Establishing orthometric heights using GNSS — Part 7

June 2, 2016  - By
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Basic procedures and tools for determining valid NAVD 88 heights for constraints

To date, the six parts of “Establishing Orthometric Heights Using GNSS” have provided the reader with basic concepts, routines and procedures for understanding, analyzing, evaluating and estimating GNSS-derived ellipsoid and orthometric heights.

In Part 5 of this series, we discussed National Geodetic Survey’s NGS 59 guidelines and methods for evaluating the results of the GNSS-derived orthometric height project. It provided methods for evaluating the results of the project and identifying stations with valid North American Vertical Datum of 1988 (NAVD 88) published heights.

In Part 6, we continued to analyze the changes in adjusted heights due to different NAVD 88 height constraints and compared the results to the published NAVD 88 orthometric heights. We demonstrated that every constraint has an influence on the final set of adjusted heights so determining valid published NAVD 88 heights is important. With that, when incorporating new geodetic data into the National Spatial Reference System (NSRS), it is important to maintain consistency between neighboring stations. If the station has moved since the last time its height was established, then not constraining the published value and superseding the height is the appropriate action to take. As it was mentioned and emphasized in Part 6, if the difference is not due to movement and is due to some other reason such as the results of a previous adjustment distribution correction then superseding the height may not be the appropriate action to take.

In this part of the series, we will look at the network design of the NAVD 88 project and estimate the potential NAVD 88 distribution correction between two benchmarks involved with the original NAVD 88 adjustment.

First, we need to address the network design in the area that was used in the General Adjustment of the North American Vertical Datum of 1988 (NAVD 88). The NAVD 88 was a major leveling network adjustment project performed by the National Geodetic Survey (NGS) that was started in the early 1970s and completed in the early 1990s. NGS provides a summary of vertical datums. The excerpt (below) from the website describes the major attributes of the NAVD 88.

Excerpt from the North American Vertical Datum of 1988 (NAVD 88)

North American Vertical Datum of 1988 (NAVD 88) consists of a leveling network on the North American Continent, ranging from Alaska, through Canada, across the United States, affixed to a single origin point on the continent:

  • Tide Station & Location = Pointe-au-Pere,Rimouski, Quebec, Canada
  • PID = TY5255
  • GSD* Designation = 54L071
  • Bench Mark = 1250 G
  • Ht above LMSL(Meters) = 6.271

* Geodetic Survey of Canada = GSD

In 1993, NAVD 88 was affirmed as the official vertical datum in the National Spatial Reference System (NSRS) for the Conterminous United States and Alaska. Although many papers on NAVD 88 exist, no single document serves as the official defining document for that datum.

View a special report of the NAVD 88 providing information about the NAVD 88, or view an abstract from the special report below.

Abstract from the NAVD 88 Special Report
Special Report
Results of the General Adjustment of the
North American Vertical Datum of 1988
David B. Zilkoski, John H. Richards, and Gary M. Young
American Congress on Surveying and Mapping
Surveying and Land Information Systems, Vol. 52, No. 3, 1992, pp.133-149

ABSTRACT. For the new general adjustment of the North American Vertical Datum of 1988 (NAVD 88), a minimum-constraint adjustment of Canadian-Mexican-U.S. leveling observations was performed holding fixed the height of the primary tidal benchmark, referenced to the new International Great Lakes Datum of 1985 (IGLD 85) local mean sea level height value, at Father Point/Rimouski, Quebec, Canada. IGLD 85 and NAVD 88 are now one and the same. Father Point/Rimouski is an IGLD water-level station located at the mouth of the St. Lawrence River, and is the reference station used for IGLD 85. This constraint satisfies the requirements of shifting the datum vertically to minimize the impact of NAVD 88 on U.S. Geological Survey mapping products, and provides the datum point desired by the IGLD Coordinating Committee for IGLD 85. The only difference between IGLD 85 and NAVD 88 is that IGLD 85 benchmark values are given in dynamic height units, and NAVD 88 values are given in Helmert orthometric height units. The geopotential numbers of benchmarks are the same in both systems. Preliminary analyses indicate differences for the conterminous United States between orthometric heights referred to NAVD 88 and to the National Geodetic Vertical Datum of 1929 (NGVD 29) range from -40 cm to +150 cm. In Alaska, the differences range from +94 cm to +240 cm. However, in most “stable” areas, relative height changes between adjacent benchmarks appear to be less than 1 cm. In many areas, a single bias factor, describing the difference between NGVD 29 and NAVD 88, can be estimated and used for most mapping applications. The overall differences between dynamic heights referred to IGLD 85 and to International Great Lakes Datum of 1955 will range from 1 cm to 40 cm. The use of Global Positioning System (GPS) data and a high-resolution geoid model to estimate accurate GPS-derived orthometric heights will be directly associated with the implementation of NAVD 88 and IGLD 85. It is important that users initiate a project to convert their products to NAVD 88 and IGLD 85. The conversion process is not a difficult task, but will require time and resources.

