Surveying Using Least Squares
Most surveyors know, from study and training if not personal experience, that “least squares” are important. That might be the extent of what they know though. Important, yes, but why? And what exactly is least squares adjustment? What exactly does it do for land surveyors and other geospatial professionals?
Least squares adjustment can be defined, as “a model for the solution of an overdetermined system of equations based on the principle of least squares of observation residuals.” For surveyors, “overdetermined systems” are the networks of related coordinates used to establish boundaries, locate points on Earth, facilitate large construction projects, etc. These networks are usually “overdetermined” in a mathematical sense because of varying randomness in observation technologies and measuring equipment. Usefully analyzing all this randomness to improve network accuracy and integrity requires a set of powerful mathematical techniques (including matrix algebra) that are collectively termed least squares adjustment.
This is an over-simplified explanation, of course, but perhaps establishes that least squares adjustment plays an important role in any surveying practice that uses or creates coordinate networks. That is to say, all surveying practices.
How can surveyors apply least squares adjustment efficiently? After all, even though surveyors tend to be extremely intelligent and well educated, they often aren’t comfortable using some of these types of advanced mathematics on a daily basis.
Fortunately, as computing power became widely available, practical least squares adjustment was one of the first applications to be developed specifically for land surveyors. One such program, MicroSurvey’s STAR*NET, has been around since the mid-1980s, and has been used by thousands of surveying firms in many different workflows. Following are examples of two firms with very different niches and business plans, both using STAR*NET for least squares adjustment in different ways to improve their client service and the quality of their deliverables.
Surveyors On Site
Surveyors On Site (SOS) was founded in 2011 as a survey firm focused on the use of the most advanced location technology available in service of advanced infrastructure development. That turned out to be an excellent business model, and SOS quickly grew to become a multinational firm headquartered in Windsor, Ontario, with two additional Ontario offices and an office in Livonia, Mich. They’re also known for extremely snazzy, blaze-orange survey vehicles.
Since 2015, the SOS commitment to advanced technology has included regular use of STAR*NET (version 9) to establish and maintain control networks for large construction projects, particularly ongoing development of the Gordie Howe International Bridge, described as a “once-in-a-generation undertaking” and one of North America’s largest infrastructure projects. Actual bridge construction is still a few years out, but preparatory work on the Herb Gray Parkway already makes this a major project, including construction of six bridges, 12 tunnels, 3 million cubic meters of cut and fill, and associated utility realignment and roadway development.
SOS established the Canadian control network for all that work, and has been doing much of the actual layout staking. “It’s a huge site, 2.5 kilometers by 1 kilometer, and very active,” says SOS Director and Partner Christopher Oyler, OLS. “I think we’re the only ones here who really appreciate the importance and difficulty of the control network — everyone else expects it to just work.”
In fact, it does “just work” for all the contractors on the site, but establishing a reliable control network, with points on both sides of the Detroit River, was a complex and formidable undertaking for SOS. Oyler says, “We used STAR*NET to adjust and reconcile close to 40 different datasets. That’s about 1,300 different GNSS and terrestrial observations, 1,497 distances and 4,700 independent measurements.” Oyler used STAR*NET from the beginning of the project for many different adjustment tasks. He describes the application as “very clickable” and adds, “It’s easy to find and apply the tool I need.
“For example, I can design individual surveys by entering instrument specifications, which computes upper and lower bounds that help us to assess the quality of our work. And it’s superb for post-processing of GNSS surveys — we bring our raw files directly into STAR*NET and processing is extremely rigorous, compared to other programs, when analyzing strength of geometry and other constellation factors.”
Oyler is also able to fine-tune the coordinates he publishes for use by other contractors on the site. “Because the site is so big, we figure it out by grid and give an average combined scale factor. But we also have separate control just for actual bridge layout. It’s proven to be the best way to make everything fit.”
Oyler says there was a “fairly steep learning curve” to use STAR*NET effectively, but he was able to get past it quickly with available videos, discussions with MicroSurvey’s support team and simply trying things out. “After some initial effort, the interface became really intuitive and easy to use.”
