Digital Terrain Modeling: Principles and Methodology - Chapter 8 ppsx

The three attributes of the source data distribution, accuracy, and density willalso have a great influence on the accuracy of the final DTM.. This is the combined effect of distribution

Accuracy of Digital Terrain Models

The accuracy of DTMs is of concern to both DTM producers and users For a DEMproject, accuracy, efficiency, and economy are the three main factors to be considered(Li 1990) Accuracy is perhaps the single most important factor to be consideredbecause, if the accuracy of a DEM does not meet the requirements, then the wholeproject needs to be repeated and thus the economy and efficiency will ultimately beaffected For this reason, this chapter is devoted to this topic

8.1 DTM ACCURACY ASSESSMENT: AN OVERVIEW

8.1.1 Approaches for DTM Accuracy Assessment

A DTM surface is a 3-D representation of terrain surface Unavoidably, some errorswill be present in each of the three dimensions of the spatial (X, Y , Z) coordinates

of the points occurring on DTM surfaces Two of these (X and Y ) are combined togive a planimetric (or horizontal) error while the third is in the vertical (Z) directionand is referred to as the elevation (or height) error

The assessment of DTM accuracy can be carried out in two different modes,that is,

1 the planimetric accuracy and the height accuracy can be assessed separately

2 both can be assessed simultaneously

For the former, accuracy results for the planimetry can be obtained separatelyfrom the accuracy of these results in a vertical direction However, for the latter, anaccuracy measure for both error components together is required

There are four possible approaches for assessing the height accuracy of theDTM (Ley 1986), namely,

1 Prediction by production (procedures): This is to assess the likely errors

intro-duced at the various production stages together with an assessment of the vertical

159

accuracy of the source materials The accuracy of the final DTM is the consequence

or concatenation of the errors involved in all these stages

2 Prediction by area: This is based on the fact that the vertical accuracy of contour

lines on a topographic map is highly correlated with the mean slope of the area

3 Evaluation by cartometric testing: This is about experimental evaluation It is

argued by many that the entire model rather than the node should be tested For such

a test, a set of checkpoints is required

4 Evaluation by diagnostic points: A sample of heights is acquired from the source

materials at the time of data acquisition and this set of data is used to check thequality of the model This can be conducted at any intermediate stage as well as atthe final stage

There are three approaches for assessing the planimetric accuracy of DTM(Ley 1986), namely:

1 No error: It is argued that a DTM provides use of a set of heights with planimetric

positions, which are inherently precise

2 Predictive: Similar to the prediction by area used for vertical accuracy.

3 Through height: To fix the positions of node heights by comparing a series of points.

However, as he also mentioned, it is difficult to bring these into practice This isperhaps the reason why the issue of planimetric accuracy is rarely addressed

An alternative approach is to simultaneously assess the vertical and horizontalaccuracies In doing so, a measure capable of characterizing the accuracy in threedimensions is required Ley (1986) suggested using a comparative measure ofthe mean slopes between the DTM surface and the original terrain surface Othershave also considered the use of other geomorphometric parameters as well as terrainfeature points and lines However, there is no consensus Most people follow thepractice of assessing the contour accuracy, that is, assessing the vertical accuracy only

a sample size of 2115 check points Some information about these experimentaltests is given in Section 8.2

To understand the distributions better, the frequency of occurrence of large errorswas also recorded.Table 8.1lists the results (Li 1990) To show how the distribu-tions deviate from the normal contribution, the theoretical values for the occurrencefrequency of large errors are also listed From this table, it is clear that curves

of the distribution of DTM errors are flatter than the standard normal distribution

N(0,1).

(b)

− 3.5 − 3.0 − 2.5 − 2.0 − 1.5 − 1.0 − 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

900 800 700 600 500 400 300 200 100 0

Figure 8.1 Distribution of DTM errors (Li 1990): (a) for Sohnstetten area (1892 checkpoints)

and (b) for Spitze area (2115 checkpoints).

