© ISO 2012 Optics and optical instruments — Field procedures for testing geodetic and surveying instruments — Part 6 Rotating lasers Optique et instruments d’optique — Méthodes d’essai sur site des in[.]
Requirements
Before commencing surveying, it is important that the operator investigates that the precision in use of the measuring equipment is appropriate to the intended measuring task.
The rotating laser and its associated equipment must be properly calibrated and maintained as outlined in the manufacturer's handbook Additionally, they should be utilized with tripods and leveling staffs as per the manufacturer's recommendations to ensure optimal performance.
The results of these tests are influenced by meteorological conditions, especially by the temperature gradient
Optimal weather conditions for measurements include an overcast sky and low wind speed, though specific requirements may differ by location It's essential to consider the actual weather at the time of measurement and the surface type involved The testing conditions should align with those anticipated during the actual measuring task, in accordance with ISO 7077 and ISO 7078 standards.
This part of ISO 17123 describes two different field procedures as given in Clauses 5 and 6 The operator shall choose the procedure which is most relevant to the project’s particular requirements.
Procedure 1: Simplified test procedure
The simplified test procedure provides an estimate as to whether the precision of a given item of rotating-laser equipment is within the specified permitted deviation, according to ISO 4463-1.
This test procedure is designed to verify the precision of a rotating laser used in area leveling applications, particularly in scenarios where measurements with varying site lengths are frequently encountered, such as in building construction.
The simplified test procedure relies on a limited set of measurements, which prevents the calculation of a significant standard deviation and standard uncertainty (Type A) For a more accurate evaluation of the rotating laser in field conditions, it is advisable to follow the comprehensive full test procedure outlined in Clause 6.
This testing procedure requires a test field with established height differences considered as accurate values In the absence of such a test field, it becomes essential to ascertain the unknown height differences.
To achieve a higher level of accuracy than that provided by a rotating laser, an optical level must be utilized, as specified in ISO 17123-2 In cases where a test field with established height differences is unavailable, it is essential to follow the complete test procedure outlined in Clause 6.
In the absence of a leveling instrument, a rotating laser can effectively determine accurate height differences between various points through central setups By rotating the laser plane 180° at each setup, two height differences should be recorded The average of these repeated measurements will yield the accepted true height differences.
Procedure 2: Full test procedure
To assess the highest level of precision achievable by a specific rotating laser and its associated equipment in real-world conditions, a comprehensive testing procedure will be implemented by a single survey team.
This test procedure is designed to assess the deflective deviation, denoted as \( a \), along with the two components \( b_1 \) and \( b_2 \) that represent the deviation of the rotating axis from the true vertical The overall deviation of the rotating laser is calculated using the formula \( b = \sqrt{b_1^2 + b_2^2} \) (refer to Figure 1).
2 © ISO 2012 – All rights reserved a) Horizontal plane (top view) b) Vertical plane through x ′ (side view)
Figure 1 — Deflective deviations a and b (see Figure 5)
The recommended measuring distances within the test field (see Figure 3) are 40 m Sight lengths greater than
For precision-in-use testing, a distance of 40 meters may be utilized, depending on project specifications or when assessing the precision range of a rotating laser at various distances.
The test procedure outlined in Clause 6 of ISO 17123 aims to assess the precision of a specific rotating laser This precision is quantified by the experimental standard deviation, s, which represents the height difference between the instrument level and a leveling staff.
(reading at the staff) at a distance of 40 m This experimental standard deviation corresponds to the standard uncertainty of Type A: s ISO-ROLAS =u ISO-ROLAS
Further, this procedure may be used to determine:
— the standard uncertainty as a measure of precision in use of rotating lasers by a single survey team with a single instrument and its ancillary equipment at a given time;
— the standard uncertainty as a measure of precision in use of a single instrument over time and differing environmental conditions;
Standard uncertainties serve as a crucial measure of precision when evaluating multiple rotating lasers This approach facilitates a comparison of the achievable precisions of these lasers under similar field conditions.
