(totalstaion )Electromagnetic Distance Measurement (EDM)

One of the fundamental measurements in surveying is distance. Obviously with a distance and an angle, one can establish a coordinate system and locate the relative positions of objects or observations. The angle defines the orientation and the distance of the scale. The direct measurement of distance in the field is one of the more troublesome of surveying operations, especially if a high degree of accuracy is desired. Indirect measurements, like the use of a stadia rod, have been developed and used extensively; however, these systems are of rather limited range and accuracy. With the advent of electromagnetic instruments, the direct measurement of distance with high precision is possible (Burnside, 1991).
When a distance is measured using an EDM instrument, some form of electromagnetic wave (in our case, infra-red--IR--radiation) is transmitted from the instrument towards a reflector where part of the transmitted wave is returned to the instrument. Electronic comparison of the transmitted and received signals allows for computation of the distance (Price, 1989). See Figure 3 for a schematic of the method of operation of an EDM.


Figure 3.1 Schematics of EDM system. From Price and Uren, 1989.


Electromagnetic waves Electromagnetic waves can be represented by a sinusoidal wave motion. The number of times in 1 second that a wave completes a cycle is called the frequency (f), and is measured in
Hz. The length of one cycle is called the wavelength (l), which can be determined as a function of the frequency from
(1)
where v is the speed of propagation of the wave.
The speed of electromagnetic waves in a vacuum is called the speed of light, c, and is taken to be 299,792,458 m s-1.(Price, 1989). The accuracy of an EDM instrument depends ultimately on the accuracy of the estimated velocity of the electromagnetic wave through the atmosphere (Burnside, 1991).
A relationship expressing the instantaneous amplitude of a sinusoidal wave is
(2)
where Amax is the maximum amplitude developed by the source, A0 is the reference amplitude, and f is the phase angle which completes a cycle in 2� radians or 360°.
In an EDM system, distance is measured by the difference in phase angle between the transmitted and received versions of a sinusoidal wave. The double path length (2D) between instrument and reflector is the distance covered by the radiation from an EDM measurement. It can be represented in terms of the wavelength of the measuring unit:
2 (3)
The distance from instrument to reflector is D, l_m is the wavelength of the measuring unit, n is the integer number of wavelengths traveled by the wave, and Dl_m is the fraction of the wavelength traveled by the wave. Therefore, the distance D is made of two separate elements. An EDM instrument using continuous electromagnetic waves can only determine Dl_m by phase comparison (Figure 4).
If the phase angle of the transmitted wave measured at the instrument is f₁, and the phase angle measured on receipt is f₂, then
(4)
The phase angle f₂ can apply to any incoming wavelength, so phase comparison will only provide a determination of the fraction of a wavelength traveled by the wave, leaving the total number, n, ambiguous (Price, 1989).


Figure 4. Phase comparison. (a) An EDM is set up at A and a reflector at B for determination of the slope length (D). During measurement, an electromagnetic wave is continuously transmitted from A towards B where it is reflected back to A. (b) The electromagnetic wave path from A to B has been shown, and for clarity, the same sequence is shown in (c), but the return wave has ben opened out. Points A and A' are effectively the same, since the transmitter and receiver would eb side by side in the same unit at A. The lowermost portion also isslustrates the ideal of modulation of the carrier wave by the measuring wave. From Price and Uren, 1989.


