Introduction

The subject of dynamic range is one that often comes up when seismologists start discussing seismic recorders and sensors. However, different manufactures sometimes mean different things when they quote dynamic range so one has to be careful. Also, people sometimes quote "noise level" in either bits or Volts, sometimes "dynamic range" in dB, sometimes "signal to noise ratio" in dB and sometimes "effective number of bits" or ENOB. Finally, a dynamic range figure is always for a certain bandwidth that is often not specified, thus making the reported value meaningless. In this technical note, I will try and explain what the various values mean.

In simple terms, we want to get an indication of the range between the biggest signal and smallest signal that can be recorded with a certain system. So lets look at what we mean by the "biggest signal" and the "smallest signal".

Biggest Signal

There are three commonly used measures of the biggest signal and they each have a place.

1. Peak-to-Peak or Full-Scale Range
   This value is the difference between the two most extreme signals
2. Zero-to-Peak
   This value is the difference between the mid-value and an extreme value
3. RMS Full-scale
   This is the root-mean-square value of the largest possible sinewave signal (=0.707 x its peak amplitude)

For example, for a 1V sine wave, the three values are:
Peak-Peak = 2.0V
Zero-Peak = 1.0V
RMS = 0.707V

Smallest Signal

There are two common measures of the smallest signal and they also both have their place.

1. RMS Noise
   This is the root-mean-square value of the signal measured with NO applied signal
2. Least-Significant-Bit (LSB)
   This is a measure used with analogue to digital converters (ADC) and corresponds to a change of one in the numerical output of the convertor

The measured RMS Noise level will depend on the bandwidth it is measured over. For example, white noise has the same energy at all frequencies. Therefore, it makes sense that if you record it over a wider bandwidth, you will record a higher signal level. In fact, for white noise, each time you double the bandwidth you increase the recorded noise level by 1.41 or 3dB. This is why you will often see noise levels quoted as some number of (e.g.) Volts/root Hertz. To convert this to an RMS value use:

RMS Noise = (Noise level in X/root Hz)*sqrt (bandwidth)

Dynamic Range and Signal to Noise Ratio

The dynamic range and signal to noise ratio of a device are most properly defined to be the same thing. That is:

Dynamic Range = SNR = RMS Full-scale/RMS Noise

and is usually quoted in dB.

Dynamic Range (dB) = SNR (dB) = 20*Log10 (RMS Full-scale/RMS Noise)

However, some manufacturers who wish to make their figures look better will either quote:

• Zero-to-Peak/RMS Noise which gives a value that is a factor of 1.41 or 3.0dB higher, or
• Peak-to-Peak/RMS Noise which gives a value that is a factor of 2.83 or 9.0dB higher

so carefully check all such specifications!

Effective Number of Bits

Analogue to digital convertors often quote the value known as Effective Number of Bits or ENOB as this allows you to directly compare one ADC with another and determine how close to "ideal" any give ADC is. ENOB is defined to be log to the base 2 of the ratio of the full-scale range (FSR) to the RMS noise. That is

ENOB = log (FSR/RMS noise)/log (2)

The relationship between ENOB and SNR is

ENOB = (SNR(in dB) - 1.76) / 6.02

this is shown in the table below.

Seismic Sensor Example

Seismometer

The plot below is a typical example of the type of plot presented by seismic sensor manufacturers. This is the plot for a Guralp CMG-40T. The horizontal axis is period (or frequency) and the vertical axis is acceleration. The green line across the bottom of the graph shows the "noise floor". The red line shows the full scale value. Because the CMG-40T is a velocity sensor, its full scale value slopes down at 45° to the right on this acceleration plot.

It is not easy to determine the instruments dynamic range from a plot such as this. The truly curious can read the following and the rest of you should skip to the next section.

The CMG-6T data sheet states that the "noise levels are peak-peak estimates, in m/s2, equal to six times the measured RMS acceleration values for a 1/3 octave bandwidth", so we will assume that this is the same here. Lets try and convert that in to English for you :-)

• "peak-peak"- we have covered and know what it means
• "six times the measured RMS" - for a white noise signal, the peak-peak value is related to the RMS only in a statistical sense. It can be shown that the p-p value will be less than 6.6 x RMS value 99.9% of the time (not 6 as Guralp is using)
• "a 1/3 octave bandwidth" - this measurement is often used with audio signals. An octave is a factor of two in frequency, e.g. from 10Hz to 20Hz or from 432Hz to 864Hz. A 1/3 octave bandwidth is defined such that if you split an octave in to three (logarithmically) even sections then each of them will be 1/3 of an octave wide. This turns out to be a factor of 1.2599 in frequency. E.g. from 10Hz to 12.6Hz, or from 12.6Hz to 15.87Hz or from 15.87Hz to 20.0Hz

So if we take the figure at 1 second = 1 Hertz as -175dB = 1.8E-9 m/s2 peak-peak. Then this must have come from a reading of 1.8E-9/6 = 3.0E-10m/s2 RMS over a bandwidth of 1.26Hz giving a noise level of 2.6E-10m/s2/rt Hz. If we do the same calculation at 0.1 seconds we get a noise level of 2.3E-10m/s2/rt Hz. Therefore, let us assume that we have a constant noise level of 2.3E-10m/s2/rt Hz over the bandwidth from 0.1 to 50Hz. The RMS noise level over this bandwidth will be approximately 2.3E-10 x sqrt (50) = 1.62E-9 m/s2. The largest RMS signal possible over this bandwidth is approximately -40dB = 0.01m/s2. Therefore, the dynamic range (or signal to noise ratio) over this bandwidth is 0.01/1.62E-9 = 6,170,000 = 136dB = 22.3 ENOB. Phew!

Accelerometer

For various reasons, most accelerometer manufacturers usually quote figures that are much easier to convert to dynamic range or ENOB. The table below lists these values for a number of common units.

ADC Examples

Most ADC data sheets specify the dynamic range of the device under a number of different operating conditions. It is usually possible to determine which of the measures described above is being reported. Most sigma-delta type ADCs have a dynamic range that reduces as the sample rate increases. The tables below show the figures for the ADC's used in various models of Kelunji recorder. For comparison purposes, I have converted all the numbers to ENOBs.

As a general rule, you want your ADC to have at least 0.5 more ENOB than the sensor connected to it.

Copyright © 2004, Environmental Systems and Services, Pty Ltd
Last modified: 2004-10-19