15.1 Survey Adjustments and Error Quantification

The January 2025 California Board for Professional Engineers, Land Surveyors, and Geologists (BPELSG) Professional Land Surveyor test plan puts Analysis and Evaluation at 26 percent of the 4-hour Prometric exam. Professional activity IV.6 is to analyze results of survey adjustments (e.g., least squares, error analysis). Knowledge X names data bias, error ellipses, and residuals. Knowledge DD names use and interpretation of significant figures for maps, plats, or reports. Independent OpenExamPrep teaching in this section is how to read an adjustment. It is not a claim of Board approval, partnership, or official sponsorship. The 2025 test plan and the Prometric candidate information bulletin do not publish a numeric PLS accuracy cut, a required ellipse size, or an official item count. Do not invent one.

What an adjustment is allowed to do

A survey adjustment is a model. Observations go in. Coordinates, orientations, and residuals come out. Least squares chooses the unknown parameters that minimize the weighted sum of squared residuals. If the weights are honest and the observations are unbiased after reduction, the adjusted coordinates are the most probable values under that model. If a prism constant, an antenna height, or a datum epoch is wrong on every observation, least squares will still converge. It will report a tight network sitting in the wrong place.

That is the exam distinction between precision (how well the observations agree with each other) and accuracy (how close the result is to the datum and to the ground). Residuals measure internal agreement. They do not certify the realization, the epoch, or a missing systematic.

Chapter 6 already covered Public Resources Code (PRC) 8813.1–8813.3 connection and documentation rules when California Coordinate System of 1983 (CCS83) values are used or established. This section is the analysis of the numbers those statutes tell you to keep: residuals, closures, and—if you claim a Federal Geographic Data Committee (FGDC) or Federal Geodetic Control Subcommittee (FGCS) accuracy—the written justification.

Worked least-squares residual example

Three independent GNSS occupations of new station P are reduced to CCS83 Zone 5, NAD83(2011) epoch 2010.00. Each occupation is treated as an uncorrelated easting and northing with equal weight. The coordinates below are a teaching set, not a published control datasheet.

The equal-weight least-squares solution for two coordinates from three uncorrelated occupations is the arithmetic mean:

• Adjusted easting = (1,984,512.318 + 1,984,512.342 + 1,984,512.309) / 3 = 1,984,512.323 m
• Adjusted northing = (562,887.105 + 562,887.090 + 562,887.120) / 3 = 562,887.105 m

Residuals v = observed − adjusted:

The easting residuals sum to 0.000 m. The northing residuals sum to 0.000 m. That is a sanity check, not a quality award. Six observations (easting and northing on three occupations) and two unknowns give n − u = 4 degrees of freedom.

Sum of squared residuals Σv² = 0.005² + 0.019² + 0.015² + 0.014² + 0.015² = 0.001032 m².

The a-posteriori reference standard deviation is sigma-0 = √(Σv² / (n − u)) = √(0.001032 / 4) = √0.000258 = 0.016 m.

That 0.016 m describes the scatter of these occupations around the mean under this equal-weight model. It is not a Board-required tolerance, and it is not FGDC network accuracy relative to the California Spatial Reference Network (CSRN). If occupation 2 had been 0.25 m away in easting, the mean would still compute, sigma-0 would jump, and a competent analysis would stop and ask whether a wrong antenna, a wrong mount point, or a different monument was occupied—not reweight the file until a flag turns green.

Error ellipse from the covariance

The simple mean above produces a nearly circular one-sigma uncertainty of about 0.009 m in each coordinate if easting and northing are taken as independent (sigma-0 / √3). Real networks are not that tidy. After a combined GNSS-plus-traverse adjustment of the same station, the software reports this stated teaching covariance (not from a Board table):

• sigma-E = 0.013 m, so variance 0.000169 m²
• sigma-N = 0.020 m, so variance 0.000400 m²
• covariance sigma-EN = 0.000120 m²

The error ellipse semi-axes are the square roots of the covariance-matrix eigenvalues:

λ = (0.000169 + 0.000400)/2 ± √[((0.000169 − 0.000400)/2)² + 0.000120²] = 0.0002845 ± 0.0001666

• λ₁ = 0.000451 → one-sigma semi-major a = 0.021 m
• λ₂ = 0.000118 → one-sigma semi-minor b = 0.011 m

Orientation: tan(2θ) = 2(0.000120) / (0.000169 − 0.000400) = −1.039, so θ ≈ 67° from +E toward +N. The semi-major axis lies about azimuth 023° (N 23° E).

FGDC-STD-007.2-1998 reports horizontal accuracy at a 95 percent confidence level. For a bivariate normal ellipse, the 95 percent contour uses √5.991 ≈ 2.45 times the one-sigma semi-axes, giving about 0.052 m by 0.027 m. If you claim an FGDC local or network accuracy, PRC 8813.2 still wants the control diagram, equipment, adjustment method, and final residuals or closures, and it requires the FGDC accuracy to be identified as local or network. The ellipse is a picture of the covariance you computed. It is not a substitute for that statute, and it is not a BPELSG-published pass size for the California-specific PLS exam.

Bias the residuals will not show

Give every occupation the same +0.025 m antenna-height blunder. In this teaching case the three plane coordinates in the table are treated as unchanged. Residuals stay 0.005, 0.024, and 0.021 m. Sigma-0 stays 0.016 m. The ellipse stays small. The ellipsoid height of P is wrong by 2.5 cm, and any orthometric height built as H = h − N from that h is equally biased.

A distance network with a −30 mm prism constant omitted on every electronic distance measurement has the same disease: the figure shrinks, the residuals can look excellent, and the ellipse can be tiny. That is data bias (knowledge X). Least squares reduces random, zero-mean noise. It does not discover a systematic that is common to every observation unless you have an independent check: a known-length baseline, a hold-out CSRN station, or a second instrument.

Significant figures after the adjustment

Knowledge DD is significant figures for maps, plats, or reports. Station P's adjusted easting is 1,984,512.323 m. Writing 1,984,512.32347 m because the printout showed extra digits is a reporting error. The occupations disagreed by up to 0.024 m in plane; sigma-0 is 0.016 m. Hundredths of a millimeter are not information.

The same rule hits a Record of Survey that inverse-pairs P to a 1924 pipe and prints 312.8472 feet. If the 1924 record was chained to the nearest tenth of a foot, the extra places describe your modern instrument, not the historic line. Section 15.3 takes that comparison to method eras. Here the rule is: the adjusted value cannot be reported more sharply than the observations and the covariance support.

What to read on an adjustment printout

PRC 8815.4 still requires equipment, procedures, closures, adjustments, and a control diagram when an FGCS or FGDC order of accuracy is claimed. Professional Land Surveyors' Act section 8771.5 still requires the control scheme on a Record of Survey that shows California Coordinate System coordinates. Analysis that ends at "the software accepted it" has not finished IV.6.