Statistical tests are essential for assessing if the experimental standard deviation, \$s\$, aligns with the theoretical standard deviation, \$\sigma\$, of the population They also help determine if two samples originate from the same population, verify if the deflective deviation, \$a\$, equals zero, and check if the deviation, \$b\$, of the rotating axis from the true vertical of the rotating laser is zero.
Configuration of the test field
To minimize the effects of refraction, it is essential to select a test area that is relatively horizontal Six fixed target points, labeled 1 through 6, should be established at varying distances between 10 m and 60 m from the instrument station S, ensuring they are positioned in approximately the same horizontal plane Additionally, the directions from the instrument to these six points should be evenly distributed across the horizon.
Figure 2 — Configuration of the test field for the simplified test procedure
To ensure reliable results, the target points shall be marked in a stable manner and reliably fixed during the test measurements, including repeat measurements.
The height differences between the six fixed points, 1, 2, 3, 4, 5 and 6, shall be determined using an optical level of known high accuracy as described in Clause 4.
The following five height differences between the t = 6 target points are known: d d d t t t
Measurements
The instrument must be securely positioned above point S, and the laser beam should stabilize before measurements begin To maintain the consistency of the laser plane throughout the measurement cycle, a fixed target will be monitored before and after each set of measurements (j=1, ).
Six separate readings, x j,1 to x j,6, on the scale of the levelling staff shall be carried out to each fixed target point,
To ensure accurate measurements, the instrument must be lifted and rotated approximately 70° between two sets of readings It should then be positioned slightly differently and relevelled Additionally, a minimum interval of 10 minutes is required between any two sets of readings.
Each reading shall be taken in a precise mode according to the recommendations of the manufacturer.
Calculation
The evaluation of the readings x t for each set j is based on the following differences: d d x x x x j j t t j j j t j t
j=1, ,5 and t=2, 6 (2) respectively d j = x j t , - x j t , -1 (3) where t is the number of the target point.
Calculating d , the mean of the differences d j , the residual vector of the height differences in set j is obtained by r j =d d- j j=1, ,5 (4)
Finally the sum of the residual squares of all five sets yields
2 j (5) ν = × − =5 (6 1 25) is the corresponding number of degrees of freedom. s r v u
The equation \$\sum 2 = ISO (6)\$ indicates that \$s\$ represents the experimental standard deviation, while \$u_{ISO}\$ denotes the standard uncertainty (Type A) of a single measured height difference, \$d_{j,t,t-1}\$, between two points in the test field This section of ISO 17123 highlights a measure of precision in relation to the standard uncertainty of a Type A evaluation, encompassing both systematic and random errors.
Configuration of the test line
To minimize the effects of refraction, it is essential to select a test area that is relatively horizontal The ground should be compact and have a uniform surface, avoiding roads made of asphalt or concrete.
If there is direct sunlight, the instrument and the levelling staffs shall be shaded, for example by an umbrella.
Two levelling points, A and B, will be established approximately 40 m apart, with levelling staffs securely positioned to ensure reliable test measurements, including repeats The instrument will be set up at positions S1, S2, and S3, with distances to the levelling points following the specifications in Figure 3 Position S1 will be selected to be equidistant from points A and B, at a distance of 20 m.
Figure 3 — Configuration of the test line for the full test procedure
Measurements
Before commencing the measurements, the instrument shall be adjusted as specified by the manufacturer.
The complete test procedure requires conducting four series of measurements, utilizing three instrument setups (S1, S2, and S3) as illustrated in Figure 3 For each setup, four sets of readings are taken, comprising two readings for rod A (\$x_{Aj}\$) and rod B (\$x_{Bj}\$) After each set, the instrument's orientation must be adjusted clockwise by 90° (refer to Figure 4) Consequently, each series yields a total of 12 readings for each rod, calculated as \$j = 3 \times 4\$ To ensure accurate alignment of the instrument deviation (\$b\$) during measurements, it is essential that the instrument is oriented in the same direction across all three positions (S1, S2, and S3) while maintaining a consistent sense of rotation.
