itself, a response spectrum considers a family of hypothetical structures, each represented as a single-degree-of-freedom oscillator. Each oscillator has a different natural period (or vibration time) and a specified damping ratio. The earthquake record is then passed through each of these hypothetical structures, and the maximum response of each one is determined. This response may be expressed as maximum acceleration, velocity, or displacement. The resulting response spectrum is a plot of maximum structural response versus natural period. In effect, the response spectrum answers the question: if many different structures were shaken by the same earthquake, which ones would respond most strongly? It therefore reveals which types of structures—whether short and stiff (short period) or tall and flexible (long period)—are most vulnerable to that particular ground motion. Because real buildings behave approximately as oscillators, response spectra are often far more useful for engineering purposes than raw ground-motion records. They provide a direct link between the characteristics of an earthquake and the expected response of structures.

Response spectra may be plotted as spectral acceleration (S_a), spectral velocity (S_v), or spectral displacement (S_d). Although all three are derived from the same earthquake record, each emphasizes different aspects of structural response. Response spectra reveal that different periods are associated with different types of response. At low frequencies (long periods), spectral displacement is nearly constant. At high frequencies (short periods), spectral acceleration is nearly constant. Between these limits lies a region of approximately constant spectral velocity. Because of this behavior, response spectra are often divided into acceleration-controlled, velocity-controlled, and displacement-controlled regions.

The response spectrum acts as a filter on the input ground motion. Rather than displaying the actual earthquake record, it displays the maximum response of oscillators with different natural periods. As a result, the frequency content of the earthquake is reflected in the shape of the response spectrum. For example, motions rich in high-frequency energy tend to produce higher spectral accelerations at short periods, whereas motions rich in long-period energy tend to produce larger spectral velocities and spectral displacements at longer periods. It is important to remember that a response spectrum does not describe the actual ground motion itself. Instead, it describes the maximum response of an entire family of hypothetical structures. This is why response spectra are so valuable in earthquake engineering: they provide a direct connection between ground motion and structural behavior, allowing engineers to identify which classes of structures are likely to respond most strongly to a given earthquake.

Predominant Period (Tp) is the period corresponding to the maximum value of the Fourier amplitude spectrum. It provides a simple single-value description of earthquake frequency content and is often determined from a smoothed spectrum to avoid the influence of individual spectral spikes. Kramer emphasizes that although predominant period is useful for characterizing frequency content, motions with very different spectral shapes may possess the same predominant period. Consequently, it should be viewed as a simplified indicator rather than a complete description of frequency content.

Bandwidth describes the range of frequencies over which significant spectral energy is present. It is commonly measured using the width of the Fourier amplitude spectrum at one-half of the maximum spectral amplitude. Narrow-band motions are dominated by a limited range of frequencies, whereas broad-band motions contain energy distributed across a much wider frequency range. Bandwidth therefore provides information about the dispersion of spectral amplitudes that is not captured by the predominant period alone.

Central Frequency (Ω) is a statistical measure derived from the power spectral density function. It represents the frequency about which the energy of a ground motion is concentrated. Unlike the predominant period, which identifies the strongest spectral peak, the central frequency reflects the overall distribution of energy throughout the spectrum. Kramer notes that central frequency is often used in conjunction with other statistical measures to characterize the frequency content of earthquake motions.

Shape Factor (δ) is a statistical parameter derived from the power spectral density function that describes the spread of energy about the central frequency. Low values indicate that energy is concentrated within a narrow frequency range, whereas larger values indicate a broader distribution of spectral energy. Shape factor is therefore a quantitative measure of spectral bandwidth and frequency dispersion.

Kanai–Tajimi Parameters are parameters used in a mathematical model of earthquake power spectral density. The model represents earthquake ground motion using a small number of variables that characterize site conditions and frequency response. The parameters determine the shape of the spectral density function and are particularly useful in stochastic simulations of earthquake ground motion. The Kanai–Tajimi model has been widely used to represent the filtering and amplification effects of local site conditions.

Velocity-to-Acceleration Ratio (v_max/a_max) is a simple frequency-content parameter based on the ratio of peak ground velocity to peak ground acceleration. Because peak velocity and peak acceleration are associated with different frequency bands, their ratio provides a rough indication of the dominant frequency characteristics of a motion. Larger ratios are generally associated with longer-period motions, whereas smaller ratios are associated with higher-frequency motions. Kramer notes that this parameter often correlates with predominant period, although the relationship is not exact because of the complex nature of real earthquake records. Seed and Idriss (1982) suggested the following representative average values of v_max/a_max for different site conditions located less than 50 km from the earthquake source:

Site Condition	vmax/amax	Equivalent Period
Rock	55 cm/sec/g	0.056 sec
Stiff soils (<200 ft)	110 cm/sec/g	0.112 sec
Deep stiff soils (>200 ft)	135 cm/sec/g	0.138 sec

Added Notes Kramer emphasizes that frequency content strongly influences the response of buildings, bridges, slopes, and soil deposits. Two earthquakes with similar amplitudes may produce dramatically different effects if their energy is concentrated in different frequency ranges. Frequency parameters therefore complement amplitude parameters such as PGA, PGV, and PGD by describing how earthquake energy is distributed through the spectrum. For engineering and archaeoseismological purposes, response spectra are often the most useful frequency-content parameter because they directly relate ground motion to the response of structures, while Fourier and power spectral measures provide a more fundamental description of the underlying motion.

Arias Intensity (I_a) is a measure of the total energy content of earthquake ground motion. Unlike parameters such as PGA, PGV, or PGD, which describe only the largest value reached during shaking, Arias Intensity incorporates both the amplitude and duration of motion. It is therefore often regarded as a measure of the total destructive potential of an earthquake record.

Arias Intensity is calculated by integrating the square of the acceleration record through time. Because acceleration is squared, large-amplitude motions contribute disproportionately to the final value. Long-duration shaking also increases Arias Intensity because energy continues to accumulate throughout the record. Consequently, two earthquakes with identical PGA values may have very different Arias Intensities if one produces strong shaking for a much longer period of time. The parameter was originally introduced by Arturo Arias and has become widely used in earthquake engineering, landslide studies, and liquefaction research because it reflects the total energy imparted to the ground. In many applications, Arias Intensity correlates more closely with slope failures and cumulative damage than peak amplitude measures alone. Kramer (1996:99-100) discusses Arias Intensity as one of several alternative ground-motion parameters and presents attenuation relationships showing how it varies with earthquake magnitude, distance from the source, and site conditions. Larger earthquakes generally produce larger Arias Intensities, whereas increasing distance from the source reduces Arias Intensity due to geometric spreading and attenuation of seismic waves.

The original definition of Arias Intensity is:

I_a = (π/2g) ∫ a(t)² dt

where:

• a(t) = ground acceleration as a function of time
• g = acceleration due to gravity
• π = Pi (Π)
• I_a = Arias Intensity

The quantity is therefore proportional to the area under the squared acceleration time history.

Kramer reproduces the attenuation relationship of Campbell and Duke (1974):

I_a (m/sec) = 313 e^{M_w(0.33M_w − 1.47)} S / R^3.79

where:

• M_w = moment magnitude
• R = distance from the center of energy release
• S = site-condition factor

Site Condition	S
Basement rock	0.57 R0.46
Sedimentary rock	1.02 R0.51
Alluvium ≤ 60 ft thick	0.37 R0.81
Alluvium > 60 ft thick	0.65 R0.74

Wilson (1993) analyzed strong-motion records from California and developed an attenuation relationship for Arias Intensity using the original Arias Intensity definition. The equation estimates how Arias Intensity decreases with distance from the fault while increasing with earthquake magnitude. Unlike simpler attenuation relationships based only on magnitude and distance, Wilson's formulation also includes terms for anelastic attenuation and the probability of exceedance. Kramer presents the relationship as an example of how Arias Intensity can be predicted for engineering applications from basic earthquake source parameters. The relationship is:

log I_a (m/sec) = M_w − 2 log R − kR − 3.990 + 0.365 (1 − P)

where:

• I_a = Arias Intensity (m/sec)
• M_w = moment magnitude
• R = √(D² + h²)
• D = minimum horizontal distance to the vertical projection of the fault plane
• h = correction factor with a default value of 7.5 km
• k = coefficient describing anelastic attenuation, with a default value of zero
• P = exceedance probability

The term −2 log R represents the geometric spreading of seismic energy away from the source. As distance increases, energy is distributed over a larger area, causing Arias Intensity to decrease. The term kR represents anelastic attenuation, the gradual loss of seismic energy due to internal friction and energy absorption within the Earth's crust. The magnitude term M_w reflects the greater energy released by larger earthquakes, while the exceedance probability term adjusts the estimate for different levels of confidence.

