Seismic Methods Geoph 465/565 ERB 5104 Lecture 3 Lee M. Liberty Research Professor Boise State University

Mid-term Presentations (Oct 19) special topics in seismology 15-minute oral presentation Metrics: presentation, style (professionalism), organization, accuracy, references Include: history of topic, theory, approach to addressing/solving topic, relevance to industry/society Topic examples: Refraction approaches– ray tracing vs tomography Earthquake relocations via HYPODD or others Volcanic seismology Topics in hydrogeophysics

Earliest known seismoscope To detect an earth quake in mouth of each dragon a bal was present if a quake was present (system disturbed) a mechanism was making that the bal was falling into the frog’s mouth. Used to indicate the occurrence of an earthquake in the year 138 (Reynolds)

Shot records distance V=slope Refracted/direct arrival time Reflections Direct Shear Surface waves

Simple harmonic motion 3 parameters describe a wave A = amplitude f = frequency = 1/T f = phase F(t) = A cos (2pft - f) f=0 =zero phase wave (cosine wave) If f=90o (p/2 radians) =sine wave F(t) = A cos (2pft – p/2) = A sin(2pft)

Waves in space Degrees: (radians) 360 (2p) 720 (4p) 1080 (6p) 360 (2p) 720 (4p) 1080 (6p) k = angular wavenumber (cycles/distance)

Waves in time Degrees: (radians) 360 (2p) 720 (4p) 1080 (6p) 360 (2p) 720 (4p) 1080 (6p) f = frequency (cycles/time) (Hz)

Space and time version of waves are related by the wave speed (v) v (m/s) = l/(t1-t0) = l/T = f l l (m/cy) = v/f f (cy/s – or Hertz) = v/l

For example: l = v/f f = v/l If a wave with a frequency of 10 Hz is traveling at a speed of 1000 m/sec, the wavelength l = v/f = 1000/10 = 100 m If we want a wave to have a wavelength of 10 m in a material with a wave speed of 1500 m/sec, we need a wave of frequency f = v/l = 1500/10 = 150 Hz

Phase shifts f Degrees: (radians) 360 (2p) 720 (4p) 1080 (6p)

If f=90o (p/2 radians) >> sine wave F(t) = A cos (2pft – p/2) = A sin(2pft)

Waves and Music 1 octave = double frequency 50Hz – 100Hz = 1 octave Low C = 128 Hz (cycles/sec) Middle C = 256 Hz High C = 512 Hz Human hearing ~ 20 Hz to 20,000 Hz Infrasonic <20 Hz Ultrasonic >20,000 Hz Wavelengths: Low C: l = v/f l = (330m/sec)/(128 cycles/sec) l = 2.6 m/cycle

Fourier Transform A mathematical operation that decomposes a signal into its constituent frequencies Transform signals between time (or spatial) domain and frequency domain The Fourier transform relates the function's time domain, shown in red, to the function's frequency domain, shown in blue. The component frequencies, spread across the frequency spectrum, are represented as peaks in the frequency domain.

Suppose we have several waves summed together? > interference

Any continuous function can be described as a series of cosine and sine waves cosine = symmetric sine =antisymmetric

Fourier transforms: Multiplication by sine and cosine waves to determine a coefficient whose amplitude is a measure of how well the wave matches the function. add multiply multiply add

Real and imaginary axes Imaginary (sin) Amplitude = sqrt(Ac2 + As2) As A f Real (cos) Ac Phase f = tan-1(As/Ac)

Fourier Transforms f(t) = (1/2p)∫F(ω) cos(ωt)dω + i∫F(ω) sin(ωt)dω F(ω) = ∫ f(t) cos(ωt)dt - i∫ f(t) sin(ωt)dt ω = 2pf = angular frequency (radians/sec) Angular frequency is a measure of how fast an object is rotating or the magnitude of the vector quantity angular velocity

But cos(w) - isin(w) = e-iw and cos(w) + isin(w) = eiw This allows us to rewrite the Fourier transforms as: f(t) = (1/2p)∫F(ω) eiωtdω F(ω) = ∫ f(t) e-iωtdt

Fourier transform results in Amplitude and phase spectra F(w)

Wavelet types: Zero phase Minimum phase Maximum phase Mixed phase All phases = 0 Minimum phase Energy is front-loaded Maximum phase Energy is back-loaded Mixed phase Most wavelets are mixed phase

Wavelets – seismic sources Short burst of energy Dynamite blast, hammer hit, airgun firing

