DITM: “tf1732_c008” — 2004/10/22 — 16:37 — page 159 — #1 CHAPTER 8 Accuracy of Digital Terrain Models The accuracy of DTMs is of concern to both DTM producers and users. For a DEM project, accuracy, efficiency, and economy are the three main factors to be considered (Li 1990). Accuracy is perhaps the single most important factor to be considered because, if the accuracy of a DEM does not meet the requirements, then the whole project needs to be repeated and thus the economy and efficiency will ultimately be affected. 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 errors will 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 to give a planimetric (or horizontal) error while the third is in the vertical (Z) direction and 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 separately from the accuracy of these results in a vertical direction. However, for the latter, an accuracy measure for both error components together is required. There are four possible approaches for assessing the height accuracy of the DTM (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 © 2005 by CRC Press DITM: “tf1732_c008” — 2004/10/22 — 16:37 — page 160 — #2 160 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY 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 ratherthanthenode 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 the quality of the model. This can be conducted at any intermediate stage as well as at the 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 is perhaps the reason why the issue of planimetric accuracy is rarely addressed. An alternative approach is to simultaneously assess the vertical and horizontal accuracies. In doing so, a measure capable of characterizing the accuracy in three dimensions is required. Ley (1986) suggested using a comparative measure of the mean slopes between the DTM surface and the original terrain surface. Others have also considered the use of other geomorphometric parameters as well as terrain feature points and lines. However, there is no consensus. Most people follow the practice of assessing the contour accuracy, that is, assessing the vertical accuracy only. 8.1.2 Distributions of DTM Errors In the field of DTM data acquisition, it is usually assumed that errors in spatial data are normally distributed. However, it is not necessarily the case for DTM errors, as shown in Figure 8.1. These two sets of data were obtained from an experimental test conducted by Li (1990). Figure 8.1(a) is the result for the Sohnstetten area with a sample size of 1892 points and Figure 8.1(b) is the result for the Spitze area with a sample size of 2115 check points. Some information about these experimental tests is given in Section 8.2. To understand the distributions better, the frequency of occurrence of large errors was also recorded. Table 8.1 lists the results (Li 1990). To show how the distribu- tions deviate from the normal contribution, the theoretical values for the occurrence frequency 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). © 2005 by CRC Press DITM: “tf1732_c008” — 2004/10/22 — 16:37 — page 161 — #3 ACCURACY OF DIGITAL TERRAIN MODELS 161 (a) (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 1000 800 600 400 200 0 −2.0 −1.5 −1.0 −0.0 0.0 0.5 1.0 1.5 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 Let f(x, y)bethe originalterrain surfaceand f (x, y)bethe constructedDTM surface, then the difference, e(x, y), where e(x, y) = f (x, y) − f(x,y) (8.1) © 2005 by CRC Press DITM: “tf1732_c008” — 2004/10/22 — 16:37 — page 162 — #4 162 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY Table 8.1 Occurrence Frequency of Large Errors in DTM Test Area Grid Interval (m) >2σ (%) >3σ (%) >4σ (%) Uppland √ 2 × 20 4.5 1.0 0.3 40 5.1 1.1 0.3 √ 2 × 40 5.2 1.3 0.3 80 5.6 1.2 0.3 Sohnstetten 20 5.6 1.7 0.8 √ 2 × 20 6.0 1.5 0.6 40 6.6 1.5 0.3 √ 2 × 40 6.1 1.5 0.3 Spitze 10 5.0 2.3 1.5 √ 2 × 10 5.8 2.7 1.2 20 5.4 2.7 1.4 N(0,1) 4.6 0.3 0.01 is the error of the DTM surface. Following a similar treatment by Tempfli (1980), the