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Presented at ISART 2002, Session III: Antennas and Propagation, 3/5/02

A Comparison of Radio Propagation Measurements and Predictions at VHF and UHF Using Univariate and Multivariate Normal Statistics Paul McKenna NTIA/ITS - ITS.E. Presented at ISART 2002, Session III: Antennas and Propagation, 3/5/02. Attenuation (function): A(q t ,q l ,q s )

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Presented at ISART 2002, Session III: Antennas and Propagation, 3/5/02

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  1. A Comparison of Radio PropagationMeasurements and Predictions at VHF and UHFUsing Univariate and Multivariate Normal StatisticsPaul McKenna NTIA/ITS - ITS.E Presented at ISART 2002, Session III: Antennas and Propagation, 3/5/02

  2. Attenuation (function): A(qt,ql,qs) qt= the fraction of time; ql = the fraction of locations; and qs = the fraction of situations. Interpretation: “In qs of like situations there will be at least ql of locations where the attenuation does not exceed A(qt,ql,qs) for at least qt of the time.” [1] [1] G.A. Hufford, A.G. Longley and W.A. Kissick, “A Guide to the Use of the ITS Irregular Terrain Model in the Area Prediction Mode”, NTIA Report 82-100, 1982. Predictive Models considered here are based on the observed statistics of measured propagation data along with some of the physical phenomena that Maxwell’s Equations imply will affect that propagation.

  3. Propagation Data: Are the Available Measurements (Data) Sufficiently Independent? • Identify Approaches to Improve Models' Predictive Accuracies The Data: C Phase 1 Data (3 Datasets) VHF: 20, 50 & 100 MHz; Vertical/Horizontal; 1, 3, 6 and 9 m mobile Rx hag, Tx ‘fixed’ • Colorado Mountains (5, 10, 20, 30 & 50 km) • Colorado Plains (5, 10, 20, 30, 50 & 80 km, no 20 or 50 MHz at 80 km) • Northeast Ohio (10, 20, 30 & 50 km, one central Tx & 5 peripheral Tx sites, no KLIR)

  4. The Data (continued): • Phase 2 Data (5 Datasets) "UHF": 230, 410, 751, 910, 1846, 4595 & 9190 MHz; Horizontal; 1-13 m ‘fixed’ Rx hag, Tx mobile • R1 (Gunbarrel Hill, NE of Boulder, CO, various distances) • R2 (Fritz Peak, W of Boulder, CO, various distances) • R3 (North Table Mountain, near Golden, CO, 1-15 m ‘fixed’ Rx hag, various distances) • R4 (Grove of Trees, near Longmont, CO, no 910 MHz, 1-24 m ‘fixed’ Rx hag, various distances, 2 seasons: summer & winter) "UHF": 76, 173, 409, 950, 2180, 4820 & 8395 MHz; Horizontal; mobile Rx hag, Tx ‘fixed’ • Virginia Piedmont (near Lynchburg, VA, 7 different Tx sites, various distances)

  5. The Data (continued): • Low Antenna Data (3 Datasets) VHF/UHF: 230 & 416 MHz; Vertical; .75 & 3 m ‘fixed’ Tx hag, .75-3 m mobile Rx hag • Idaho (various distances) • Washington (various distances) • Wyoming (various distances) • Ft. Huachuca Data (1 Dataset) VHF: 60 MHz; Vertical; 10 m ‘fixed’ Tx hag, 2 m mobile Rx hag; various distances, 12-16 km

  6. The Data (continued): • TASO Data (1 Dataset or 19 Datasets) VHF & UHF TV Bands; Horizontal; high ‘fixed’ Tx hag, 9 m mobile Rx hag, 30 m run; various distances out along radials from Tx’s Potentially Correlated Measurements: • WCBS, WABC, WUHF (also 2 seasons: summer, winter) • WHYY (FM), WHYY-U • WCRV, WCAU • WMAR, WBAL • WBRE, WBRE-U • KFRE, KJEO • WHYN, WHYN-U • WBRZ, WAFB • WISC, WMTV Probably Uncorrelated Measurements: • WGR, WJBK, KWK, WLAC, WNAC, WFAA, WIS, WISN, WBUF, WNOK

  7. The Data: Summary • Good News • Over 41,000 Measurements • Many Different Terrain Types, Path Lengths, Frequencies, Antenna Heights • Both Polarizations are Available • Bad News • Within Individual Datasets, All Measurements for a Given Path are Correlated to a Greater or Lesser Extent and are, therefore, not Independent. Stated another way: If the propagation conditions on a given path were good, then they were typically good for all frequencies, antenna heights and polarizations for which the measurements were made. Similarly, if the propagation conditions were poor, then they were typically poor for all frequencies, antenna heights and polarizations for which the measurements were made.

  8. The Models: ITM (Point-to-Point Mode): • User Inputs • Terrain Profile (see below) • Distance Between the Terminals • Terminal Heights Above Ground • Frequency/Wavenumber • Minimum Monthly Mean Surface Refractivity • Effective Earth's Curvature • Surface Transfer Impedance of the Ground (or Polarization, Dielectric Permittivity and Electrical Conductivity of the Ground) Radio Climate • Computes Long-Term (Hourly) Median Excess Loss (Reference Attenuation), Then Adjusts Median Loss For Desired Quantiles of Time, Location and Situation Variabilities • Three Excess Loss Regimes: Line-of-Sight, Diffraction, Troposcatter • Choice of regime is based on path length and other inputs • Loss is continuous across regime transitions • Automated Routines Estimate the Terrain Irregularity Parameter, Effective Antenna Heights, the Radio Horizon Distances and Horizon Elevation Angles from the Terrain Profile but, Strictly Speaking, These are User Inputs

  9. The Models (continued): TIREM: • User Inputs • Terrain Profile • Distances of Points on the Terrain Profile • Terminal Heights Above Ground • Frequency • Surface Refractivity • Polarization • Dielectric Permittivity of the Ground • Electrical Conductivity of the Ground • Humidity • Computes Long-Term (Hourly) Median Basic Transmission Loss, Then Adjusts Median Loss For Desired Quantiles of Time Variability • Three Loss Regimes: Line-of-Sight, Diffraction, Troposcatter Line-of-Sight to Diffraction/Troposcatter Transition Occurs when Transmitter Radio Horizon Distance is Less Than the Path Length Diffraction to Troposcatter Transition Occurs Whenever Diffraction Losses Exceed Troposcatter Loss

  10. A Comparison of the Algorithms Used in theITM and TIREM Propagation Models • TIREM uses Antenna Structural Heights only. • ITM uses Antenna Effective Heights Above Dominant Reflecting Plane, Terrain Irregularity Parameter. • Troposcatter Range: • Both Models Use Equivalent Formulations, After a Lot of Work.

  11. LOS Range: TIREM Determines this Range Based on Whether or Not this Distance is Less than or Equal to the Radio Horizon Distance, Using Re (Must Examine Profile in Every Case) Two-Ray Model with Reflected Ray Chosen to Occur at Deepest Terrain Intrusion Into Direct Ray’s Path. Reflection Coefficient is -1 and Empirical Expression for Rough Surface Scattering. ITM Determines this Range Based on Whether or Not this Distance is Less than or Equal to the Smooth Earth Radio Horizon Distance, Using Re (Profile is Examined for h, Eff. Heights, Radio Horizons, Take-off Angles, etc.) Computes Two-Ray Short Range Loss ( 2 Distances Well Within Radio LOS). Reflection Coefficient Depends on Polarization, Zg and a Different Expression for Rough Surface Scattering. These Two Points and a Third Point, at the Smooth Earth Radio Horizon Distance, Computed Using Diffraction Formulas, are Fitted to a Function (Constant, Distance, Log(Distance)) to yield Excess Loss.

  12. Diffraction Range: TIREM Determines this Range Based on Whether or Not this Distance is Greater than or Equal to the Radio Horizon Distance, Using Re, and Less than the Troposcatter Range Locates All Knife Edges that Support the Convex Hull, Except That It Treats Smooth Earth Support Specially. Sums Losses Over Each KE, Inclusive of Ground Reflections. It also Computes a Smooth Earth Diffraction Loss and a Smooth Earth Surface Wave Loss and Derives the Diffraction Loss based on Empiricisms. ITM Determines this Range Based on Whether or Not this Distance is Greater than or Equal to the Smooth Earth Radio Horizon Distance, Using Re, and Less than the Troposcatter Range Locates the Radio Horizons for Both Terminals. Then Computes Diffraction Loss (Convex Combination of Double KE + Smooth Earth Diffraction) at Two Distances Well Beyond the Smooth Earth Radio Horizon Distance, Preserving the Radio Horizon Distances. Fits the Loss to a Straight Line Function of Distance Using the Point-Slope Formula

  13. Comparison of the Two Models’ Predictionsto the Measurements • Provide Identical Inputs to Both Models • Relative Dielectric Permittivity of the Ground = 15. • Electrical Conductivity of the Ground = .008 mhos/m • Surface Refractivity = 310 N-units • Humidity = 0.0 g/m3 • Use Terrain Profile Extraction Resolution Not Greater Than 200 m • Prediction Error: L = Lpredicted – Lmeasured • Lpredicted is the (Computed) Median Excess Loss for the Model in Question

  14. Li= Prediction Error for a Given Propagation Measurement, i (e.g., Frequency of Operation, Terminal Height(s), Path Distance and Intervening Path Profile, etc.): Li = Lipredicted -Limeasured (db) where Lipredicted = Model’s Predicted (Excess) Loss Relative to the Free Space Loss (dB) Limeasured = Measured (Excess) Loss Relative to the Free Space Loss (dB) (Univariate Normal) Statistics were Computed for each Model’s Predictions: N = Number of Measurements in the Sample Sample’s Mean Prediction Error, m: (dB) Variance of the Sample’s Prediction Error, v: (dB2) If the Liare iid samples taken from a univariate normally distributed population, then m and vare Maximum Likelihood Estimates of  and 2.

  15.  Dataset  No. of meas. ITM mean (dB)  TIREM mean (dB)  ITM std. dev. (dB)  TIREM std. dev. (dB)  ITM skew- ness  TIREM skew- ness  ITM excess  TIREM excess  CO. MTNS.   550 (286) -17.1 +/- .7 (-22.8) -4.4 +/- .6 (-4.5)  16.2 +/- .5 (12.1)  13.7 +/- .4 (15.0) .6 +/- .1  .0 +/- .1  .5 +/- .2  -.1 +/- .2 CO. PLNS.   1983 (1983)  -14.9 +/- .2 (-16.7)  -4.4 +/- .2 (-5.6)  10.2 +/- .2 (10.3)  9.9 +/- .2 (9.8)  0.0 +/- .1  .1 +/- .1  .4 +/- .1  .4 +/- .1  NE OH.   1787 (1787)  -10.1 +/- .2 (-12.7)  0.0 +/- .2 (-.2)  9.2 +/- .2 (8.7)  9.6 +/- .2 (8.7) 0.0 +/- .1  .1 +/- .1 .4 +/- .1 .1 +/- .1  R-1   6780  2.0 +/- .2  1.2 +/- .1  13.9 +/- .2  12.0 +/- .1  .8 +/- 0.0  -.2 +/- 0.0  2.5 +/- .1  .8 +/- .1  R-2   2458  -7.5 +/- .5  -18.4 +/- .4  25.7 +/- .3  20.8 +/- .3  -.2 +/- 0.0  -.6 +/- 0.0  -.3 +/- .1 .0 +/- .1 R-3   5149 1.9 +/- .2  2.8 +/- .2  11.6 +/- .2  11.4 +/- .1  .9 +/- 0.0  .2 +/- 0.0  3.1 +/- .1  .4 +/- .1  R-4   9498  -12.8 +/- .2  -14.1 +/- .2  16.6 +/- .2  16.6 +/- .1  -.4 +/- 0.0  -.8 +/- 0.0  1.3 +/- .1  .3 +/- .1  VA.   1655 (1871)  -.9 +/- .3 (-3.7)  -.2 +/- .4 (1.8)  13.2 +/- .3 (9.6)  15.6 +/- .3 (10.8)  -.2 +/- .1  -.2 +/- .1 1.5 +/- .1  .2 +/- .1  ID.   435 (435)  -17.5 +/- .7 (-15.4)  -10.9 +/- .6 (-8.7)  14.5 +/- .4 (12.7)  11.9 +/- .4 (11.3)  -.3 +/- .1  -.4 +/- .1  -.4 +/- .2  -.2 +/- .2  WA.   892 (892)  -2.4 +/- .4 (-5.7)  5.1 +/- .4 (4.7)  12.8 +/- .3 (11.7)  12.0 +/- .3 (13.2)  .3 +/- .1  .1 +/- .1  .1 +/- .2 .2 +/- .2  WY.   704 (704)  -11.9 +/- .6 (-15.8)  -6.8 +/- .5 (-5.5)  14.6 +/- .4 (12.6)  12.5 +/- .3 (9.7)  -.1 +/- .1  0.0 +/- .1  0.0 +/- .2  -.2 +/- .2  Ft. Hua.   372 (420)  -3.0 +/- .6 (-5.4)  11.4 +/- .3 (7.8)  11.5 +/- .3 (11.3)  6.0 +/- .2 (5.9)  -.5 +/- .1  -.2 +/- .1  -1.2 +/- .3  -.9 +/- .3  TASO   8865 -3.2 +/- .1 -1.2 +/- .1  12.5 +/- .1  14.0 +/- .1  -.2 +/- 0.0  -.5 +/- 0.0  2.3 +/- .1  .7 +/- .1 Table 1: Comparison of the Overall Dataset Prediction Error’s Statistics for ITM and TIREM

  16. Dataset No. of meas. ITM mean (dB) TIREM mean (dB) ITM std.dev. (dB) TIREM std.dev. (dB) ITM skew- ness TIREM skew-ness ITM excess TIREM excess CO. MTNS.  550  2.3 +/- .5 -4.4 +/- .6  12.5 +/- .5  13.7 +/- .4  .4 +/- .1  .0 +/- .1  1.2 +/- .2 .1 +/- .2 CO. PLNS.  1983  .7 +/- .2 -4.4 +/- .2  8.5 +/- .1  9.9 +/- .2  .1 +/- .1  .1 +/- .1  0.0 +/- .1  -.4 +/- .1 NE OH.  1787  1.7 +/- .2  0.0 +/- .2  8.2 +/- .1  9.6 +/- .2  .1 +/- .1  .1 +/- .1  .4 +/- .1  .1 +/- .1 R-1  6780  14.8 +/- .2  1.2 +/- .1  18.0 +/- .2  12.0 +/- .1  .8 +/- 0.0  -.2 +/- 0.0  2.1 +/- .1  .8 +/- .1 R-2  2458 4.4 +/- .5  -18.4 +/- .4  23.5 +/- .3  20.8 +/- .3  .2 +/- 0.0 -.6 +/- 0.0  -.4 +/- .1  .0 +/- .1 R-3  5149  29.3 +/- .2  2.8 +/- .2  16.5 +/- .2  11.4 +/- .1  .2 +/- 0.0  .2 +/- 0.0  .1 +/- .1  .4 +/- .1 R-4 9498  -3.1 +/- .2 -14.1 +/- .2  21.2 +/- .2  16.6 +/- .1  .4 +/- 0.0  -.8 +/- 0.0  1.5 +/- .1  .3 +/- .1 VA.  1655  6.3 +/- .3  -.2 +/- .4  12.7 +/- .3  15.6 +/- .3  -.4 +/- .1  -.2 +/- .1  1.0 +/- .1  .2 +/- .1 ID.  435  -3.8 +/- .5  -10.9 +/- .6  10.9 +/- .4  11.9 +/- .4  -.1 +/- .1  -.4 +/- .1  -.1 +/- .2 -.3 +/- .2 WA.  892  10.0 +/- .4  5.1 +/- .4  13.1 +/- .3  12.0 +/- .3  .4 +/- .1  .1 +/- .1  .2 +/- .2  .2 +/- .2 WY.  704  -2.2 +/- .4  -6.8 +/- .5  11.7 +/- .3  12.5 +/- .3  -.1 +/- .1  0.0 +/- .1  .4 +/- .2  -.2 +/- .2 Ft. Hua.  372  15.2 +/- .4  11.4 +/- .3  8.7 +/- .2  6.0 +/- .2  0.0 +/- .1  -.2 +/- .1  -1.1 +/- .3  -.9 +/- .3 TASO  8865  3.1 +/- .2  -1.2 +/- .1  15.0 +/- .2  14.0 +/- .1  .7 +/- 0.0  -.5 +/- 0.0  2.9 +/- .1  .7 +/- .1 Table 2: Comparison of the Overall Dataset Prediction Errors' Statistics for ITM (with the Effective Antenna Heights Set to the Terminal Heights Above Ground) and TIREM

  17. Elements of ITM and TIREM Prediction Error Correlation Matrices for the Three Phase 1 Datasets

  18. Are the Data from our Samples iid (Independent, Identically Distributed)? Not if the Data are Correlated: Stated Simply, Li Depends on Li for some I … j What is the Cause of the Correlation? Multiple Measurements (e.g., Frequencies, Terminal Heights, Polarizations) Were Attempted on the Same Path If the Propagation Conditions were Favorable for Any Measurement on a Given Path, Then Typically They were Favorable for All Measurements on that Path (and v.v.). Analysis Approach for Models’ Prediction Error Statistics in the Presence of Correlation: Group all p Measurements (and, likewise, Predictions and Prediction Errors) Taken on the Same Path Together as the Components of an Observation Vector, xi, but Assume that is xi Independent of xi for i … j. (Note the notation change, i now refers to the path index, not to the measurement index used above!) The Individual Components of Any Given Observation Vector are not Assumed to be Independent of One Another, but Measurements Made on Different Paths are Assumed to be Independent.

  19. A Simple Example to Illustrate the Analysis Approach Phase 1 CO Plains Dataset p=12 (Number of Measurements per Path) and N=184 (Number of Paths) The p(=12) Components of the ith Observation Vector, xi (1  i N (=184) ): Xi (1) = 20 MHz, 1 m height, vertical polarization, for the ith path; Xi (2) = 50 MHz, 1 m height, vertical polarization, for the ith path; Xi (3) = 50 MHz, 3 m height, vertical polarization, for the ith path; Xi (4) = 100 MHz, 3 m height, vertical polarization, for the ith path; Xi (5) = 100 MHz, 6 m height, vertical polarization, for the ith path; Xi (6) = 100 MHz, 9 m height, vertical polarization, for the ith path; Xi (7) = 100 MHz, 3 m height, horizontal polarization, for the ith path; Xi (8) = 100 MHz, 6 m height, horizontal polarization, for the ith path; Xi (9) = 100 MHz, 9 m height, horizontal polarization, for the ith path; Xi (10) = KLIR, 3 m height, at the ith path’s receiver location; Xi (11) = KLIR, 6 m height, at the ith path’s receiver location; and Xi (12) = KLIR, 9 m height, at the ith path’s receiver location.

  20. What Happens to Our Statistics? Consider Multivariate Normal Statistics: N = Number of paths in the sample p = Number of Measurements per path Np = Number of measurements in the sample Generalize the Mean, m, and the Variance, v: Sample Mean Vector, (dB) Sample Covariance Matrix, (dB2) where if xi (j) denotes the jth Component of the ith Observation Vector and Cjkdenotes the Element of C in the jth Row and the kth Column, then (dB2) Note that is a Real, Symmetric and, if Non-Singular, Positive-Definite Matrix

  21. How are m and v Related to and C ? if (very often the case for our datasets) where is the Eigenvalue of C Thus v is (approximately) an Average Variance (over p, the Number of Measurements per Path). The Total Variance ( ) is (approximately) pv. and then

  22. What is the Meaning of this Result? • Computation of the Variance using the Formula for can lead to Serious Underestimates of the Underlying Variances when the Data are Correlated. • It is Important to Know the Distribution of the Eigenvalues of the Covariance Matrix when the Data are Correlated. • If the Range of the Eigenvalues is Disparate, Only the Few Largest have a Significant Contribution to the Total Variance

  23. The Eigenvalues of the Covariance Matrix, C, Satisfy the Equality Cx = x where xis a Column Vector of Length p (an Eigenvector of C)  is a Multiplicative Constant (an Eigenvalue of C) There are p Real Values (Not Necessarily Distinct) and Vectors that Satisfy the Equality: (j,ej) such that Cej = jej for j = 1,…, p The Eigenvectors can be made to be Orthonormal. In fact, It can be shown that There Exists a Real Orthogonal Transformation, , Q such that Q' CQ =  = diag (1,…, p) The Columns of Q are the Eigenvectors (e1,…,ep). The Eigenvalues are Computed Using the Symmetric QR Algorithm with Wilkinson Shift. The Corresponding Eigenvectors are Computed Using the Method of Inverse Iteration.