More than one million kilometers of leveling data were analyzed during the NAVD 88 project. The design of the leveling network involved in the NAVD 88 project is shown in Figure 1.

Figure 1. Leveling Network Design Used in the General Adjustment of the North American Vertical Datum of 1988 (Figure 3 from the report).

Figure 1. Leveling Network Design Used in the General Adjustment of the North American Vertical Datum of 1988 (Figure 3 from the NAVD88 report).

Not all of the leveling data depicted in Figure 1 were used in the general adjustment. Some of the older leveling data were not consistent with the newer data so these older data were not included in the adjustment. When proper procedures are followed, leveling data is very precise and accurate over short distances but the leveling network design usually does not provide a lot of redundancy. That’s why it is important to design a leveling network with many connecting loops. The loops provide the redundancy required to ensure that the leveling data does not contain any remaining significant systematic errors and/or blunders. At a minimum, the connected loops help to control and/or localize the remaining errors. Some of the older leveling data that were not included in the general adjustment were incorporated into the NAVD 88 after the general adjustment and were loaded into the NGS database. These stations are denoted as POSTed monuments on the NGS datasheet, shown in the highlighted section below in the excerpt labeled “NAVD 88 General Adjustment: What Does This Really Mean?”

Excerpt from the NAVD 88 Special Report

NAVD 88 General Adjustment: What Does This Really Mean?

The general adjustment of NAVD 88 was completed in June 1991. All heights from the general adjustment were loaded into the NGS geodetic database in September 1991. This means that benchmarks included in the NAVD 88 Helmert blocking phase (approximately 80% of the total) have final NAVD 88 heights available for distribution to the public.

The remaining 20% of the benchmarks in “stable” areas were removed from the adjustment (denoted as “POSTed” benchmarks), because older data were inconsistent with newer data. NAVD 88 heights for these posted benchmarks will be determined from these older data during 1992-93. This task involves analyzing the data associated with the posted benchmarks to determine the best estimate of their NAVD 88 heights.

“POSTed” benchmarks in large crustal movement areas (e.g., southern Alaska, southern California, Phoenix, Houston, and southern Louisiana) will be published as special reports. This is a long-term task that started in January. It is important to note that some benchmarks in crustal-movement areas (i.e., benchmarks that were included in the NAVD 88 Helmert blocking phase) are available now. The heights of these benchmarks were usually based on the latest available data, but still may be influenced by crustal movement effects. In some areas, these benchmarks were not based on the latest available data, because this would have forced large distribution corrections into good, but older, adjacent leveling data.

In addition, there are approximately 500,000 USGS third-order benchmarks for which NGS does not yet have any data.

The NGS datasheet provides the date the station’s NAVD 88 orthometric height was adjusted so a user can determine if the station was part of the general adjustment of NAVD 88 or if the station was readjusted or incorporated in the NAVD 88 after the general adjustment. Station V 49 (PID = FA0151) is an example of a station that was involved in the general adjustment and published in 1991. The highlighted statement “The orthometric height was determined by differential leveling and adjusted by the NATIONAL GEODETIC SURVEY in June 1991” in the text portion of the datasheet indicates that this station’s adjusted height was established in the general adjustment of NAVD 88, as shown in the highlighted section in excerpt from “NGS datasheet for station V 49″ below.

NGS-1991

Station Phaniel is an example of a station that was incorporated into NAVD 88 after the general adjustment. Phaniel’s datasheet has the following statement, highlighted below: “The orthometric height was determined by differential leveling and adjusted by the NATIONAL GEODETIC SURVEY in January 2005.”

NGS-Phaniel

So why is this important?