mean square error (mse) can be used as a measure for DTM accuracy, where mse = e 2 (x, y) dxdy (8.2) 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 (e max and e min ), the mode (the most likely value), the median (the frequency center), and the mathematical expectation (weighted average). To measure the dispersion, some parameters such as the 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 following parameters can also be used to measure DTM accuracy: Range: R = e max −e min (8.3) Mean: µ = e N Standard deviation: σ = (e −µ) 2 N − 1 (8.4) 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 this measure might be objectionable, that is, 1. The value R depends on only two values of the random variable and others are all ignored. 2. The probability of the values in e(x, y) is ignored. © 2005 by CRC Press DITM: “tf1732_c008” — 2004/10/22 — 16:37 — page 163 — #5 ACCURACY OF DIGITAL TERRAIN MODELS 163 Therefore, the combination of mean and standard deviation is preferred although the distribution of DTM errors is not necessarily normally, as shown in Figure 8.1. This is because most of the probability distribution is massed with 4σ distance from µ, according Chebyshev’s theorem (Burington and May 1970). Chebyshev’s theorem states that the probability is at least as large as 1 −1/k 2 that 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 k 2 (8.5) where k 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 three attributes (accuracy, density, and distribution) of the source data (Li 1990, 1992a). Mathematically, A DTM = f(C DTM , M Modeling , R Terrain , A Data , D Data ,DN Data , O) (8.6) where A DTM is the accuracy of the DTM; C DTM refers to the characteristics of the DTM surfaces; M Modeling is the method used for modeling DTM surfaces; R Terrain is the roughness of the terrain surface itself; A Data , D Data , and DN Data are the three attributes (accuracy, distribution, and density) of the DTM source data; and O denotes other elements. The roughness of the terrain surface determines the difficulty of DTM represen- tation of terrain. If the terrain is simple, then only a few points need to be sampled and the surface to be used for reconstruction will be very simple. For example, if the terrain is flat, only three points are essential and a plane can be used for modeling this piece of terrain surface. On the other hand, if the surface is complex, then more points need to be measured and higher-order polynomials may have to be used for modeling this terrain. The descriptors for the complexity of terrain surfaces have already been introduced in Chapter 2. Among the various descriptors, slope is the most important one widely used in the practice of surveying and mapping and will be used later in the development of the DTM accuracy model. A DTM surface can be constructed by two methods. One is to construct it directly from 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. The accuracy 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 interpolation process. As discussed in Chapter 4, three types of DTM surfaces are possible, discontinu- ous, continuous, and smooth. It has been found that the continuous surface consisting of a series of contiguous linear facets is the least misleading (or the most trustable). © 2005 by CRC Press DITM: “tf1732_c008” — 2004/10/22 — 16:37 — page 164 — #6 164 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY The three attributes of the source data (distribution, accuracy, and density) will also 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 was discussed in Chapter 2. The third attribute, the accuracy of the source data, will be discussed 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 in Chapter 3 that aerial photographs and existing topographical maps are the main data sources for digital terrain modeling. The accuracy 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 camera with 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., contouring and profiling), the accuracy is much lower. The speed