For many EDM instruments, an accuracy in measurement between 1 and 10 mm is specified at short ranges, and a phase resolution of 1 in 10,000 is normal. Assuming an accuracy of at least 1 mm, therefore, a measuring wavelength (l_m) of 10 m is required. Approximating the speed of propagation, v, by 3 x 10⁸ m s^-1, 10 m corresponds to a frequency of 30 MHz. Adequate propagation of an electromagnetic signal of 30 MHz frequency for EDM purposes is not practical, so a higher frequency carrier wave is used and modulated by the measuring wave (Figure 5, (Price, 1989)). In the case of our instrument, the carrier wave is infra-red, with a wavelength of 0.835 �m, corresponding to a frequency of 3.6 x 10⁵ GHz.
The return signal is usually amplified, and then the phase difference is determined digitally. The signal derived from the modulation triggers off the counting mechanism every time the signal changes from negative to positive. The signal derived from the reflected ray stops the counting mechanism (Burnside, 1991). For our instrument, the integer ambiguity (n) of wavelengths is determined using the coarse measurement frequency of 74,927 Hz, equivalent to 2000 m, and the amplitude modulation of 4,870,255 Hz (30.7692 m) provides the fine measurement (Dl_m). In order to achieve the stated accuracy of 5 mm, the phase measurements are accurate to 1 part in 6,154.
Errors in distance measurement The path of electromagnetic energy is the true distance that is measured by the EDM, and will be determined by the variability in the refractive index through the atmosphere.
(5)
Since the medium is air, the velocity is nearly the same as that of a vacuum, and so the refractive index is nearly one, and for standard conditions may be taken to be 1.000320. The exact value of the refractive index is dependent on the atmospheric conditions of temperature, pressure, water vapor pressure, frequency of the radiated signal, and composition. Therefore, for measurements of the highest accuracy, adequate atmospheric observations must be made. Other potential error sources are from path curvature (similar to the curvature of the earth).(Burnside, 1991).
Scale error One source of error that is adjusted for in our instrument is a scale correction in units of ppm that adjusts for slight errors in the reference frequency and in the accuracy of the average group refractive index along the line of measurement. The ppm value is set to 0 on the EDM, and adjusted in the theodolite.
Atmospheric correction is one scale error adjustment that takes into account both atmospheric pressure and temperature. It is an absolute correction for the true velocity of propagation, and not a relative scale correction like reduction to sea level (see below). To determine the atmospheric correction to an accuracy of 1 ppm, measure the ambient temperature to an accuracy of 1°C and atmospheric pressure to 3 mb. For most applications, and approximate value for the atmospheric correction (within about 10 ppm) is adequate. This can be obtained by taking the average temperature for the day and the height above mean sea level of the survey site. A temperature change of about 10°C or a change in height above sea level of about 350 m (= 35 mb) varies the scale correction by only 10 ppm. The atmospheric correction is computed in accordance with the following formula:
(6)
where: DD₁ = atmospheric correction (ppm), p = atmospheric pressure (mb), and t = ambient temperature (°C). For extreme conditions of 30°C temperature change and 100 mb pressure change, one can expect variations in scale error of 50 ppm in a day. This maximum value is ten times greater than the stated initial accuracy of the instrument, and therefore should be accounted for by adjusting the scale error on the theodolite occasionally during the day's surveying effort (see Figure 6, graph 1 to determine DD₁).
The correction in ppm for the reduction to mean sea level is based upon the formula:
(7)
where DD₂ = reduction to MSL in ppm, H = height of EDM above MSL, and R = 6378 km (earth radius). This correction is a constant and should be determined at the beginning of the survey effort by consulting Figure 6, graph 2a or 2b.
Corrections in ppm may be made for map projections as well.
Reduction formula The theodolite computes slope distance according to the following formula:
D = D₀x(1+deltaDx10^-6) + mm
where D is the corrected slope distance in mm, D₀ is the measured (uncorrected) slope distance in mm, (deltaD) = sum of scale corrections (n is the number of scale corrections) in ppm, and mm = prism constant in mm.
The theodolite computes horizontal distance (D_h) and height difference (e_h) by accounting for earth curvature and mean refractive index. These corrections are on the order of 10^-8 or 0.01 ppm of D₀ and therefore not significant compared to the scale correction (~50 to 100 ppm).
Figure 6. Scale correction graphs from Theodolite manual. Use these to determine DD (sum of scale corrections).
Infra-red radiation from a GaAs lasing diode The source of the IR radiation for our instrument is a Gallium Arsenide (GaAs) lasing diode. This device is made from a small chip of semiconducting material, and is similar in size and appearance to other semiconducting devices (Figure 7). Driven by a forward biased voltage and maintained by different electrostatic potentials in the two halves of the diode, an electron population inversion between the two halves of the diode will provide the energy level transition for stimulated emission of photons by electrons as they fall to the lower energy state (Price, 1989). The energy difference is emitted as radiation (and thus the process and device are called a laser--Light Amplification by the Stimulated Emission of Radiation). The process of stimulated emission enables the laser to emit an intense, monochromatic radiation that travels as a narrow beam for considerable distances before it spreads out (Price, 1989). The intensity of the IR radiation is nearly linearly proportional to the current flow and with virtually an instantaneous response (Burnside, 1991). If an alternating voltage is superimposed upon the normal operating voltage of the GaAs diode, the intensity of its emitted radiation varies in sympathy with the alternating voltage (Figure 8, Price, 1989). This provides a simple and inexpensive means of directly modulating the infra-red beam.
The GaAs diode is widely used in surveying. The small dimensions of the junction in which the radiation is emitted gives rise to poorer collimation of the radiation. Therefore, the GaAs laser emits a beam with a relatively large elliptical spread and the brightness of the GaAs laser is lower than that of other lasers. The spectral width of the radiation emitted is usually 2-3 nm compared with 0.001 nm of the visible light, HeNe gas laser and thus the GaAs laser lacks the monochromacity of other lasers, contributing error to the effective propagation velocity. However, GaAs lasers can be made to operate at orders of magnitude greater efficiency than other lasers, can be made much smaller and more rugged, and are considerably less expensive than other lasers (Price, 1989).
Because of the relatively low power radiated, a beam of sufficient power will not be reflected from an unprepared surface. A special reflector is therefore used in order to ensure a good return signal. A plane mirror can be employed, but it requires accurate setting, so in practice, a corner cube reflector is most often used. This reflector will return a beam along a path parallel to the incident path over a wide range of angles of incidence onto the front surface. A cube of glass is usually used with its edges ground into a corner with accuracies of grinding to within a few degrees of arc (Burnside, 1991). The path length of the signal within the reflector must be corrected for, and that is the reason for the setting of the prism constant within the theodolite. The constant for Wild circular prisms is 0. Note that over short distances, the "cat-eye" type reflector commonly used for bicycles will adequately reflect highly oblique incident signals.
Figure 7. Gallium arsenide diode characteristics. From Burnside, 1991.


