Transcript Seismic Methods Geos 465/GEOPH565 ERB 2104 January

Seismic Methods
Geos 465/GEOPH565
ERB 2104
January 30, 2017
Lecture 5 –bandwidth, sampling and amplitudes
Lee M. Liberty
Research Professor
Boise State University
Seismic unix script
#RVSP script to display channel 1
channel=1
#Read and display the data
segyread tape=1100.sgy | segyclean | \
suwind key=fldr min=1102 max=1270|\
suwind key=tracf min=$channel max=$channel tmax=.10|\
sushw key=sdepth a=44.0 b=-.25 |\
sugain agc=1 wagc=.01|\
supswigb perc=95 nbpi=300 title="Channel "$channel f2=44. d2=-.25 \
labelsize=8 label1="time (s)" label2="Depth (m)" >RVSP$channel.ps
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.
trace
Sampling a time function
• digitizing interval = dt (sample rate=1/dt)
• duration (period) = T
T
1.5
Amplitude
1
0.5
0
-0.5
-1
-1.5
0
dt
0.1
0.2
0.3
time (sec)
0.4
0.5
1.00
-0.31
-0.81
0.81
0.31
-1.00
0.31
0.81
-0.81
-0.31
1.00
-0.31
-0.81
0.81
0.31
-1.00
0.31
0.81
-0.81
-0.31
1.00
-0.31
-0.81
0.81
0.31
-1.00
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.
1.5
1.5
Amplitude
Amplitude
1 1
0.5
0.5
0 0
-0.5
-0.5
-1
-1
-1.5
-1.5
0 0
0.10.1
0.2
0.2
0.3
0.3
time(sec)
(sec)
time
0.4
0.4
0.5
0.5
Aliasing
• Time series must be sampled so that the highest frequency is
sampled at least twice.
• dt = 1/2fmax
fmax = 1/(2dt)
1.5
Amplitude
1
0.5
0
-0.5
-1
-1.5
0
0.1
0.2
0.3
time (sec)
What sample rate/frequency did we use to record the VSP data?
0.4
0.5
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
frequency
fN
Example
• dt = 0.04
fN = 12.5 Hz
10 Hz
15 Hz
1.5
Amplitude
1
0.5
0
-0.5
-1
-1.5
0
0.1
0.2
0.3
time (sec)
0.4
0.5
Resampling
• Before resampling (down sample) 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.
• What about sampling to a faster sample rate?
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.
Range which can be measured using different number
of bits:
8-bit (28):
1 mV - 256 mV
24-bit (224) :
1 mV - 16 V (16 million samples)
Dynamic range is expressed in dB:
 A max 
20 log 

 A min 
 256mV 
Examples: 20 log 
  48dB
 1mV 
 16V 
  144dB
20 log 
 1mV 
Geometrics Geode seismograph
• Configurations: 3, 6, 8, 12, 16 or 24 channels
• A/D Conversion: 24 bit result using Crystal Semiconductor sigma-delta converters
• Dynamic Range: 144 dB (system), 110 dB (instantaneous, measured) at 2 ms, 24 dB.
• Distortion: 0.0005% @ 2 ms, 1.75 to 208 Hz.
• Bandwidth: 1.75 Hz to 20 kHz. 0.6 and DC low frequency option available.
• Crosstalk: -125 dB at 23.5 Hz, 24 dB, 2 ms.
• Noise Floor: 0.20 uV, RFI at 2 ms, 36 dB, 1.75 to 208 Hz.
• Stacking Trigger Accuracy: 1/32 of sample interval.
• Maximum Input Signal: 2.8V PP, 0 dB.
• Input Impedance: 20 kOhm, 0.02 uf.
• Preamplifier Gains: 24 and 36 db, jumpered for software selectable 12 and 24 dB or can be jumpered in four channel
blocks as a single fixed gain of 0 dB for high-voltage devices.
• Anti-alias Filters: -3 dB at 83% of Nyquist frequency, down 90 dB.
• Acquisition and Display Filters:
• Low Cut: OUT, 10, 15, 25, 35, 50, 70, 100, 140, 200, 280, 400 Hz, 24 or 48 dB/octave, Butterworth. Notch: 50, 60, 150,
180 Hz and OUT, with the 50 dB rejection bandwidth 2% of center frequency. High Cut: OUT, 32, 64, 125, 250, 500 or
1000 Hz, 24 or 48 dB/ octave.
• Sample Interval: 0.02, 0.03125, 0.0625, 0.125, 0.25, 0.5, 1.0, 2.0, 4.0, 8.0,16.0 ms.
• Record Length: 16,384 samples standard, 65,536 samples optional
• Pre-trigger Data: Up to full record length.
• Delay: 0 to 100 sec in steps of 1 sample interval.
Delta sigma A/D
• The modulator is a chip circuit that operates on a 256,000 Hz clock
(0.004 ms) – or – 1 bit per 0.004 ms
• For a 2 ms sample rate (or 512 clock cycles), 512 samples goes into
the final measurement
(for a 1 ms sample rate, 256 samples goes into the final
measurement)
• but, we need to remove signals above the Nyquist and near DC
• For a 24-bit system (144 dB) with system noise and at 2 ms SR, we
can get about 110 dB (18 bits) of usable signal
• geophone equation is a second-order linear
ODE characteristic of damped harmonic
oscillation
•
V(t) is the seismograph voltage input, and vB(t) is the
vertical particle velocity.
•
The constants h, C, and w0 are the total damping constant,
total transduction constant, and natural frequency
•
natural frequency w0 can be adjusted by changing
•
the stiffness K of the spring, or the mass m of the coil
Geophone specs
Geophone response (4.5 Hz)
A/D measured displacements
 350,000mV 
20
log