This equation illustrates why Arias Intensity is often considered an energy-based ground-motion parameter. The dominant controls are earthquake size and distance from the source. Because Arias Intensity incorporates both amplitude and duration, it tends to scale strongly with magnitude and often provides a better indication of cumulative damage potential than peak ground-motion measures alone. Compared with PGA, which may be dominated by a brief spike of shaking, Arias Intensity reflects the total energy delivered throughout the earthquake record. This characteristic makes it particularly useful for evaluating landslides, liquefaction potential, and other processes that depend on the cumulative effects of shaking rather than on a single peak value.

A useful way to think about Arias Intensity is that PGA, PGV, and PGD tell us how large the strongest part of the motion became, whereas Arias Intensity tells us how much total energy was delivered during the entire earthquake. In this sense it is analogous to comparing the highest wave in a storm with the total energy of all waves combined. For many engineering and geologic applications, especially landslides and cumulative damage processes, the total energy delivered by shaking may be more important than the single largest acceleration value.

• from Kramer (1996:98)

Root Mean Square Acceleration (RMS Acceleration or a_rms) is a measure of the average amplitude of earthquake acceleration throughout a time history. Unlike PGA, which describes only the single largest acceleration reached during an earthquake, RMS acceleration reflects the overall level of shaking sustained throughout the record. Because it is based on averaging acceleration over time, RMS acceleration is less sensitive to isolated spikes and provides a more representative measure of the typical intensity of ground motion. Mathematically, RMS acceleration is obtained by squaring the acceleration values, averaging them over the duration of the record, and then taking the square root of the result. This procedure prevents positive and negative accelerations from cancelling one another and yields a quantity that is directly related to the average energy level of the motion. As a result, RMS acceleration is often regarded as a better indicator of cumulative shaking intensity than peak values alone. Because earthquake damage depends not only on peak amplitudes but also on the persistence of strong shaking, RMS acceleration can provide useful information about the damaging potential of ground motions. Two earthquakes with identical PGA values may have very different RMS accelerations if one contains sustained strong shaking while the other is dominated by a brief acceleration pulse. RMS acceleration therefore occupies a conceptual middle ground between peak-amplitude parameters and energy-based measures such as Arias Intensity. Kramer cites the attenuation relationship developed by Hanks and McGuire (1981) from California earthquakes of magnitude 4.0–7.0 at hypocentral distances between 10 and 100 km. The relationship predicts RMS acceleration as a function of source frequency characteristics and distance from the earthquake source. Hanks and McGuire (1981) obtained:

a_rms = 0.119 √(f_max/f_c) / R

where:

• a_rms = root mean square acceleration


• f_c = corner frequency
• f_max = cutoff frequency
• R = hypocentral distance in kilometers

This relationship shows that RMS acceleration increases as the frequency bandwidth of the motion increases and decreases approximately in inverse proportion to distance from the earthquake source. Ground motions containing a broad range of frequencies therefore tend to produce larger RMS accelerations than motions dominated by a narrow frequency band.

RMS acceleration is particularly useful when the goal is to characterize the overall intensity of shaking rather than the most extreme instantaneous value. Because it averages motion over time, it is less affected by short-lived peaks than PGA. However, unlike Arias Intensity, RMS acceleration does not explicitly account for the duration of motion. A long-duration earthquake and a short- duration earthquake may have similar RMS accelerations if their average shaking amplitudes are comparable. Consequently, RMS acceleration is best viewed as a measure of average shaking level, whereas Arias Intensity measures the total energy delivered during the entire earthquake record.

Stress Drop (Δσ) is the average decrease in shear stress on a fault during an earthquake. It represents the difference between the stress acting on the fault immediately before rupture and the stress remaining after slip has occurred. Stress drop is commonly expressed in megapascals (MPa) or bars (1 MPa = 10 bars). Stress drop is one of the principal controls on the frequency content of earthquake ground motion. Earthquakes with larger stress drops tend to radiate more high-frequency energy, produce higher corner frequencies, and generate stronger short-period shaking. Lower stress-drop earthquakes generally produce relatively greater low-frequency motion and lower corner frequencies. In Brune's source model, stress drop controls the relationship between seismic moment, source radius, and corner frequency. For a given seismic moment, a larger stress drop implies a smaller rupture area, more rapid slip, and a higher corner frequency. Conversely, a smaller stress drop implies a larger rupture area and lower corner frequency. Typical tectonic earthquakes commonly exhibit stress drops of roughly 1–10 MPa (10–100 bars), although values outside this range are observed. Because stress drop strongly influences the generation of high-frequency seismic waves, it is an important parameter in strong-motion prediction equations and earthquake source models.