It is important to realize that just because the leveling data is newer than the rest of the leveling network around it, it doesn’t necessarily mean its absolute height value is more accurate or more reliable than the stations it was established from. The newer leveling data most likely is associated with an older leveling survey used in the general adjustment of NAVD 88. This older leveling data may have been affected by crustal movement and could be inconsistent with its neighbors 5-15 kilometers away. If proper procedures were adhered to, such as the FGCS geodetic leveling procedures, then the new leveling should have been connected to the NAVD 88 through a two- or three-mark leveling validation check leveling procedure, shown in the excerpt from “FGCS Specifications and Procedures to Incorporate Electronic Digital/Bar-Code Leveling Systems” below.

Page 1 from FGCS Specifications and Procedures Document

FGCS Specifications and Procedures to Incorporate Electronic Digital/Bar-Code Leveling Systems*

3.5 Geodetic Leveling

Geodetic leveling is a measurement system comprised of elevation differences observed between nearby rods. Geodetic leveling is used to extend vertical control.

Network Geometry


Order
Class
First
I
First
II
Second
I
Second
II
Third
Bench mark spacing not more than (km) 3 3 3 3 3
Average bench mark spacing not more than (km) 1.6 1.6 1.6 3.0 3.0
Line length between networkcontrol points not more than (km) 300a 100a 50a 50a 25b
Minimum bench mark ties 6 6 4 4 4

aElectronic Digital/Bar-Code Leveling Systems, 25 km
bElectronic Digital/Bar-Code Leveling Systems, 10 km


As specified in above table, new surveys are required to tie to existing network bench marks at the beginning and end of the leveling line. These network bench marks must have an order (and class) equivalent to or better than the intended order (and class) of the new survey.

First-order surveys are required to perform valid check connections to a minimum of six bench marks, three at each end. All other surveys require a minimum of four valid check connections, two at each end.

A valid “check connection” means that the observed elevation difference agrees with the published adjusted elevation difference within the tolerance limit of the new survey. Checking the elevation difference between two bench marks located on the same structure, or so close together that both may have been affected by the same localized disturbance, is not considered a proper check.

In addition, the survey is required to connect to any network control points within 3 km of its path. However, if the survey is run parallel to existing control, then the following table specifies the maximum spacing of extra connections between the survey and the existing control.

When using Electronic Digital/Bar-Code Leveling Systems for area projects, there must be at least 4 contiguous loops and the loop size must not exceed 25 km. (Note: This specification may be amended at a future date after sufficient data have been evaluated and it is proven that there are no significant uncorrected systematic errors remaining in Electronic Digital/Bar-Code Leveling Systems.)


* NGS’ analyses of the data will be the final determination if the data meet the desired FGCS order and class standards.

The validation check leveling procedure ensures that the new leveling is consistent with the local stations it’s connected to. However, if the local area around these monuments all moved together than the validation check leveling procedure may meet the allowable tolerances but the new heights could still be inconsistent with neighbors 5 to 15 kilometers away. Similarly, if the validation check leveling stations were involved in a large distribution correction in the NAVD 88, than, once again, the validation check leveling may meet the allowable tolerances but the new heights could still be inconsistent with neighbors 5-15 kilometers away. This is not to say that the older leveling or published heights of the stations are bad or incorrect; all it is ensuring is that the new leveling is consistent with the adjusted heights in the local area surrounding the new leveling project.

Another statement on the NGS datasheet that should be explained is “No vertical observational check was made to this station,” shown in the highlighted statement from the excerpt of Phaniel’s datasheet, below. This means that the station was determined on a leveling line that is known as a spur level line. This means that the leveling data were not involved in a loop. This is important because the lack of redundancy means that there is no check on the adjusted heights of these stations other than the checks performed during the double running procedure. The double-running procedure is very important but the procedure may not detect, reduce, and/or eliminate all systematic errors and/or blunders. The GNSS-derived values may be the first check on the published height of these stations. When performing GNSS-derived orthometric height adjustments the users should investigate all stations that seem to be inconsistent with its neighboring stations especially stations that their published datasheet contains the statement “No vertical observational check was made to this station” such as station Phaniel.

When analyzing GNSS projects, it is helpful to understand how the NAVD 88 height of the station was established and what year it was leveled. Figures 2 and 3 depict the original leveling network design used in the general adjustment of the NAVD 88 in the Rowan County, North Carolina, project area, and Figures 4 and 5 depict the current NAVD 88 leveling network design. Looking at Figures 2 and 3, it appears that the leveling network used in the general adjustment of NAVD 88 in Rowan County was fairly sparse and mostly consisted of leveling data observed in the 1930s and 1960s.