8.1.3 Measures for DTM Accuracy

Letf (x, y) be the original terrain surface and f
(x, y) be the constructed DTM surface,

then the difference,e(x, y), where

e(x, y) = f
(x, y) − f (x, y) (8.1)

Table 8.1 Occurrence Frequency of Large Errors in DTM Test Area Grid Interval (m) >2σ(%) >3σ(%) >4σ(%)

e(x, y) is a random variable in statistical terms (Li 1988) and magnitude and spread

(dispersion) are the two characteristics of random variable To measure the magnitude,some parameters can be used such as the extreme values (emaxandemin), the mode(the most likely value), the median (the frequency center), and the mathematicalexpectation (weighted average) To measure the dispersion, some parameters such asthe range, the expected absolute deviation, and the standard deviation can be used

To summarize, in addition to the mse which is in common use, the followingparameters can also be used to measure DTM accuracy:

The use of range, R, may lead to a specification of DTM accuracy something

like the American National Map Accuracy Standard But some characteristics of thismeasure might be objectionable, that is,

1 The valueR depends on only two values of the random variable and others are all

ignored

2 The probability of the values ine(x, y) is ignored.

Therefore, the combination of mean and standard deviation is preferred although thedistribution of DTM errors is not necessarily normally, as shown inFigure 8.1.This is because most of the probability distribution is massed with 4σ distancefromµ, according Chebyshev’s theorem (Burington and May 1970) Chebyshev’s

theorem states that the probability is at least as large as 1− 1/k2that an observation

of a random variable (e) will be within the range from µ − k × σ to µ + k × σ , or

P (|e − µ| > k × σ ) < 1

wherek is any constant greater than or equal to 1 If the normal distribution is used

to approximate the distribution of e(x, y), the standard deviation computed from

Equation (8.4) has the special meaning that is familiar to us

8.1.4 Factors Affecting DTM Accuracy

The accuracy of the DTM is a function of a number of variables such as the roughness

of the terrain surface, the interpolation function, interpolation methods, and the threeattributes (accuracy, density, and distribution) of the source data (Li 1990, 1992a).Mathematically,

ADTM= f (CDTM,MModeling,RTerrain,AData,DData, DNData,O) (8.6)whereADTM is the accuracy of the DTM;CDTM refers to the characteristics of theDTM surfaces;MModelingis the method used for modeling DTM surfaces;RTerrain

is the roughness of the terrain surface itself;AData,DData, and DNDataare the threeattributes (accuracy, distribution, and density) of the DTM source data; andO denotes

other elements

The roughness of the terrain surface determines the difficulty of DTM tation of terrain If the terrain is simple, then only a few points need to be sampledand the surface to be used for reconstruction will be very simple For example, if theterrain is flat, only three points are essential and a plane can be used for modeling thispiece of terrain surface On the other hand, if the surface is complex, then more pointsneed to be measured and higher-order polynomials may have to be used for modelingthis terrain The descriptors for the complexity of terrain surfaces have already beenintroduced inChapter 2.Among the various descriptors, slope is the most importantone widely used in the practice of surveying and mapping and will be used later inthe development of the DTM accuracy model

represen-A DTM surface can be constructed by two methods One is to construct it directlyfrom the measured data and the other is indirect In the latter, the DTM surface is con-structed from grid data that are interpolated via a random-to-grid interpolation Theaccuracy of the DTM surface constructed indirectly will be lower than the accuracy

of that constructed directly, due to accuracy loss in the random-to-grid interpolationprocess

As discussed inChapter 4,three types of DTM surfaces are possible, ous, continuous, and smooth It has been found that the continuous surface consisting

discontinu-of a series discontinu-of contiguous linear facets is the least misleading (or the most trustable)

The three attributes of the source data (distribution, accuracy, and density) willalso have a great influence on the accuracy of the final DTM If there are a lot of points

in the smooth or flat areas and few points in the rough areas, then the result will not

be satisfactory This is the combined effect of distribution and density, which wasdiscussed inChapter 2.The third attribute, the accuracy of the source data, will bediscussed in detail in this section Undoubtedly, errors in source data are propagated

to the final DTM during the modeling process

It has already been discussed inChapter 3that aerial photographs and existingtopographical maps are the main data sources for digital terrain modeling Theaccuracy of photogrammetric data is affected by the following factors:

1 the quality and scales of the photographs

2 the accuracy and physical conditions of the photogrammetric instruments used

3 the accuracy of measurement

4 the stereo geometry of aerial photographs

Generally, the accuracy of photogrammetric data is 0.07 to 0.1H ‰ if acquired by

using an analytical photogrammetric plotter or 0.1 to 0.2H‰ if acquired by using an

analog photogrammetric plotter Here,H is the flying height, that is, the height of the

aerial camera when the photographs were taken (usually with a wide-angle camerawith a focal length of 152 mm and a frame of 23 cm×23 cm) It refers to the accuracy

of static measurement However, if the measurement is dynamic (e.g., contouringand profiling), the accuracy is much lower The speed of measurement is also animportant factor Various experimental tests (e.g., Sigle 1984) reveal that the accuracy

of photogrammetrically measured data is about 0.3H‰ Some experiments (e.g.,Gong et al 2000) also reveal that the accuracy of photogrammetric data acquired by afully digital photogrammetric system is not as high as that from an analytical plotter.The accuracy of contouring data obtained from digitization is affected by thefollowing factors:

1 the accuracy and physical condition of the digitizer

2 the quality of the original map

3 the accuracy of measurement

The accuracy of contours can be written as:

mc= mh+ mp× tan α (8.7)wheremhrefers to the accuracy of height measurement;mpis the planimetric accuracy

of the contour line;α is the slope angle of the terrain surface; and mcis the overallheight accuracy of the contours, including the effect of planimetric errors

Usually, the accuracy specifications for contours all appear in the form ofEquation (8.7) A summary of such specifications is given inTable 8.2 Accuracyloss during the digitization process is about 0.1 mm in point mode and 0.2 to 0.25 mm

in stream mode In any case, the overall accuracy of digitized contour data will bestill within a 1/3 contouring interval

Table 8.2 Some Examples of Contour Accuracy Specifications Country Scale Accuracy of Contours (m)

Ground measurement (including GPS) 1–10 cm Digitized contour data About 1/3 of contouring interval

8.2.1 Strategies for Experimental Tests

As stated previously, the accuracy of a DTM is the result of many individual factors,that is,

1 the three attributes (accuracy, density, and distribution) of the source data

2 the characteristics of the terrain surface

3 the method used for the construction of the DEM surface

4 the characteristics of the DEM surface constructed from the source data

Accordingly, six strategies for an experimental testing of DEM accuracyare possible (Li 1992a), in each of which only one of the six factors is used asthe independent variable and the other five as controlled variables:

1 The accuracy of the source data could be varied while all the other factors remain

unchanged This can be achieved by using different data acquisition techniques

such as GPS, photogrammetry, and other methods It can also be achieved by usingthe same type of data acquisition techniques but with different accuracies.

2 The density of the source data could be varied while all other factors remain

unchanged This can be achieved by using different sampling intervals or dataselection methods Alternatively, resampling without involvement of interpolation,

as discussed inChapter 4, can be applied to a set of data with finer resolutions(i.e., smaller intervals) to coarser resolution (i.e., larger intervals)

3 The distribution of source data could be varied while all other factors remain

unchanged This can be achieved by using different sampling patterns or dataselection methods In digital terrain modeling practice, grid and contour data arethe two types of basic data patterns that have been widely used Another two types

of data are also widely used, that is, with or without feature points (i.e., top of hills,bottom of valleys, points along ridge lines, points along ravine lines, points alongthe edge of terrace, saddle points, etc.)