Figure 8. Modulation of a Gallium arsenide diode. From Price and Uren, 1989.

Selected Technical Data--Leica (Wild) DI4L

Standard deviation of distance measurement	5 mm + 5 mm/km
Breaks in beam	result not affected
Range with one reflector	1.2 km in strong haze 2.5 km in average atmospheric conditions (the maximum range I have shot is 1.8 km)
Carrier wavelength	0.835 �m infra-red
Fine measurement	4,870,255 Hz = 30.7692 m
Coarse measurement	74,927 Hz = 2000 m
Weight--DI4L Counterweight Container Total weight	1.1 kg (2.4 lb) 0.8 kg (2.0 lb) 3.8 kg (8.4 lb) 5.7 kg (12.8 lb)
Beam width at half power	4' (12 cm at 100 m)

Theodolite
The Theodolite is an accurate horizontal and vertical angle measuring device with a telescope and on board electronics for data storage and EDM operation. Fortunately for us, the angles are measured and recorded accurately and electronically, avoiding the need for us to read a vernier and record data manually as is typical on transits and optical theodolites. This electronic theodolite contains circular encoders which sense the rotations of the vertical and horizontal spindles of the telescope, and converts those rotations into horizontal and vertical angles electronically, and displays the values of the angles on a Liquid Crystal Display (LCD) (Moffitt, 1987).
The integrated EDM/Theodolite combination is often called a "Total Station" or "Total Geodetic Station." Output from the horizontal and vertical circular encoders and from the EDM are stored in a data collector. The instrument may convert the data (horizontal and vertical angles and the slope distance) electronically into Easting and Northing coordinates, height difference, and horizontal distance (Moffitt, 1987).

Selected Technical Data--Leica (Wild T-1000)

Standard deviation of angular measurement	3.0' (seconds of a degree) horizontal and vertical
Telescope	Erect image
Magnification	30x
Shortest focusing range	1.7 m
Field at 1000 m	27 m
Displays	2 LCD displays each for 8 digits, sign, decimal point and symbols for user guidance
Keyboard	Weatherproof, 14 multiple function keys, contact pressure 30 g
Automatic power off	About 3 minutes after last keystroke
Angle measurement	Continuous, by absolute encoder
Updates	0.1 to 0.3 seconds
Optical plummet (in tribrach)	Focusing
Magnification	2x
Temperature Range	-20°C to +50°C
Weight--T1000	4.5 kg (9.9 lb)
Container	3.9 kg (8.6 lb)
Total weight	8.4 kg (18.5 lb)

Determination of Easting, Northing, and Elevation
The actual observations made by the Total Station are the horizontal and vertical angles (Hz and V), and the slope length (D)--these are called fundamental measurements. Clearly, from these data, one can determine the relative coordinates of the instrument and reflector (Figure 9). The instrument makes its own reductions based only upon the fundamental measurements. The horizontal angles are made relative to a backsight or reference azimuth or known orientation. This may be determined by a careful compass sighting to a distant, but stable and easily identifiable landmark, or it may be arbitrary. Once the backsight is made, the reference azimuth, (hz0) may be set on the Theodolite. The vertical angle may be called a zenith angle and is measured in a vertical plane down from the upward direction of a vertical or plumb line (Moffitt, 1987).
Instrument Station Reference Location The point over which the Total Station is set up is called an instrument station. Such a point should be marked as accurately as possibly on some firm object. On many surveys, each station is marked by a wooden stake, spike, or piece of REBAR driven flush with the ground and into the top of which a small (~ mm diameter) dimple is marked (a tack may be put in the wooden stake) (Moffitt, 1987). The Total Station measurements and reductions are made relative to the instrument station (E₀, N₀, and H₀--where E refers to distance East, N distance North, and H elevation, and the subscript ₀ indicates a reference value). These values must be predetermined by other means (triangulation, previous survey location, Global Positioning System, etc., or they may be arbitrary). In fact, for most local surveys, with only one survey set up, the instrument station is commonly taken to be (0, 0, 0).
Determinations in the vertical plane In the lower portion of Figure 9, the geometric reductions in a vertical plane are indicated. From the fundamental measurements and a few more observations, the horizontal distance (HD) and elevation (H) may be determined:


(9)
H^* is the elevation difference between instrument and reflector, IH is instrument Height, and PH is the Prism Height.
(10)
Figure 9. Surveying geometry.

Determinations in the horizontal plane
In the upper portion of Figure 9, the map view reductions from the fundamental measurements are shown. Because we are interested in the map relations of the surveyed objects or observations, the distances East (E) and North (N) may be determined in the following way:
(11)
(12)
Foresight and Backsight: the traverse
One other thing that people have trouble with and that I have only recently gotten to the bottom of is the traverse or how you move your station to a new position. The figure below shows how one starts of at the reference position E0, N0, H0, works, and then moves to the new station with a reference position E1, N1, H1. What you have to do is shoot from the first location to the second (foresight) and record the E1, N1, H1, and the bearing Hz0. Then, move to the new location, set up, tell the total station you are at E1, N1, H1, and set the horizontal circle to Hz1 or Hz0+180 degrees (backsight). Then shoot to the reflector set up at the first station and record the position. It should be within a few mm of the original E0, N0, H0.
Precision
The angular accuracy at one standard deviation of the Theodolite is 3", and the linear accuracy at one standard deviation of the EDM is 5 mm � 5 ppm of the slope length. The upper portion of Figure 10 shows a map view of the 1 standard deviation error volume (remember that the horizontal and vertical angles have the same precision). In reality, this volume is really an ellipsoid if we assume that the errors are normally distributed. The significance of this figure and the plot in the lower portion of Figure 10 is that one can anticipate the precision and its changes with slope length, and consider them accordingly. For example, for a 100 m shot, or slope length D = 100 m, the linear error, l, is 5.5 mm, and the angular error, a, is 1.5 mm, and therefore, the error volume is 11 mm along the shot, and 3 mm wide along horizontal and vertical arcs. Note that this is much smaller than reflector placement error.
Planning and executing a surveying/mapping project
The important concern when planning a mapping project is the question that is being asked, or problem being addressed. While mapping and surveying may be intrinsically fascinating and interesting of themselves, unless you want to become a surveyor, they are only methods used to collect data in order to address a geologic problem. Therefore, the technique and methods should be appropriate for the problem. You should collect data at the appropriate scale and precision that will most efficiently shed light upon the problem. Check (Compton, 1985) for an inspiring and essential text to guide you in the field.
Basic detailed mapping procedure
This technique is appropriate for outcrop to kilometer scale mapping (1:10 to 1:1000 scale) for which no adequate base map exists. The basic idea is to shoot in control points with the Total Station, establish a base map, and then use tape and compass and triangulation to interpolate and locate features between the control points.
Figure 10. Surveying precision.
1) Recon the area. Consider the problem, and instrument locations.
2) Flag points. Place flags (numbered) on important features (along large fractures or contacts for example). The density depends upon the scale of the problem and the map. If the flags are too close, you will spend a long time shooting them in and then may be confused while mapping. If they are too far apart, you will spend a long time measuring off distances and bearings between points. I suggest a radial distance between flags of 5 to 10 m.
3) Shoot points with Total Station. Make sure that the point number corresponds either directly or in some noted way with the flag numbers.
4) Produce basemap. This may be done in several ways. It may be done in the field or the office by manually plotting the location and recording the elevation of the point on grid paper. Be sure to use metric grid paper. The other way to produce the basemap is to download the data as described above, and contour and plot the points with their numbers. Print the contour map out to an appropriate size for your mapping. I do this step in Deltagraph Pro by importing a four column (point number, E, N, H) Excel file, and then plot an XYZContour chart (select the data by using the arrow in between the two label words:

c	Label
Label

Show symbols, and adjust the size of the chart by selecting Axis Attributes under the Axis choice under the Chart menu. Make sure that the X and Y (E and N) axes are the same scale. Choose an appropriate contour interval. Show the labels of the points (point numbers, for example) by selecting the chart, and then under the Chart menu, select Show Values, and choose a location besides None, and the Text should be Category. This will plot the items in the corresponding cell in the Labels Column adjacent to the symbol.
5) Mapping. The above method should provide a basemap with the control points displayed and labeled. Tape the base map or a portion of it to your map board and overlay a piece of vellum or mylar. Mark the control points and a few more index marks, especially if you will use more than one page for the base map. As you map, observe the locations and orientations of objects and features by noting the bearing and distance (using a Brunton Compass and a tape measure or a well calibrated pace) from one or more control points. You may also shoot a bearing to two or more control points, plot the angle on the map and triangulate your location. Mark your observations carefully with a sharp pencil. As you map, record the topography. If you plotted contour lines, trace and modify them in the field and include the subtleties that you can. In the evenings or whenever you are at a stable point, trace your lines in ink using a very fine pen.
Examples
Below are a few examples of different mapping projects our members of our group have completed recently. These are meant to illustrate the different solutions to different problems that have been achieved.
Topography
My own interest in the geomorphic responses to active faulting provides the following example. During the June 28, 1993 Landers California earthquake, spectacular faulted landforms were produced all along the surface rupture. These landforms provided an important opportunity to document the original shapes and initial modification of faulted landforms. This project posed several interesting survey challenges: 1) producing detailed scarp profiles and longitudinal profiles of gully main and tributary channels, and detailed topographic contour maps of faulted knickpoints; 2) establishing a cheap but hopefully stable control network; and 3) reoccupying the network.
Scarp profiles--coordinate transformation by rotation These profiles are made perpendicular to the fault scarp and are ideally planar. In the field, we marked the upper and lower ends of the profiles with steel rods, and walk between, shooting points about every 50 cm until we are at the free face of the scarp, where points are shot about every 10 - 20 cm. These data are down loaded as described above, and then the profile is projected to a plane perpendicular to the scarp. The coordinates are transformed by a rotation about the origin so that one axis is the distance along the profile and the other is the deviation from the plane of the profile. The equations for such a coordinate transformation are as follows:
(13)
(14)
The figure above and to the right illustrates this geometry.

The following two figures illustrate the initial and rotated coordinate systems for a scarp profile. In the lower plot, the coordinate system has been rotated by -36° (counterclockwise), and the horizontal axis used as the horizontal distance along the profile. The vertical axis indicates that the deviation from a plane is a maximum of 5 m over 30 m.

Longitudinal profiles
The gully longitudinal profiles were determined by observing points along the lowest portion of the active channel in a given gully. The distance along the profile is the sum of the distances between individual points. In other words, the longitudinal profile is not projected to a plane, rather the run, or distance along the channel, is stretched out and plotted as the horizontal axis for the longitudinal profile, while the vertical axis is the corresponding elevation measurements. This plot is important since it indicates the effective slope for the flow, which clearly may not be ideally contained within a single plane.
Contour maps - topographic mapping Because of the precision and rapidity of data acquisition of our Total Station, as well as the opportunity to quantitatively document the three-dimensional (i.e. volumetric) changes in scarp form, particularly where the gully channels were disrupted by the earthquake surface rupture, we made detailed observations of small portions of the gully channel surfaces. We have observed after one year, and expect significant change within a few years. The data were observed over a relatively small area (125 to 250 m²) at a fairly high density: 1 measurement every 0.5 to 2.5 m². The lower density shots seemed to optimally address the problem, and we achieved them by setting a goal of 100 shots to complete the map. This seems to be a better psychological tactic than simply observing every apparently important break in slope.
We used the program Deltagraph Pro to contour our data. This program has two advantages: 1) it can contour a set of data with an irregular boundary, enclosing the area of interest with a relatively close fitting polygon, instead of a rectangle, as is common with most contouring programs; and 2) it uses triangle-based terrain modeling (commonly called a Triangular Irregular Network--TIN; Kennie, 1990). This method essentially models the surface as a series of planar, triangular elements, each of which contains three neighboring data points (Figure 11). As you are surveying it is important for the reflector person to attempt to visualize this triangle network in order to assess the appropriateness of a given observation point. The points where contour lines intersect the lines between neighboring points are determined by direct, linear interpolation. The contour lines are then determined by connecting those intersections. Because the contour lines are not smoothed, this method provides a basic contour map that honors each data point directly. However, it does generate a somewhat jagged map which incidentally appears more appropriate for semi-arid landscapes rather than smoother humid-temperate landscapes.
Given a method of plotting the map and taking it in the field, the subtleties of the contours could be adjusted manually. We adjusted our maps slightly in the office, honoring the topography as best as possible, by adding a few more vertices to the contour lines in the drawing program Canvas.