  24. Why are the Eigenvectors Useful? From the Result Above and the Definition of the Covariance Matrix, C, we have that Q' CQ =  = diag (1,…, p) = ??????????????????? So, if we Transform our Sample Data According to , then it is Uncorrelated and Independent. The Variances of the Transformed Data are the Eigenvalues, 1,…, p, with respect to the Transformed Sample Mean Vector . Differences Between the Eigenvectors Corresponding to the Largest Eigenvalues Can Yield Useful Information in Model Improvements

  25. Dataset  ITM Total Variance TIREM Total Variance  ITM 1  TIREM 1  ITM 2  TIREM 2 ITM 3  TIREM 3  CO. MTNS. (12)  2.89 x 103  2.61 x 103  1.87 x 103 (64.7%)  1.51 x 103 (57.9%)  6.33 x 102 (21.9%) 5.21 x 102 (20.0%)  1.75 x 102 (6.0%)  3.28 x 102 (12.6%)  CO. PLNS. (12)  1.17 x 103  1.20 x 103  7.10 x 102 (60.5%)  7.71 x 102 (64.2%)  2.37 x 102 (20.2%)  1.93 x 102 (16.0%)  8.38 x 101 (7.1%)  9.24 x 101 (7.7%)  NE OH. (9) 8.13 x 102  7.30 x 102  5.81 x 102 (71.5%)  4.78 x 102 (65.5%)  7.75 x 101 (9.5%)  7.90 x 101 (10.8%)  6.03 x 101 (7.4%)  6.41 x 101 (8.8%)  R-1 (91)  2.13 x 104  9.01 x 103  1.48 x 104 (69.6%)  3.62 x 103 (40.2%)  3.50 x 103 (16.4%)  2.45 x 103 (27.2%) 7.80 x 102 (3.7%)  7.74 x 102 (8.6%)  R-2 (91)  4.55 x 104  3.29 x 104  4.10 x 104 (90.2%)  2.84 x 104 (86.2%)  4.99 x 103 (11.0%)  2.94 x 103 (8.9%)  1.49 x 103 (3.3%)  1.14 x 103 (3.5%)  R-3 (105)  2.24 x 104  9.97 x 103  2.01 x 104 (89.7%)  6.84 x 103 (68.6%)  1.66 x 103 (7.4%)  1.75 x 103 (17.5%)  5.60 x 102 (2.5%)  5.52 x 102 (5.5%)  R-4 (144) 5.44 x 104 2.10 x 104  4.85 x 104 (89.3%)  1.33 x 104 (63.4%)  1.89 x 103 (3.5%) 2.25 x 103 (10.7%)  1.34 x 103 (2.5%)  1.49 x 103 (7.1%)  VA. (7)  1.39 x 103  8.42 x 102  1.11 x 103 (80.2%) 5.32 x 102 (63.2%)  1.23 x 102 (8.9%)  1.39 x 102 (16.6%)  5.39 x 101 (3.9%)  6.82 x 101 (8.1%)  ID. (16)  1.84 x 103  1.76 x 103  1.74 x 103 (94.6%)  1.58 x 103 (89.8%)  5.63 x 101 (3.1%)  9.57 x 101 (5.4%)  3.60 x 101 (1.9%)  4.94 x 101 (2.8%)  WA. (16)  2.89 x 103  2.64 x 103  2.78 x 103 (96.3%)  2.39 x 103 (90.7%)  6.98 x 101 (2.4%)  1.48 x 102 (5.6%)  3.80 x 101 (1.3%)  5.52 x 101 (2.1%)  WY. (16)  2.42 x 103  1.97 x 103  2.21 x 103 (91.3%)  1.65 x 103 (83.6%)  1.15 x 102 (4.7%)  1.26 x 102 (6.4%)  3.58 x 101 (1.5%)  1.10 x 102 (5.6%) Table 3: Summary of the ITM and TIREM Prediction Error Results at 100 m Terrain Extraction Interval Using Available-Case Data