• High stress drop
  • more abrupt release of energy
  • higher corner frequency
  • more high-frequency shaking


• Low stress drop
  • gentler release of energy
  • lower corner frequency
  • relatively more low-frequency shaking

• from Kramer (1996:83-84)

Characteristic Intensity (I_c) is a composite parameter that combines the effects of shaking amplitude and duration into a single measure. Kramer notes that it is defined as:

I_c = a_rms^1.5 T_d^0.5

where a_rms is the RMS acceleration and T_d is the duration of shaking. According to Ang (1990), characteristic intensity is linearly related to an index of structural damage based on maximum deformation and absorbed hysteretic energy. Because it incorporates both amplitude and duration, it can provide a better indication of cumulative damage potential than amplitude measures alone.

Cumulative Absolute Velocity (CAV) is the total area beneath the absolute acceleration time history and is defined as:

CAV = ∫ |a(t)| dt

Unlike PGA, which considers only the largest acceleration, CAV measures the cumulative amount of acceleration experienced during an earthquake. Kramer notes that CAV has been found to correlate well with structural damage potential. Benjamin and Associates (1988) noted that a threshold value of 0.30 g-sec (obtained after filtering out frequencies above 10 Hz) corresponds to the lower limit for MMI VII shaking.

Response Spectrum Intensity (SI) was introduced by Housner (1959) to represent the overall severity of earthquake shaking for structures whose natural periods fall within a specified range. It is defined as the area under the pseudo-velocity response spectrum between periods of 0.1 and 2.5 seconds:

SI(ξ) = ∫_0.1^2.5 PSV(ξ,T) dT

where PSV is the pseudo-velocity response spectrum and ξ is the damping ratio. Because many structures have natural periods within this range, SI captures important aspects of both the amplitude and frequency content of earthquake motion.

Velocity Spectrum Intensity is a modified form of response spectrum intensity suggested by Von Thun et al. (1988). It is based on the velocity response spectrum and was proposed as a particularly useful parameter for evaluating the response of earth dams and other structures whose natural periods commonly fall between 0.6 and 2.0 seconds. Because it integrates response over a range of periods rather than focusing on a single period, it provides a broader representation of potential structural effects.

Acceleration Spectrum Intensity (ASI) was introduced by Von Thun et al. (1988) for structures characterized by relatively short natural periods, specifically concrete dams. It is defined as the area under the acceleration response spectrum between periods of 0.1 and 0.5 seconds:

ASI = ∫_0.1^0.5 S_a(ξ=0.05,T) dT

where S_a is the spectral acceleration. Because the integration range corresponds to short periods, ASI is particularly sensitive to the high-frequency components of ground motion that affect stiff structures. note that ξ=0.05

Effective Peak Acceleration (EPA) and Effective Peak Velocity (EPV) were introduced by the Applied Technology Council (1978) to provide smoothed measures of ground motion for engineering design. EPA and EPV are derived from response spectra rather than directly from accelerograms. EPA is approximately equal to the average spectral acceleration over the period range 0.1–0.5 seconds divided by 2.5 (the standard amplification factor for 5% damping spectrum), while EPV is approximately equal to the average spectral velocity over the period range 1.0–2.5 seconds divided by 2.5. By averaging across a range of periods, these parameters reduce the influence of isolated spectral spikes and provide more stable estimates for design purposes. Kramer notes that EPA and EPV were widely used in the specification of smoothed design spectra in building codes.

Kramer emphasizes that no single ground-motion parameter can adequately characterize an earthquake. Some parameters primarily describe amplitude (PGA, PGV, PGD), others emphasize frequency content (response spectra, predominant period, bandwidth), and others incorporate duration and cumulative effects (Arias Intensity, CAV, characteristic intensity). The suitability of a parameter depends on the engineering problem being considered. Different structures respond to different aspects of ground motion, which is why earthquake engineering commonly relies on multiple parameters rather than a single measure of shaking severity.