Figures 4 and 5 show the amount of leveling data incorporated into the NAVD 88 after the general adjustment. The red stars on Figure 4 are the stations that have been incorporated into the NAVD 88 since the general adjustment. Figure 5 depicts the dates of the leveling lines that were used to establish the new NAVD 88 heights. All of these new stations will have adjustment dates after June 1991. Having a different adjustment date than the general adjustment date of 1991 is not an issue, it’s just a way of informing the user that the station was incorporated into NAVD 88 and constrained to previously published NAVD 88 heights. The user should know the adjustment date of the control they are using in their GNSS project because the accumulated NAVD 88 distribution correction could be large especially between stations with different adjustment dates in areas with old leveling data and large loops.

Figure 2. Leveling Network Design Used in the General Adjustment of the North American Vertical Datum of 1988 (Green stations are stations established in the NAVD 88 and published in June 1991).

Figure 2. Leveling Network Design Used in the General Adjustment of the North American Vertical Datum of 1988 (Green stations are stations established in the NAVD 88 and published in June 1991).

Figure 3. Dates of the Original Leveling Network Design in the Vinicity of the Rowan County, North Carolina, Height Modernization Project.

Figure 3. Dates of the Original Leveling Network Design in the Vinicity of the Rowan County, North Carolina, Height Modernization Project.

Figure 4. Leveling Network Design Incorporated into the General Adjustment of the North American Vertical Datum of 1988 (Red stars are stations that were incorporated in NAVD 88 after June 1991).

Figure 4. Leveling Network Design Incorporated into the General Adjustment of the North American Vertical Datum of 1988 (Red stars are stations that were incorporated in NAVD 88 after June 1991).

Figure 5. Dates of the Current Leveling Network Design in the Vinicity of the Rowan County, North Carolina, Height Modernization Project.

Figure 5. Dates of the Current Leveling Network Design in the Vinicity of the Rowan County, North Carolina, Height Modernization Project.

As depicted in Figure 3, the original leveling data used in NAVD 88 in southern Rowan County, NC, was an east-west leveling line performed in 1935. It was connected at both ends of the line to leveling data performed in the 1970s. The validation check leveling procedure was performed and met the required tolerances. The loops that the 1935 leveling line was involved in are fairly large, around 175 kilometers. The leveling data involved in the loops consists of first- and second-order data. The allowable loop closure would have been based on the amount of leveling of each order and class involved in the loop. The allowable loop closure for the older second-order, class 0 leveling line would have been based on 8.4 mm times the square root of the length of loop in kilometers. In this case, a loop 175 kilometers would have an allowable closure of 111 mm. The allowable loop closure for first-order, class 2 leveling is 4 mm times the square root of the length of loop in kilometers. In this case, a loop 175 kilometers would have an allowable closure of 53 mm. Since this is based on a mixture of order and classes of leveling data, the allowable loop closure would have been somewhere in between.

For this column, I decided to estimate the NAVD 88 distribution correction between two benchmarks involved with the older leveling lines in southern Rowan County. The observed Helmert orthometric height difference between station V 49 and T 78 is -6.850 meters, and the Published NAVD 88 Helmert orthometric height difference from the NAVD 88 general adjustment is -6.891 meters. This means that the distribution correction between stations V 49 (FA0151) and T 78 (FA0295) is 0.041 meters (4.1 cm).

Figure 6 depicts the location of the stations and the leveling route used to estimate the NAVD 88 distribution correction. Since the leveling distance between these two stations is approximately 60 kilometers, the distribution correction is less than 1 mm per kilometer (0.7 mm/km). This is a very reasonable distribution correction because it only modifies each leveling section observation by about 1 mm per kilometer allowing users to check their local leveling projects. This, however, may be an issue with some GNSS surveys that extend over a large area were the leveling network consists of old leveling data with large loops. The GNSS-derived orthometric heights may be more accurate than the leveling-derived orthometric heights. As shown in Figure 6, stations V 49 and T 78 are involved in large loops and were established using older leveling data in the original NAVD 88 resulting in a distribution correction of 4.1 cm.

Figure 6. Example of an estimate of the NAVD 88 Distribution Correction between two stations established with old leveling data and large loops.

Figure 6. Example of an estimate of the NAVD 88 distribution correction between two stations established with old leveling data and large loops.

Station V 49 was used in this analysis because the station was occupied during the Rowan County GNSS project. The shortest leveling distance between station V 49 and T 78 was used to estimate the NAVD 88 distribution correction. Station T 78 was selected because it is the junction station for the leveling line that was used to incorporate station Buffalo 2 into the NAVD 88 in January 2005. Since T 78 was the junction station and its height changed 4.1 cm, 4.1 cm was applied to station Buffalo 2’s height to obtain its modified height. This is not the most rigorous way to estimate the effects of the distribution correction but it provides a quick method to determine an estimate of the NAVD 88 distribution correction between two stations.

Figure 7 is a plot that depicts the differences at station Buffalo 2 using the modified NAVD 88 height. The difference between the GNSS-derived orthometric adjusted height and the new NAVD 88 height decreased from 3.5 cm to -0.6 cm. This difference agrees to within 1 cm with the results of station V 49 (see Figure 7). It should be noted that one of the recommendations in the National Geodetic Survey’s NGS 59 document is to occupy valid NAVD 88 stations every 20 km. Following this procedure can help reduce the number of stations that need to be investigated due to NAVD 88 distribution corrections from the general adjustment.

Figure 7. Example of the possible effect of the NAVD 88 distribution correction on an adjusted GNSS-derived orthometric height.

Figure 7. Example of the possible effect of the NAVD 88 distribution correction on an adjusted GNSS-derived orthometric height.

Three stations were identified as potential outliers in Part 6 — Phaniel, Plaza, and Row 3. As mentioned in Part 5 (February 2016), station Phaniel has a large difference between the adjusted GNSS-derived orthometric height and the published NAVD 88 orthometric height value (-4.2 cm); indicating an issue with the ellipsoid height and/or orthometric height (see Figure 8). However, Phaniel’s published NAD 83 (2011) ellipsoid height and the Rowan County minimum-constraint adjusted height of Phaniel only differed by 0.8 cm. The comparison of adjusted ellipsoid heights and published ellipsoid heights for the Rowan County GNSS project were provided in Part 4 (December 2015). This is an indication that the GNSS-derived ellipsoid height of station Phaniel is not an issue and that the station hasn’t moved since the original GNSS survey and the 2015 Rowan County GNSS survey. It should be noted that the leveling project used to incorporate station Phaniel into NAVD 88 was performed in 2001 which was in between the two GNSS surveys.

Two other stations (Row 17 and Row 16) were leveled on the same leveling line as Phaniel and their adjusted GNSS-derived orthometric height and the published NAVD 88 orthometric height values agree to 1.6 cm and 1.7 cm respectively; this is an indication that the leveling data and GNSS data are consistent from the main level line to these two stations. Phaniel’s datasheet has the statement “No vertical observational check was made to this station,” indicating the station’s height was established on a spur leveling line and therefore has a lack of redundancy and reliability. Based on the information up to now, I would not recommend constraining station Phaniel in the final adjustment. Saying that, before it is superseded by the GNSS project, the benchmarks between Phaniel and Row 17 should be re-leveled to determine if a leveling error was made between these stations in 2001.

Figure 8. NAVD 88 leveling network design involving station Phaniel.

Figure 8. NAVD 88 leveling network design involving station Phaniel.

The geodetic data and information for station Plaza is listed below:

  • As described in Part 6 (April 2016), station Plaza and station Fifth have a large relative difference between the adjusted GNSS-derived orthometric height and the published NAVD 88 orthometric height value (-3.2 cm); (See Figure 9.);
  • Four other stations in the vicinity have small relative differences between the adjusted GNSS-derived orthometric heights and the published NAVD 88 orthometric heights values, 37 DRD (0.6 cm), Midtown (-0.1 cm), Midway (1.0 cm), and J 181 (1.1 cm) – indicating a problem with station Plaza;
  • Station Fifth and Plaza are only 400 meters apart, and their adjusted heights were established in two different adjustments: station Fifth was leveled in 2013 (adjustment date of March 2015) and station Plaza was leveled to in 1989 (adjustment date of September 1997) – indicating a potential inconsistency between adjustments;
  • Plaza’s datasheet states that “the station was recovered as described in 2012 except the area between the curb and sidewalk has been filled with concrete. Mark is now part of the sidewalk but does not appear to have been disturbed.”