of measurement is also an important 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 a fully digital photogrammetric system is not as high as that from an analytical plotter. The accuracy of contouring data obtained from digitization is affected by the following 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: m c = m h +m p ×tan α (8.7) where m h refers totheaccuracy ofheight measurement; m p is theplanimetricaccuracy of the contour line; α is the slope angle of the terrain surface; and m c is the overall height accuracy of the contours, including the effect of planimetric errors. Usually, the accuracy specifications for contours all appear in the form of Equation (8.7). A summary of such specifications is given in Table 8.2. Accuracy loss 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 be still within a 1/3 contouring interval. © 2005 by CRC Press DITM: “tf1732_c008” — 2004/10/22 — 16:37 — page 165 — #7 ACCURACY OF DIGITAL TERRAIN MODELS 165 Table 8.2 Some Examples of Contour Accuracy Specifications Country Scale Accuracy of Contours (m) France 1:5000 0.4 + 3.0 ×tan α Switzerland 1:10,000 1.0 + 3.0 ×tan α Britain 1:10,560 1.8 2 +(3.0 ×tan α) 2 Italy 1.8 + 12.5 ×tan α France 1:25,000 0.8 + 5.0 ×tan α Finland 1.5 + 3.0 ×tan α America 1:50,000 1.8 + 15 ×tan α Switzerland 1.5 + 10 ×tan α Table 8.3 Comparison of the Accuracy of DTM Data Obtained by Different Techniques Methods of Data Acquisition Accuracy of Data Ground measurement (including GPS) 1–10 cm Digitized contour data About 1/3 of contouring interval Laser altimetry 0.5–2 m Radargrammetry 10–100 m Aerial photogrammetry 0.1–1 m SAR interfereometry 5–20m For convenience of reference, the accuracy of DTM source data from various sources is summarized in Table 8.3. 8.2 DESIGN CONSIDERATIONS FOR EXPERIMENTAL TESTS ON DTM ACCURACY 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 accuracy are possible (Li 1992a), in each of which only one of the six factors is used as the 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 © 2005 by CRC Press DITM: “tf1732_c008” — 2004/10/22 — 16:37 — page 166 — #8 166 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY such as GPS, photogrammetry, and other methods. It can also be achieved by using the 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 data selection methods. Alternatively, resampling without involvement of interpolation, as discussed in Chapter 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 data selection methods. In digital terrain modeling practice, grid and contour data are the 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 along the 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 smooth surfaces 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 two heights at each point is obtained. These differences are used to compute statistical values, as discussed in Section 8.1. It is clear that the final DTM accuracy figures are 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 three attributes can be used to characterize the set of checkpoints (Li 1991). First, the required sample size (number) of the set of checkpoints will be considered. From statistical theory it can be found that this is related to the following two factors: 1. the degree of accuracy required for the accuracy figures (i.e., the mean µ and standard deviation σ) 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 given degree 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 © 2005 by CRC Press DITM: “tf1732_c008” — 2004/10/22 — 16:37 — page 167 — #9 ACCURACY OF DIGITAL TERRAIN MODELS 167 the test area or the size of the data set. Similarly, the higher the given degree of accuracy 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 in Section 8.1, the distribution of DTM errors is not necessarily normal and, therefore, a new random variable with approximate normal distribution needs to be selected for further discussion. Let H 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; and M be the true value of the random variable. Then, the ratio