Zero-phase Symmetric vibroseis

Why is phase important? Interpretation Zero-phase (Vibroseis) reflection is at the peak Minimum/mixed phase (dynamite, airguns) reflection is first deflection Maximum/mixed phase (water guns) Reflection is later, but we usually correct this

Why is phase important? Deconvolution (predictive error filtering) We will independently alter the phase spectrum and the amplitude spectrum to change the source wavelet to a more desirable shape. i.e., we can change minimum phase to zero phase, etc. But: maximum-phase wavelets can create unstable inverse filters – you can delay the energy, but moving the energy forward in time is more difficult

Bandwidth Bandwidth is the frequency range contained in a source wavelet or seismic trace. Mono-frequency waves extend forever (e.g. sine wave). Waves with an infinite bandwidth (all frequencies) can be infinitely short Dirac Delta function = spike

Zero-phase 1 Hz What happens to wavelet as we increase or decrease the range of frequencies? 5.5 Hz

side lobes

Bandwidth Mono-frequency waves extend forever. Waves with an infinite bandwidth (all frequencies) can be infinitely short Dirac Delta function = spike Frequency spike = infinite time function infinite frequencies = time spike

Bandwidth Broader bandwidth = sharper wavelet >>For seismic reflection data, our resolving power (ability to separate two thin layers) increases as the bandwidth increases. Therefore, we would like to get a source that produces the largest bandwidth possible. Minimum of 2 octaves is suggested, 3 is better!! (5-40 Hz; 10-80 Hz; 20-160 Hz; etc.)

How thin is a thin bed? Widess (1973) showed the effects of bandwidth on resolution ¼ wavelength rule (not thin bed) If you want better resolution, you need higher frequencies

Bandwidth = higher frequencies The same bandwidth at higher frequencies has the same number of side lobes. For distinguishing thin layers, it is better to have more bandwidth, even if freqs are lower.

Bandwidth Bandwidth is the frequency range contained in a source wavelet or seismic trace. Mono-frequency waves extend forever (e.g. sine wave). Waves with an infinite bandwidth (all frequencies) can be infinitely short Dirac Delta function = spike

side lobes

Bandwidth = higher frequencies The same bandwidth at higher frequencies has the same number of side lobes. For distinguishing thin layers, it is better to have more bandwidth, even if freqs are lower.

Sampling frequency We want to minimize the sample rate (maximize the digitizing interval) so that we minimize the computer storage requirements and the processing time. What is the largest digitizing interval (minimum sample rate) that we can use? For a given sample rate, what is the highest frequency wave that is correctly sampled? Fourier transforms – we must be able to accurately take FTs of the time function for filtering and other processes.

Sampling a time function trace digitizing interval = dt (sample rate=1/dt) duration (period) = T

Aliasing If not sampled frequently enough, the time series does not provide an accurate representation of the wave. Other frequencies also fit the time series.

Aliasing Time series must be sampled so that the highest frequency is sampled at least twice. dt = 1/2fmax fmax = 1/(2dt)

Aliasing – frequency domain fmax is called the aliasing frequency, the folding frequency, or the Nyquist frequency fN. Frequencies above the Nyquist frequency are “aliased” or “folded” to lower frequencies. Amp fN frequency

Example dt = 0.04 fN = 12.5 Hz 15 Hz 10 Hz

Resampling Before resampling a seismic trace to a larger digitizing interval, or when you collect seismic data, you MUST use an anti-alias filter first to prevent aliasing! >>>It’s not just signal – you also MUST sample the noise properly, or filter it out before sampling.

Sampling constraints (for a harmonic Fourier Series) The time sampling interval and the maximum frequency are related. dt=1/2fmax fmax=1/(2dt) Similarly, the frequency sampling interval (df) and the maximum time (T) are related. df = 1/T T=1/df Lowest frequency wave = 1 cycle/period Fundamental mode

df = 1/T 2df 3df .

A/D converter An analog-to-digital converter (A/D) is a device that converts a continuous quantity to a discrete time digital representation. converts an input analog voltage or current to a digital number proportional to the magnitude of the voltage or current. A/D converter is defined by bandwidth (the range of frequencies it can measure) and its signal to noise ratio (how accurately it can measure a signal relative to the noise it introduces). The actual bandwidth is characterized primarily by its sampling rate.

Dynamic range Range which can be measured using different number of bits: 8-bit : 1 mV - 256 mV 24-bit: 1 mV - 16 V Dynamic range is expressed in dB: Examples: