2.2 Signal Amplitude Sampling
Amplitude sampling is often not within our control since commercial equipment will have an A-to-D converter that operates with a fixed, finite number of bits. The amplitude values are recorded as binary numbers where each digit is a 0 or 1. Each binary digit is called a bit; the binary number represented by a sequence of bits is called a word. A word is used to represent the amplitude of the signal at a particular time or space. An example of binary representation and its relation to the familiar decimal numbers is given here for those who might have forgotten the basis of our numbering system:
The decimal number 19 means
1 x 101 + 9 x 100 = 19;
the corresponding binary representation is
10011 = 1 x 24 + 0 x 23 + 0 x 22 + 1 x 21 + 1 x 20 = 19.
The number 19 requires a 5 bit binary word. A 5 bit binary word can record a maximum binary number of 11111 which is 31. Since zero is the lowest number in the 5 bit binary sequence, a five bit word has 25 = 32 quantization levels, i.e., 00000, 00001, 00010, 00011,…, 11110, 11111. When the recorded signal has both positive and negative values, the left-most bit can be used as a sign bit (0 for + and 1 for -). There will still be the same number of quantization levels, but, e. g., an unsigned five bit word would accommodate positive decimal integers 0,1,…, 31 positive and the decimal values -16, -15, …, -1, 0, +1, …, +15 would result from a signed, five bit word. Non-integer decimal numbers are expressed using negative powers of two that are set-off from other bits by a radix point which is the equivalent of a decimal point in the decimal system, e. g. , the decimal number 3.25 is represented in binary as
11.01 = 1 x 21 + 1 x 20 + 0 x 2-1 + 1 x 2-2 = 3.25.
For linear quantization (the usual kind), the input range of whatever is being recorded is divided into 2N equal steps where N is the number of bits used in actual numerical recording within the system. Examples of input ranges are a full scale of 20,000 nT to 70,000 nT (nanoteslas) for magnetic field measuring equipment or —5 to +5 V (volts) recorded by an electromagnetic system. The quantization step size, Ds (or amplitude resolution) is the input range, R divided by 2N, i.e., Ds = R/2N.
Geophysical recording systems are often rated by the number of bits used in A-to-D conversion. Modern systems typically have 16-24 bits. The magnetic data presented above from Los Alamos (Figure 1.5 or 2.2a) were recorded using a Geometrics 826 cesium vapor magnetometer using 18 bits. Another common way of describing the performance of digital equipment is to use the ratio of the full recording range to the step size (R/Ds). This ratio of the largest possible recorded signal to the smallest one simply equals 2N. It is called the dynamic range and is expressed in decibels or dB (named for American Alexander Graham Bell who invented the telephone in 1879) as
(2.2.1)This is the same logarithm scale used when complaints are made of the noise levels at airports or friends brag about their sound systems. Increasing the ratio by a factor of 2 (i.e., increasing the number of bits, N by 1) results in approximately 6 dB increase in dynamic range.
In the illustrations above for combinations c and d (Figures 2.2c and 2.2d) we clearly see the problem associated with an A-to-D converter with a low dynamic range (20 log 8 = 18.06 dB). Since we are replacing the actual, continuous amplitude by quantized steps, we are effectively adding noise to the continuous signal. Other problems inherent in A-to-D converters such as unequal step sizes and improper positioning of steps can cause a reduction of the dynamic range, e.g., a 24 bit system may have an effective range of 22-23 bits. Equipment specification sheets often reveal such information.
Unless you are building your own equipment, you must accept the dynamic range available in commercial equipment. The current 24 bit systems are usually up to the task of faithfully recording the amplitude information required for most geophysical applications. Such a system has a dynamic range of 20 log 224 = 144.5 dB. As a result, we will treat all recorded amplitude values as though they are continuous (undistorted) throughout the remainder of our discussions. So, you can think of geophysical data as having continuous amplitude values with discrete space or time sampling (combination b, Figure 2.2b above). There are still very nasty problems ready to rear their ugly heads during the process of space or time sampling which we will now address.