Each time a new reference direction is established using the reference marks on the tripod head, the instrument must be carefully relevelled If the instrument includes a compensator, it is essential to ensure its proper functioning It is advisable to designate the four orientations of the instrument on the ground plate The twelve measurements for each set can be numbered as illustrated in Figure 4, and all readings should be taken in precise mode following the manufacturer's recommendations.
Calculation
The possible deviations of a rotating laser may be modelled as shown in Figure 5.
Figure 5 — Model of instrument deviations
To establish a horizontal sighting in the specified measuring setup, the readings on the leveling staffs for chosen sighting distances can be adjusted to account for deviations a and b (refer to Table 1).
Table 1 — Corrections of the readings
From the observation formulae for the i th series, the residuals, r 1 to r 12, are obtained (see Table 2).
Table 2 presents the observation formulae for the i-th series, detailing various calculations based on different weighting factors (p = 2.0 and p = 0.5) The formulas include height differences (h) between levelling staffs B and A, with specific equations for each series For instance, the formula for r1 is given by \$r_1 = -h b_1 (x_{B,1} - x_{A,1})\$, while r2 is expressed as \$r_2 = +h b_2 - (x_{B,2} - x_{A,2})\$ Each subsequent formula follows a similar structure, incorporating the respective height differences and weighting factors, emphasizing the importance of accurate measurements in levelling operations.
In this study, we analyze an over-determined system with 12 observations and four unknown parameters: h, a, b₁, and b₂, which necessitates a parametric adjustment Given that the observation formulas are linear, we can represent the data in matrix notation as follows: \[r = Ay - x\]Here, \( r \) is the (12 × 1) residual vector comprising \( r_j \) for \( j = 1, \ldots, 12 \), while \( x \) is the (12 × 1) quasi-observation vector of height differences This vector \( x \) is defined as \( x = B - x A \), where \( x_A \) and \( x_B \) are the (12 × 1) reading vectors from levelling staffs A and B, respectively, with \( x_{Aj} \) and \( x_{Bj} \) for \( j = 1, \ldots, 12 \).
y (4 × 1) is the vector of the unknown parameters.
(8) the solution vector of the unknown parameters is
is given by diag (pj) = (2 2 2 2 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5) (10) Inserting Formulae (8) and (10) into Formula (9) the solution vector will be obtained finally by
Regarding Formula (7) the experimental standard deviation for a sighting distance of 40 m is given by s= r Pr T ν with ν 4 8− = (12)
From all series i = 1, …, 4 of observations we can derive the mean values of the parameters
Finally we get the total deviation of the rotating axis from the true vertical of the rotating laser, referenced to a sighting distance of 40 m: b= b 1 2 +b 2 2 (14)
With Formula (12) the overall experimental standard deviation of all series i = 1, …, 4 yields s s i r Pr i i T i i
The standard uncertainty (Type A) of the height difference, denoted as \( h \), between the instrument level and a levelling staff reading, is referenced to a sighting distance of 40 m, expressed as \( u_{\text{ISO-ROLAS}} = s \) (16).
The experimental standard deviation for the parameters of all series can be calculated by s y( ) =s 1diag Q
Thus the standard deviations and the standard uncertainties (Type A), respectively, of the parameters are given by s h =u h( ) 0,14= s (19) s a =u a( ) 0,25= s (20) s b 1 =s b 2 =s b 12 =0,20s (21)
Applying the law of variance covariance propagation on Formula (14), the experimental standard deviation of the parameter b can be written as s b = b1 b s b +b s b
Statistical tests
Statistical tests are recommended for the full test procedure only.
For the interpretation of the results, statistical tests shall be carried out using
— the experimental standard deviation, s, of a height difference, h, between the instrument level and a levelling staff (reading at the levelling staff) referenced to a sighting distance of 40 m,
— the deflective deviation, a, referenced to a sighting distance of 40 m and its standard deviation, s a , and
The total deviation, \( b \), of the rotating axis from the true vertical of the rotating laser at a sighting distance of 40 m, along with its standard deviation, \( s_b \), is analyzed to determine if the calculated experimental standard deviation, \( s \), for a single reading at a leveling staff is less than the manufacturer's specified value, \( \sigma \), or any other predetermined value, \( \sigma \).