4 The type of terrain could be varied while all other factors remain unchanged.

This is achieved by using terrain surface with various types of relief

5 The type of DTM surface could be varied while all other factors remain unchanged.

This is achieved by using different types of discontinuous, continuous, and smoothsurfaces for DTM surface reconstruction

6 Two types of modeling methods are used to construct two types of surfaces,

that is, direct modeling using triangulated networks and indirect modeling using

a random-to-grid interpolation to form a grid network

8.2.2 Requirements for Checkpoints in Experimental Tests ∗

In experimental tests on DTM accuracy, a set of checkpoints is used as the ground

truth Then, the points interpolated from the constructed DTM surface are checked

against the corresponding checkpoints After that, the difference between the twoheights at each point is obtained These differences are used to compute statisticalvalues, as discussed in Section 8.1 It is clear that the final DTM accuracy figuresare definitely affected by the characteristics of the set of checkpoints In other words,the final estimates may be affected by the three attributes of the set of checkpoints,that is, accuracy, sample size (number of points), and distribution, because the threeattributes can be used to characterize the set of checkpoints (Li 1991)

First, the required sample size (number) of the set of checkpoints will beconsidered From statistical theory it can be found that this is related to the followingtwo factors:

1 the degree of accuracy required for the accuracy figures (i.e., the meanµ and

standard deviationσ ) to be estimated

2 the variation associated with the random variable, that is, the height differences

in the case of DTM accuracy tests

The smaller the variation, the smaller the sample size needed to achieve a givendegree of accuracy required for accuracy estimates For an extreme example, if the

σ of the height differences is 0, then one checkpoint is enough no matter how large

∗ Largely extracted from Li 1991, with permission from ASPRS

the test area or the size of the data set Similarly, the higher the given degree ofaccuracy requirement for the accuracy estimates, the larger the sample size needed.The relationship between the sample size, the value σ, and the given degree of

accuracy required needs to be established

If the distribution is normal, the discussion is simpler However, as discussed inSection 8.1, the distribution of DTM errors is not necessarily normal and, therefore,

a new random variable with approximate normal distribution needs to be selectedfor further discussion LetH be the random variable of height differences e(x, y)

in discrete space; µ be the mean of a random sample of size n from a particular

distribution; andM be the true value of the random variable Then, the ratio

Y = µ − M

is a standardized variable and has approximately the normal distributionN(0, 1),

even though the underlying distribution is not normal, as long asn is large enough

(Hogg and Tanis 1977) Suppose theσ of a distribution is known but the M is unknown,

then for the probabilityr and for a sufficiently large value of n, a value Z can be found

from the statistical table forN(0, 1) distribution, such that the probability that Y will

be within the range from−Z to Z is approximately equal to r, or approximately,

The closeness of the approximate probabilityr to the exact probability depends on

both the underlying distribution and the sample size If the distribution is unimodal(with only one mode) and continuous, the approximation is usually quite good foreven a small value ofn (e.g., 5) If the distribution is “less normal” (i.e., badly skewed

or discrete), a large sample size is required (e.g., 20 to 30 points)

Substituting Equation (8.8) into Equation (8.9) and rearranging it, the followingexpression can be obtained:

For a given constantS, the percentage of the probability, (100r)%, of the random

intervalµ ± S including M is called the confidence interval, where S is the specified

degree of accuracy for the mean estimate,µ in this case In general, if the required

confidence interval(100r)% = 100(1−α)%, then the sample size n can be expressed

as follows:

n = Z r2× σ2

S2 = Z2

r ×σ S

2

(8.11)whereZ r is the limit value within which the values of the random variableY will

fall with probabilityr Its value can be found in the statistical table for the N(0, 1)

distribution The mathematical expression is as follows:

and the commonly used values are as follows:

Z r=0.95 = 1.960, Z r=0.98 = 2.326, Z r=0.99= 2.576For example, if the accuracy required for the mean estimate is 10% of the standarddeviation of the DTM errors (i.e., σ ), and the confidence level is 95%, then the

required sample size is

n = Z2

r ×σ S

2

= 1.962×

10010

For example, if the accuracyσ σ required for the standard deviation estimateσ is

10% ofσ , then the required sample size is 51.

The variation of DTM accuracy estimate values with the number of checkpointsused has been intensively tested by Li (1991) The number of checkpoints was reducedsystematically from 100 to 1% to produce a number of new sets of checkpoints Thesenew sets of checkpoints were then used to assess the DTM accuracy and produce newsets of DTM accuracy estimates The test results confirm the relationships expressed

by Equations (8.11) and (8.14)

Equations (8.11) and (8.14) can be used to estimate the number of checkpointsrequired In such calculations, it is implicit that the checkpoints are free of errors.However, this is not the case in practice If the accuracy of the set of checkpoints

is lower than the expected DTM accuracy, then the result of the DTM accuracyestimated from the height differences is meaningless This means that the relationshipbetween the required accuracy of checkpoints and the given degree of accuracy forthe DTM accuracy estimate should be established In this discussion, the accuraciesare discussed in terms of the standard deviations

Let H2 be the error involved in the checkpoints and H1 the true heightdifference Then,

representative ofσ H1 As expressed in Equation (8.13), the standard deviation of

σ H1 has a variance approximately as follows:

It is much more convenient to use a single value, so the square root of these two terms

is used as the representative value because they are independent Then, the followingequation can be obtained:

is usually specified in terms of RMSE, then RMSE might be used to replaceσ in

Equation (8.20)

The last consideration is the distribution of the checkpoints An intensive test

as to whether random distribution is as good as even distribution (e.g., in gridform) was conducted by Li (1991) Two test areas (see Section 8.3) were used.The numbers of checkpoints for the areas were 1892 and 2314 From each set ofcheckpoints, 15 subsets of checkpoints, each with 500 points, were randomly gener-ated The randomness of selection was achieved by using a set of random numbersfrom a uniform distribution generated by a computer subroutine for random num-bers In the generation of random numbers, the range was determined by the totalnumber of points in the original set of checkpoints After this, those checkpoints withthe same numbering as the generated random numbers were taken from the originalset to form the sample As expected, there were differences among the 15 accuracyestimates However, the variation was very small and well within the acceptable range.Therefore, it might be assumed that the random selection of checkpoints is acceptable

if the selection is over the whole test area

8.3 EMPIRICAL MODELS FOR THE ACCURACY OF

THE DTM DERIVED FROM GRID DATA

From the literature it can be seen that many experimental investigations into the acy of DTM have been conducted by many researchers The best known investigationwas the international test organized by the International Society for Photogrammetryand Remote Sensing (ISPRS) in the early 1980s (Torlegard et al 1986) A number

accur-of institutions all over the world participated in the acquisition accur-of DTM source data

by using the photogrammetric method Six areas with different types of terrain weretested However, this international test failed to produce any empirical model for DTMaccuracy In the early 1990s, a systematic investigation into the relationship betweensampling intervals and DTM accuracy was conducted by Li (1990, 1992a, 1994) usingthree sets of the ISPRS test data Through this testing, an empirical model for DTMaccuracy prediction was produced Cases both with and without terrain features wereconsidered and a different model for each was produced Recently, in the community

of geo-information, similar tests have also been conducted (e.g., Gong et al 2000;Tang 2000) This section is based mainly on the tests by Li (1990, 1992a, 1994)

8.3.1 Three ISPRS Test Data Sets

The three ISPRS data sets used were for the Uppland, Sohnstetten, and Spitze areas.The basic characteristics of these test areas are described in Table 8.4 A set ofphotogrammetrically measured contour data, a set of square-grid data, and a set ofF-S data for each of these areas were used Some information about the test data isgiven inTable 8.5 The checkpoints were measured from much larger-scale aerialphotographs and therefore have much higher accuracy then the test data points Someinformation about these checkpoints is given inTable 8.6

Figure 8.2shows the contour maps of these areas The corresponding F-S dataare superimposed onto each of these maps The Uppland area is relatively flat, with

a few mounds In the Sohnstetten area, a valley runs through the middle of the area,

so most of the F-S points are along the edges and ravines In the Spitze area, a roadjunction cuts through the right side of the area, so the F-S points are those along thebreak lines caused by these roads

8.3.2 Empirical Models for the Relationship between DTM Accuracy and Sampling Intervals

A triangulation-based modeling system was used in this experiment and linear polation was used to avoid any misleading fluctuation on the constructed surface

inter-Table 8.4 Description of the ISPRS Test Areas Test Area Terrain Description Height Range (m) Mean Slope ( ◦ )

Sohnstetten Hills with moderate height 538–647 15

Table 8.5 Description of Test Data Parameter Uppland Sohnstetten Spitze

a Accuracy is represented in terms of RMSE.

b The mean planimetric CI is equal to CI cotα, whereαis the mean slope angle.