Y = µ −M σ/ √ n (8.8) is a standardized variable and has approximately the normal distribution N(0, 1), even though the underlying distribution is not normal, as long as n is large enough (Hogg andTanis 1977). Supposetheσ ofa distributionis known but theM is unknown, then for the probability r and for a sufficiently large value of n, a value Z can be found from the statistical table for N(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, P(−Z ≤ y ≤ Z) ≈ r (8.9) The closeness of the approximate probability r 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 for even a small value of n (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 following expression can be obtained: P µ − Zσ √ 2 ≤ y ≤ µ + Zσ √ 2 ≈ r (8.10) For a given constant S, the percentage of the probability, (100r)%, of the random interval µ ±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 2 r ×σ 2 S 2 = Z 2 r × σ S 2 (8.11) where Z r is the limit value within which the values of the random variable Y will fall with probability r. Its value can be found in the statistical table for the N(0, 1) distribution. The mathematical expression is as follows: (Z) = 1 −α/2 (8.12) © 2005 by CRC Press DITM: “tf1732_c008” — 2004/10/22 — 16:37 — page 168 — #10 168 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY and the commonly used values are as follows: Z r=0.95 = 1.960, Z r=0.98 = 2.326, Z r=0.99 = 2.576 For example, if the accuracy required for the mean estimate is 10% of the standard deviation of the DTM errors (i.e., σ ), and the confidence level is 95%, then the required sample size is n = Z 2 r × σ S 2 = 1.96 2 × 100 10 2 = 384 Similarly, there is also a relationship between the accuracy specified for the stand- ard deviation estimate σ and the required sample size. According to Burington and May (1970), the variance of the standard deviation estimate from a sample can be expressed as follows: σ 2 σ = σ 2 2(n − 1) (8.13) that is, n = σ 2 2σ 2 σ +1 (8.14) For example, if the accuracy σ σ required for the standard deviation estimate σ is 10% of σ , then the required sample size is 51. The variation of DTM accuracy estimate values with the number of checkpoints used has been intensively tested by Li(1991). The number of checkpoints was reduced systematically from 100 to 1% to produce a number of new sets of checkpoints. These new sets of checkpoints were then used to assess the DTM accuracy and produce new sets 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 checkpoints required. 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 accuracy estimated from the height differences is meaningless. This means that the relationship between the required accuracy of checkpoints and the given degree of accuracy for the DTM accuracy estimate should be established. In this discussion, the accuracies are discussed in terms of the standard deviations. Let H 2 be the error involved in the checkpoints and H 1 the true height difference. Then, H = H 1 +H 2 (8.15) By applying the error propagation law to Equation (8.15), the following expression can be obtained: σ 2 = σ 2 H 1 +σ 2 H 2 (8.16) The value of σ itself is not of interest but the value of σ H 2 is. The attempt is made here to find a critical value for σ H 2 so that the σ is still acceptable as being the © 2005 by CRC Press [...]... Composite Data Tested Result (m) Difference (m) Uppland 40 √ 40 2 80 1.04 1. 18 1. 38 0.76 0.93 1. 18 0. 28 0.25 0.20 0.66 0.70 0 .80 0. 38 0. 48 0. 58 Sohnstetten 20 √ 20 2 40 √ 20 2 0.74 0. 98 1. 38 1.77 0.56 0 .87 1.45 2.40 0. 18 0.11 −0.07 −0.63 0.43 0.56 0. 78 1. 08 0.31 0.42 0.60 0.69 Spitze 10 √ 10 2 20 0.29 0.37 0. 48 0.21 0. 28 0.35 0. 08 0.09 0.13 0.16 0.17 0. 18 0.13 0.20 0.30 It is interesting to note that accuracy... C, and D; and the actual accuracy value varies with the positions of I and J between the two nodes and the characteristics of the terrain surface Therefore, the average © 2005 by CRC Press DITM: “tf1732_c0 08 — 2004/10/22 — 16:37 — page 179 — #21 180 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY B C I E J A ∆ d D d Figure 8. 