Manufacturers typically express precision through the deflective angle from the horizontal, which should be interpreted in relation to the standard deviation, σ, at a distance of 40 meters Additionally, it is important to determine whether two experimental standard deviations, s and s', derived from different measurement samples, belong to the same population, given that both samples share the same degrees of freedom, ν.
The experimental standard deviations, s and s, may be obtained from
— two samples of measurements by the same instrument at different times, or
— two samples of measurements by different instruments. c) Is the deflective deviation, a, equal to zero? d) Is the total deviation, b, of the rotating axis from the true vertical equal to zero?
For the following tests, a confidence level of 1−α = 0,95 and, according to the design of the measurements, a number of degrees of freedom of ν = 32 are assumed.
Question Null hypothesis Alternative hypothesis a) s=σ s>σ b) σ σ= σ σ≠ c) a=0 a≠0 d) b≠0
The null hypothesis stating that the experimental standard deviation, s, is smaller than or equal to a theoretical or a predetermined value, σ, is not rejected if the following condition is fulfilled: s≤ ⋅σ χ − ν ν α
Otherwise, the null hypothesis is rejected.
In experiments involving two distinct samples, a test determines if the experimental standard deviations, \( s \) and \( \tilde{s} \), originate from the same population The null hypothesis, \( \sigma = \tilde{\sigma} \), remains accepted when specific conditions are met.
Otherwise, the null hypothesis is rejected.
The null hypothesis, stating that the deflective deviation, a, of the rotating laser is equal to zero, is not rejected if the following condition is fulfilled: a s t≤ a ⋅ 1 − α / 2 ( )ν (34) a s t≤ a ⋅ 0 975 , ( )32 (35) s a =0,25⋅s (36) t 0 975 , ( )32 =2 04, (37) a ≤ ⋅0 51, s (38)
Otherwise, the null hypothesis is rejected.
The null hypothesis posits that the total deviation, \( b \), of the rotating axis from the true vertical of the rotating laser is not rejected if the condition \( b_{st} \leq b \cdot 1 - \frac{\alpha}{2} \) is satisfied Specifically, this translates to \( b_{st} \leq b \cdot 0.975 \) Additionally, with \( s_b = 0.20 \cdot s \) and \( t_{0.975, (32)} = 2.04 \), it follows that \( b \leq 0.42 \cdot s \).
Otherwise, the null hypothesis is rejected.
NOTE In practice, the parameters a and b may significantly influence the height readings.
7 Influence quantities and combined standard uncertainty evaluation
The standard uncertainty \( u_{\text{ISO-ROLAS}} \), as calculated in section 6.3, serves as a precision measure for the specified test configuration under current conditions To realistically estimate the accuracy of a quantitative measure for a specific application, it is essential to consider all potential influencing factors that may impact the measurement outcome This approach allows us to derive uncertainty components, potentially utilizing a Type A evaluation method.
According to ISO 17123-1, the evaluation of standard uncertainties, including Type A and Type B evaluations, is governed by specific rules Additionally, the potential influence quantities associated with the use of a rotating laser are systematically compiled.
Table 4 — Typical influence quantities of the rotating laser
Sources of uncertainty (influence quantity) Evaluation Distribution
I Relevant sources of the rotating laser
Typical measure of precision for measuring distances up to 40 m Type A u ISO-ROLAS normal
Expanded measurement range Type A normal
Non-orthogonality between rotating axis and laser path, parameter a Type B rectangular
Inclination of the rotating laser plane or standing (rotating) axis, parameter b Type B triangular
Instability of laser beam, variation of the parameters a and b by temperature Type B rectangular
II Error sources of the staff
Tilts of the staff, maladjustment of spot bubble Type B rectangular
III Error patterns of the electronic reading
Round-off error Type B rectangular
Dependence on light intensity Type B rectangular
Sources of uncertainty (influence quantity) Evaluation Distribution
Dependence on atmospheric conditions Type B rectangular
Error due to subsidence of instrument Type B rectangular
Curvature of the earth Type B rectangular
To determine the standard uncertainties of individual influence quantities based on their distributions, it is often recommended to establish upper and lower limits Furthermore, it is essential to specify the probability that the value significantly falls within this range Detailed guidance on this process can be found in ISO/IEC Guide 98-3:2008, section 4.3.