Table 8.6 Description of Checkpoints

Test Area Photo Scale

Flying Height (m)

Number of Points RMSE (m)

Largest Error (m)

cor-To test the accuracy of DTM with sampling intervals (i.e., the grid intervals

in this case were due to regular grid sampling), a number of new data sets withgrid intervals larger than the interval of the original grid were produced by simpleresampling without interpolation, as discussed inChapter 4.The test results are shown

inTable 8.7,which lists the variation of DTM accuracy with grid interval and changes

in accuracy after F-S data are added

The results for the Uppland and Sohnstetten areas are plotted in Figure 8.3

(a) (b)

(c)

Figure 8.2 Contour maps of the test areas (photogrammetrically measured) superimposed

with feature-specific points: (a) Uppland area (CI = 5 m); (b) Sohnstetten area (CI = 5 m); and (c) Spitze area (CI = 1 m), where the large blank area was not measured due to difficulties.

8.3.3 Empirical Models for DTM Accuracy Improvement

with the Addition of Feature Data

From Equation (8.23) it can be found that the difference between the DTM accuracywith and without additional feature points is the second-order term Therefore, thedifference could be expressed as follows:

σ = σDTM-g− σDTM-c= a + b × d2

(8.24)The regression result is shown inFigure 8.4.It is clear that the curves fit the experi-mental data very well In fact, in Figure 8.4, the ratio of the grid interval to its smallestinterval (d/d0) is used instead of the absolute value ofd The regression results also

Table 8.7 The Relationship between the Accuracy of DTM and Grid Intervals

Standard Error (σ) (m) Test Area

Grid Interval (m) No F-S Data With F-S Data

Difference in

σValue (m)

Ratio

in Grid Interval

0.1 0.0

(b)

Figure 8.3 Variation of DTM accuracy with sampling interval (grid interval in this case)

(Reprin-ted from Li 1994, with permission from Elsevier): (a) for Uppland area and (b) for Sohnstetten area.

reveal that the constanta in Equation (8.24) is close to 0; therefore, Equation (8.24)

ON SLOPE AND SAMPLING INTERVAL ∗

Since the early 1970s, attempts have been made to establish a mathematical model forthe prediction of DTM accuracy through experimental analysis A number of such

∗ The materials included in this section were first published in Photogrammetric Record (Li 1993a, 1993b).

2 Ratio of grid interval (d/d0)

0 0.5

1.0

L2

L1

Figure 8.4 Relationship between the difference in DTM accuracy values (with and without

F-S points) and the ratio of grid interval The dot and square points represent the test result; the continuous curves are for regression results L1and L2 are

for Uppland and Sohnstetten, respectively d / d0is the ratio of the grid interval d to the smallest grid interval d0(Li 1994).

models have been developed Most of them are either not reliable or not practicalenough In this section, the theories behind these models will be outlined The modeldeveloped by Li (1990, 1993b) will be presented in detail, because it is similar totraditional map accuracy specification, that is, making use of slope and samplinginterval

8.4.1 Theoretical Models for DTM Accuracy: An Overview

It is understandable that a terrain profile can be expanded by a Fourier series Throughthe analysis of these individual sine and cosine waves, the accuracy loss due tosampling and surface reconstruction from sinusoidal functions could then be estim-ated (Makarovic 1972) The fidelity of the reconstructed surface is represented bythe ratio of the mean value of the magnitude of the linearly constructed sinusoidalwaves to the amplitude of the input waves, as shown inFigure 8.5 In this figure,the profile ABCDEF, reconstructed by linear interpolation, is an approximation tothe sinusoidal input;x is the sampling interval; and δy is the height error at X i,