Based on the available information to date, I would not recommend constraining the published height of station Plaza in the final adjustment. Once again, this station’s published height should not be superseded by the GNSS project until new leveling has been performed between station Fifth and Plaza.

Figure 9. NAVD 88 leveling network design involving station Plaza.

Figure 9. NAVD 88 leveling network design involving station Plaza.

Figure 10 depicts the leveling network involving station Row 3. As described in Part 6 (April 2016), station Row 3 has a large difference between the adjusted GNSS-derived orthometric height and the published NAVD 88 orthometric height value, -3.8 cm (see Figure 10.). Except for station AE4540 (382 JAS), all of the differences between the adjusted GNSS-derived orthometric height and the published NAVD 88 orthometric height value at the other nearby stations are all less than 1.7 cm; as a matter of fact, most of the differences are less than +/- 0.5 cm.

I could not find any leveling data in NGS’ database involving station AE4540 (382 JAS). (See Figure 11.) As far as I could determine, this station was not leveled to by NGS and leveling data were not submitted to NGS for inclusion in the NAVD 88. You can retrieve all project identifiers for those projects with observations to or from a station using the stations’s PID. The station’s PID is provided on the NGS datasheet. The input and output for PID AE4540 is shown below. There are no identifiers listed under the sections labeled “Vert_Obs,” “Lev_Obs,” or “Level_Obs” indicating that this station does not have any leveling observations in NGS database.

Input for PID AE4540 from http://www.ngs.noaa.gov/cgi-bin/SURV_ID/mark_sources.prl.

Output for PID AE4540 from http://www.ngs.noaa.gov/cgi-bin/SURV_ID/mark_sources.prl.

Figure 10. NAVD 88 leveling network design involving station row 3.

Figure 10. NAVD 88 leveling network design involving station row 3.

 

Figure 11. NAVD 88 leveling network design involving station 382 JAS.

Figure 11. NAVD 88 leveling network design involving station 382 JAS.

Based on the available information so far, I would not recommend constraining the published heights of station Row 3 or 382 JAS (AE4540) since they will distort the adjusted heights of surrounding stations (see Part 6, Figure 10). If no supporting leveling data can be found for station 382 JAS then I would recommend superseding that station’s height with the GNSS-derived value. As for station Row 3, I wouldn’t recommend superseding the published height with the GNSS-derived height until a leveling check has been made between Row 3 (DG5673) and a nearby station such as station 384 JAS (FA0564).

I realize that by not constraining a station and not superseding the published height that an inconsistency between the leveled NAVD 88 height and the NAVD 88 GNSS-derived orthometric height may occur. This information needs to be noted in the project report with an explanation of why you made certain decisions in your final adjustment. The analysis and plots provided in these columns are the types of information that should be provided in the final report.

All of the analysis and recommendations have been based on using the latest scientific geoid model xGeoid15b. However, in practice, GNSS-derived orthometric heights are incorporated into the NAVD 88 using the latest hybrid geoid model GEOID12B. I recommend first performing the analysis using the scientific geoid model because the hybrid geoid model has been warped to be consistent with the published NAVD 88 values. This was described in detail in my Part 3 (October 2015). The analysis using the scientific geoid should be included in the report especially if the user finds significant differences between the results using the two different geoid models. Saying that, maintaining consistency between closely spaced stations is extremely important when incorporating data into an existing network. Based on the information so far and the results using GEOID12B, I would not recommend constraining the published NAVD 88 heights of stations Phaniel and Plaza in the final NAVD 88 GNSS-derived orthometric height adjustment. These two stations resulted in significant changes in relative adjusted heights when they were constrained. (See Part 6, April 2016.)

It was noted in Part 5 (February 2016) that ten of the 2015 GNSS Rowan County Height Modernization project’s stations have published NAVD 88 GNSS-derived orthometric heights. These station are important because they are on the edge of the network where there’s a void of published NAVD 88 leveling-derived orthometric heights. In the next column, we will look at these stations and the differences between their minimum-constraint least squares adjusted GNSS-derived orthometric heights and their published NAVD 88 GNSS-derived orthometric height.

These columns have provided a lot of routines and procedures for analyzing and estimating GNSS-derived orthometric heights. My intent was to provide the analyst with tools for documenting the results of the analysis and providing a basis for making recommendations associated with the GNSS project. A future column will address what information should be included in a project report.

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