After calculating the combined standard uncertainty, it is often beneficial to express the expanded uncertainty This approach provides a higher level of confidence, which enhances the accuracy of the measurement tolerance indication.
An example for the calculation of a typical uncertainty budget of the rotating laser is given in Annex C.
Example of the simplified test procedure
A.1 Configuration of the test field
A level of known sufficient accuracy is used to determine the reference heights (relative heights) of the six target points of the test field.
The experimental standard deviation and the standard uncertainty, respectively, of a single height difference is determined according to the full test procedure as given in ISO 17123-2:2001, Clause 6. s x =u x =0 2, mm
The relative heights of the six target points and the height differences were obtained as x 1 02 2, m x 2 21 4, m d 2 1 , = −0 180 8, m x 3 =1 637 6, m d 3 2 , = +0 116 2, m x 4 12 4, m d 4 3 , = +0 074 8, m x 5 610, m d 5 4 , = −0 151 4, m x 6 =1 608 8, m d 6 5 , = +0 047 8, m
Table A.1 contains the measured values, x j t , , in columns 1 to 3, and the height differences, d t t , −1 , in column 5, as given in A.1.
Instrument type and number: NN xxx 630401
The height differences, denoted as \$d_{tt,-1}\$, are calculated using Formula (2) as shown in column 4 of Table A.1 Subsequently, the residuals \$r_{tt,-1}\$ are derived from Formula (4), detailed in column 6 of the same table The sum of squares of the residuals, \$\Sigma r^2\$, amounts to 20.80 mm², as indicated in the last line of column 7 With the degrees of freedom \$v\$ set at 25, the standard deviation and standard uncertainty for the height difference \$d_{tt,-1}\$ are computed using Formula (6), resulting in \$s = \frac{20.80}{\sqrt{25}} = u\$ 0.9.
There are two arithmetic checks in Table A.1:
— the value in the last line of column 3 shall be equal to the sum of column 4: − 0,469 =− 0,469;
— five times the sum of column 5, minus the sum of column 4 shall be the sum of column 6:
Example of the full test procedure
Table B.1 contains the measured values, x A j and x B j , of the i th series of measurements (the series of measurements numbers 2, 3 and 4 were not printed).
Instrument type and number: NN xxx 630401
Table B.1 — Measurements and residuals of series No 1
The four unknown parameters, h, a, b 1 and b 2 are calculated according to Formula (11):
Following Formula (6), the residuals can be calculated to r =
According to Formula (12) the standard deviation of the first series over a distance of 40 m (standard deviation of the weighting unit) yields s 1 1174 2
The corresponding results of the second, third and fourth series are for the parameters
m and for the standard deviations s 2 =0 9, mm s 3 , mm s 4 =0 9, mm
For every series, the following holds: ν ν= 1 =ν 2 =ν 3 =ν 4 =8
The standard uncertainty (Type A) and the parameters over all series are calculated according to Formulae (13),
2 2 2 2 mm) mm) mm) mm) 1,0 mm u ISO-ROLAS , mm h= −0 245 1 0 244 8 0 245 4 0 245 4− − − = −
With Formulae (19), (20) and (24), the experimental standard deviations and the standard uncertainties, respectively, of the parameters are s h =u h( )=0,14 1,0=0,14⋅ mm s a =u a( )=0,25 1,0=0,25⋅ mm s b =u b( )=0,20 1,0=0,20⋅ mm
B.3.1 Statistical test according to question a) (see 6.4.1)
If the manufacturer has stated for σ= 2,0 mm and it was obtained for s = 1,0 mm with v2 we get according to Formula (29)
The null hypothesis, which asserts that the empirically determined standard deviation of \$s = 1.0 \, \text{mm}\$ is less than or equal to the manufacturer's value of \$\sigma = 2.0 \, \text{mm}\$, is upheld at a 95% confidence level.