After learning on the job with the Gordie Howe project, SOS now uses STAR*NET fairly routinely. “We don’t use it for everything — it can seem like overkill on smaller projects — but any job where we’re working with a lot of points, or going back a lot, we definitely process with STAR*NET and sometimes that’s caught some big potential errors,” Oyler says. “After all, we’re professionals, and we want our work to stand out for accuracy and reliability. Using STAR*NET [for least squares adjustment], for us, has become a professional obligation.”
Johnson-Frank & Associates
Johnson-Frank & Associates is a very different firm than Surveyors On Site. Founded in 1975, they’re about 36 years older. They’re a full service firm, but with considerably more emphasis on boundary and cadastral surveying, typically for public agencies or municipalities. They’re based in Southern California, so it’s no surprise that one of Johnson-Frank’s specialties is retracement of Government Land Office (GLO) surveys.
They’ve also been using STAR*NET since 1987, and they use it a little differently — and more often. “We’re using version 9 now, and it’s in every step of everything we do,” says Alan D. Frank, PLS, and president of Johnson-Frank & Associates. “For us, it’s a quality check that applies to a boundary survey, or topo survey, or whatever. I’d estimate a 5 percent quality improvement across the board, which is worth it and good for our reputation. And, now and then, it saves our butt, which is priceless.”
“It finds errors, and we certainly don’t want any errors going out,” adds Roger A. Frank, PLS, vice president and founder (and Alan’s father). “Sure, using STAR*NET adds a little time to our projects, but it’s worth it — just like taking a little time to buckle your seatbelt is worth it.”
In addition to using STAR*NET as a QC procedure and to process survey data and adjust networks, like most users, Johnson-Frank also applies the power of least squares adjustment to boundary surveying, and especially to the art of preparing to do field work for retracements of GLO surveys and of the Mexican and Spanish Rancho surveys that are a special challenge for land surveyors in Southern California. In fact, the package Roger Frank bought in 1987 was “either the first or second” package ever sold, and he immediately put it to work on the difficult boundary surveys that his firm was famous for. He shared what he discovered in several articles over the years. Frank recalls:
- In late 1995, I assisted a local surveyor to locate the boundaries of portions of seven sections in southwestern South Dakota. During the course of that project, I developed a system using least squares and particularly STAR*NET software to develop initial State Plane search coordinates based on the original survey data and USGS quad sheet topography, and then continually refine those coordinates on a daily basis, based on both the record information available and the physical evidence found in the field. This method of developing increasingly better search coordinates, teamed with [then] newly developed capabilities of RTK GPS, allowed us to complete this project rapidly and effectively.
- In 1996, our firm was contracted to monument the boundary of Marine Corps Logistics Base in Barstow, Ca. In this survey, I continued to develop the “least squares-RTK” search system devised on the Dakota survey by including not only the data from the original PLSS survey notes, but also all of the other record information in the area. On this project, we combined the original record information from the 1850s, as well as more modern information from the 1940s through the 1990s. All of the record information was weighted in accordance with the normal errors and accuracy to be expected from the equipment and procedures of the time period in which the survey was made. Additionally, STAR*NET has an entry mode in which bearing data from record maps can be entered, and the software converts the bearing data to angles. Using the resulting angles and distances, the adjustment file is not hindered by basis of bearings differences. This results in the ability to easily attach all of the record data to the State Plane coordinate system to produce coordinates in that system for ease of search using GPS. Using RTK GPS attached to a small helicopter, in four days of search operation, we were able to locate 20 original PLSS corners, which had not been seen since 1857.
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To boundary surveyors who have actually spent weeks in the field beating through brush and climbing steep hillsides looking for the remnants of stone mounds, quarts of charcoal, glass rods, scribed trees, or some other 100+ year old monument set by GLO surveyors, the techniques described by Roger Frank sound revolutionary. Reliably reducing search circle diameters from 100-feet or more down to a few feet makes it possible to actually find far more crucial original monuments. And it saves days in the field — finding 20 original PLSS corners in four days, in rough country, is world class surveying.