8 Bilinear interpolation of point E by use of four nodes (A, B, C, and. .. 0.64 0 .85 1.24 0.63 0.76 0.93 1. 18 −0.09 −0.13 −0. 08 0.06 0.51 0.56 0.66 0 .81 0.59 0.66 0.70 0 .80 −0. 08 −0.10 −0.04 0.01 Sohnstetten 20 √ 20 2 40 √ 40 2 0.63 0.97 1.56 2. 58 0.56 0 .87 1.45 2.40 0.07 0.10 0.11 0. 18 0.45 0.63 0 .87 1.23 0.43 0.56 0. 78 1. 08 0.02 0.07 0.09 0.15 Spitze 10 √ 10 2 20 0.17 0.25 0. 38 0.21 0. 28 0.35 −0.04 −0.03 0.03 0.12 0.15 0.20 0.16 0.17 0. 18 −0.04 −0.02 0.02 Test Area Uppland... “tf1732_c0 08 — 2004/10/22 — 16:37 — page 188 — #30 ACCURACY OF DIGITAL TERRAIN MODELS Table 8. 11 The Improvement of DTM Accuracy with the Addition of F-S Data into Contour Data Uppland Parameter RMSE (m) µ (m) σ (m) +Emax (m) −Emax (m) CI/σ 189 With F-S Data Without F-S Data 0.93 0.47 0 .80 3.25 −5. 18 6.25 Reduction in σ (%) Sohnstetten 1.74 1.05 1.39 5.91 −5. 18 3.60 With F-S Data Without F-S Data 0.35... in Table 8. 10 These results reveal that this set of mathematical models produce reasonably reliable estimates of DTM accuracy © 2005 by CRC Press DITM: “tf1732_c0 08 — 2004/10/22 — 16:37 — page 187 — #29 188 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY Table 8. 10 Comparison of the Predicted Accuracy and the Test Results Grid Data Only Composite Data Grid Interval (m) √ 20 2 40 √ 40 2 80 Predicted... Interval With F-S Data 0.63 0.76 0.93 1. 18 0.59 0.66 0.70 0 .80 0.04 0.10 0.23 0. 38 1 2 2 √ 2 2 20 2 × 20 40 √ 2 × 40 0.56 0 .87 1.44 2.40 0.40 0.55 0.77 1. 08 0.16 0.32 0.67 1.32 1 2 2 √ 2 2 10 2 × 10 20 Uppland No F-S Data 0.21 0. 28 0.36 0.14 0.15 0.16 0.07 0.13 0.20 1 1 √ 2 2 √ Sohnstetten √ Spitze (a) 0.4 √ √ (H 0/00) (b) 2.0 1.6 ata 0.2 No 0 .8 -S data 0.1 ta 1.2 d F-S (H 0/00) 0.3 With F -S o da F N... #15 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY Difference in σ values (σg – σc) 174 1.0 L2 0.5 L1 0 1 2 3 Ratio of grid interval (d/d0) 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 L1 and L2 are for Uppland... DITM: “tf1732_c0 08 — 2004/10/22 — 16:37 — page 183 — #25 184 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY 8. 4.3.3 A Practical Consideration Regarding Emax and σ The three extreme values identified previously belong to three different types of distribution Eb applies to grids taken across faults or break lines; Er is related to grids taken across peaks, pits, ridges, and ravines; and Ec is used... σT,r = (8. 50) For a DTM constructed from composite data, the accuracy loss formula is as follows: σT,c = d tan α Ec,max = K 4K (8. 51) Substituting Equations (8. 51) and (8. 50) into Equation (8. 37), the accuracy models of the DTM linearly constructed from composite data and grid data only, respectively, © 2005 by CRC Press DITM: “tf1732_c0 08 — 2004/10/22 — 16:37 — page 186 — # 28 ACCURACY OF DIGITAL TERRAIN. .. surface Table 8. 4 Description of the ISPRS Test Areas Test Area Uppland Sohnstetten Spitze Terrain Description Farmland and forest Hills with moderate height Smooth terrain Height Range (m) Mean Slope (◦ ) 7–53 5 38 647 202–242 6 15 7 © 2005 by CRC Press DITM: “tf1732_c0 08 — 2004/10/22 — 16:37 — page 170 — #12 ACCURACY OF DIGITAL TERRAIN MODELS Table 8. 5 171 Description of Test Data Parameter Uppland Sohnstetten . Area Grid Interval (m) Predicted Accuracy (m) Tested Result (m) Difference (m) Tested Result (m) Difference (m) Uppland 40 1.04 0.76 0. 28 0.66 0. 38 40 √ 2 1. 18 0.93 0.25 0.70 0. 48 80 1. 38 1. 18 0.20 0 .80 0. 58 Sohnstetten 20 0.74 0.56 0. 18 0.43 0.31 20 √ 2 0. 98 0 .87 0.11 0.56 0.42 40 1. 38 1.45 −0.07 0. 78 0.60 20 √ 2. “tf1732_c0 08 — 2004/10/22 — 16:37 — page 176 — # 18 176 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY 024 681 0121416 182 0 0 10 20 30 40 50 60 70 Semivariogram for Uppland 20 0 2 4 6 8 10121416 182 0 0 50 100 150 200 250 300 400 350 Semivariogram. as follows: (Z) = 1 −α/2 (8. 12) © 2005 by CRC Press DITM: “tf1732_c0 08 — 2004/10/22 — 16:37 — page 1 68 — #10 1 68 DIGITAL TERRAIN MODELING: PRINCIPLES AND METHODOLOGY and the commonly used values