B.3.2 Statistical test according to question b) (see 6.4.1) s, mm
The null hypothesis, which asserts that the experimental standard deviations of \( s = 1.0 \, \text{mm} \) and \( s' \, \text{mm} \) originate from the same population, is rejected at a 95% confidence level due to the failure to meet the required condition.
B.3.3 Statistical test according to question c) (see 6.4.1) s, mm v2 a= −3 4, mm s a =0 25, mm
Since the above condition is not fulfilled, the null hypothesis stating that the deflective deviation, a, is equal to zero is rejected at the confidence level of 95 %.
B.3.4 Statistical test according to question d) (see 6.4.1) s, mm v2 b= −2 7, mm s b =0 20, mm
The null hypothesis, which asserts that the total deviation \( b \) of the rotating axis from the true vertical is zero, is rejected at a 95% confidence level due to the failure to meet the necessary condition.
The rotating laser should therefore be adjusted.
Example for the calculation of an uncertainty budget
C.1 The measuring task and measuring conditions
The rotating laser used for measuring was thoroughly tested according to ISO 17123, with results detailed in Annex B It was employed in airfield construction for distances up to 120 m, necessitating a special test since the standard uncertainty for applications up to 40 m was approved Under favorable meteorological conditions, repeated measurements of the height difference yielded a standard uncertainty of \( u_{120} = 90 \, \text{mm} \) The construction occurred on a sunny day with a temperature of 32°C, which introduced strong scintillation effects The supervising surveyor identified and quantified various sources of uncertainty for operations up to 120 m, as outlined in Table 4.
— Expanded measurement range (120 m) Type A u 120 =9mm
— Parameter a = - 1,5 (±1) mm/40 m, estimated after adjustment, for 120 m: a 120 =−4 5 3, ( ) mm±
— Parameter b = - 1,5 (±1) mm/40 m, estimated after adjustment, for 120 m: b 1 20 =−4,5 ( ) mm±3
— Tilts of staff, impact on the height difference: dh 1 = (+)0,4 mm
— Dependence on atmospheric conditions dh 4=± 6 mm
— Subsidence of instrument, dh 5= (-)2 mm
— Curvature of the earth dh 6 = (+)1,1 mm
C.2 Calculation of the uncertainty budget
Table C.1 — Uncertainty budget In pu t q ua nt ity X i In pu t e st im at es a x i [d im]
St an da rd un ce rt ai nt y u ( x i ) [d im]
D is tr ib ut ion Se ns iti vi ty co ef fic ien ts cfx ii≡∂∂/ [d im] uxcux tt t ()≡⋅() [m m]
The evaluation types and sources of uncertainty are outlined in ISO 17123-1, referencing the probability of measurement outcomes The expanded range test series includes dimensions such as 12.09 mm, 19.0 mm, and 1.74 mm, with a focus on rectangular measurements of 12.61 mm.
B, deflective deviation ua()according to equ (57) a+=−1,5 a - =−7,5 a =−4,5 p = 100 % b-4,5 mm0,92 mmtriangular10,92
B, deflective deviation ub()according to equ (58) a += 0,0 a - =−4,5 a = 2,25 p = 100 % dh 1 0,4 mm0,12rectangular10,12
B, tilt of staff udh()1according to equ (57) a += 0,4 a - = 0,0 a = 0,2 p = 100 % dh 2 0,5 mm0,14rectangular10,14
B, zero error, general knowledge udh()2according to equ (57) a += 0,5 a - = 0,0 a = 0,25 p = 100 % dh 3 - 2,0 mm0,58rectangular10,58
The junction error, denoted as B, is defined by the equation \( a += 0.0 \), \( a -= 2.0 \), and \( a = 1.0 \) with a probability \( p = 100\% \) This represents the maximum estimated value without any correction for the measured input quantity \( h \).
In pu t q ua nt ity X i In pu t e st im at es a x i [d im]
St an da rd un ce rt ai nt y u ( x i ) [d im]
D is tr ib ut ion Se ns iti vi ty co ef fic ien ts cfx ii≡∂∂/ [d im] uxcux tt t ()≡⋅() [m m]