  110dB
110 dB:
1mV


• 1.5 mm max geophone displacement
• ~4 nm detection threshold
• But, a typical geophone only has ~70 dB of
instantaneous dynamic range or ~500 nm
70 dB:
 3,300 mV 
20 log
  dB
 1mV 
• But, we can gain our instruments to record greater
sensitivity
Seismic Amplitudes
Amplitude Corrections
•
•
•
•
•
Spherical divergence
Array directivity (beam forming)
Scattering
Absorption
Amplitude versus offset (Zoeppritz
equations)
• Multiples
• Tuning effects
(constructive/destructive
interference)
Amplitude attenuation (from Sheriff, 1975)
Spherical divergence
Waves spread in all directions to form a sphere
with a surface area of 4pR2
energy density or E = 1/ 4pR12
The amplitude equals the square root of the
energy density, so the ratio is:
A1 /A0 = sqrt(4pR02 )/sqrt(4pR12 )
A1 /A0 = R0 /R1
If we assume R0 is a unit radius, the amplitude
at radius R is
A1 =A0/R
A1
A0
R
Attenuation
• Amplitude also is lost due to friction as the wave moves through the material.
• A1 = A0e-pf t/Q
• f = frequency, t = traveltime
• Q = dissipation (attenuation) constant
• Soils, sediment: Q ~ 20 to 100
• Sedimentary rock: Q ~ 50-500
• Crystalline rock: Q ~ 200-2000
• Smaller Q results in faster damping (greater deviation from elastic case)
• Frequency-independent Q damps high frequencies more than low frequencies
• Q = 2π (total energy/energy lost during one cycle)
Higher freqs
are
attenuated
with depth.
Time (seconds)
Attenuation
Attenuation
125
Q=100
Q=250
Q=500
Q=1000
Amplitude (%) .
100
75
50
25
0
0
2
4
6
traveltime (seconds)
8
10
Attenuation + spherical spreading
120
1/R
Q=100
Q=250
Q=500
Q=1000
Amplitude (%) .
100
80
60
40
20
0
0
2
4
6
traveltime (seconds)
8
10
Correcting for spherical spreading and
attenuation
• Automatic gain control (equalize running average)
The input trace is subdivided into fixed time gates. The amplitude of each
sample in a gate is squared. The mean of these values is then computed and
its square root is taken (rms amplitude over that gate). The ratio of a desired
rms amplitude to the actual rms value is assigned as the value of the gain
function at the center of the gate. Hence, the scaling function g(t) at the gate
center is given by
• Inverse correction
A = At2
Amplitude corrections
Correcting the
signal for spherical
spreading and
attenuation will
cause the noise to
also increase.
A = A 0t
Automatic Gain Control (AGC)
(like a running average amplitude correction)
AGC “shadow”
Automatic Gain Control (AGC)
(like a running average amplitude correction)
AGC “shadow”
AGC
• Because it is not predictable:
• Changes relative amplitudes of reflectors in an
unpredictable way (i.e., it ruins true amplitude information
like direct hydrocarbon indicators)
• Degrades processes like multiple suppression
(deconvolution) that rely on a constant amplitude ratio
between the primary and the multiple (e.g. Backus water
filter)
• >>much better to use a linear gain function
• Can front load or back load an AGC operator