  26. Dataset  ITM Total Variance  TIREM Total Variance  ITM 1  TIREM 1  ITM 2  TIREM 2 ITM 3  TIREM 3  CO. MTNS. (12)  2.67 x 103  2.50 x 103  1.69 x 103 (63.4%)  1.47 x 103 (58.6%) 6.09 x 102 (22.8%)  4.71 x 102 (18.8%)  1.67 x 102 (6.2%) 2.96 x 102 (11.8%)  CO. PLNS. (12)  1.07 x 103  1.12 x 103  6.71 x 102 (62.8%)  6.83 x 102 (61.2%)  1.80 x 102 (16.9%)  1.93 x 102 (17.3%)  8.13 x 101 (7.6%)  8.61 x 101 (7.7%)  NE OH. (9)  7.54 x 102  6.90 x 102  5.37 x 102 (71.2%)  4.44 x 102 (64.3%)  6.75 x 101 (8.9%)  7.64 x 101 (11.1%)  6.03 x 101 (8.0%)  6.48 x 101 (9.4%)  R-1 (91)  1.82 x 104 8.48 x 103  1.20 x 104 (66.0%)  3.17 x 103 (37.3%)  3.19 x 103 (17.5%)  2.43 x 103 (28.7%)  7.72 x 102 (4.2%)  7.39 x 102 (8.7%)  R-2 (91)  4.60 x 104  2.91 x 104  4.44 x 104 (96.5%)  2.61 x 104 (89.5%)  3.65 x 103 (7.9%)  2.54 x 103 (8.7%)  8.26 x 102 (1.8%)  1.00 x 103 (3.4%)  R-3 (105)  1.74 x 104  9.09 x 103  1.55 x 104 (89.3%)  6.05 x 103 (66.6%)  9.93 x 102 (5.7%)  1.69 x 103 (18.6%)  5.87 x 102 (3.4%)  5.01 x 102 (5.5%)  R-4 (144)  4.77 x 104  2.05 x 104  4.14 x 104 (86.8%)  1.29 x 104 (62.7%)  1.86 x 103 (3.9%)  2.18 x 103 (10.6%)  1.61 x 103 (3.4%)  1.45 x 103 (7.1%)  VA. (7)  1.23 x 103  8.40 x 102  9.98 x 102 (81.4%)  5.24 x 102 (62.4%)  8.32 x 01 (6.8%)  1.44 x 102 (17.1%)  4.90 x 101 (4.0%)  6.79 x 101 (8.1%)  ID. (16)  1.86 x 103  1.47 x 103  1.77 x 103 (95.0%)  1.31 x 103 (89.5%)  7.28 x 101 (3.9%)  7.15 x 101 (4.9%)  1.85 x 101 (1.0%)  3.96 x 101 (2.7%)  WA. (16)  2.96 x 103  2.61 x 103  2.88 x 103 (97.5%)  2.39 x 103 (91.6%)  5.32 x 101 (1.8%)  1.19 x 102 (4.5%)  3.26 x 101 (1.1%)  5.72 x 101 (2.2%)  WY. (16)  2.30 x 103  2.27 x 103  2.12 x 103 (92.0%)  2.00 x 103 (88.1%)  9.94 x 101 (4.3%)  1.04 x 102 (4.6%)  3.61 x 101 (1.6%)  8.59 x 101 (3.8%) Table 4: Summary of the ITM and TIREM Prediction Error Results at 200 m Terrain Extraction Interval Using Available-Case Data

  27. Dataset ITM Total Variance TIREM Total Variance  ITM 1 TIREM 1  ITM 2  TIREM 2 ITM 3 TIREM 3  CO. MTNS. (12)  2.64 x 103  2.22 x 103  1.78 x 103 (67.5%)  1.33 x 103 (59.8%)  5.12 x 102 (19.4%)  4.06 x 102 (18.3%)  1.38 x 102 (5.2%)  2.40 x 102 (10.8%)  CO. PLNS. (12)  1.01 x 103  1.12 x 103  6.42 x 102 (63.4%)  6.80 x 102 (60.8%)  1.63 x 102 (16.1%)  1.90 x 102 (17.0%)  7.47 x 101 (7.4%)  9.33 x 101 (8.4%)  NE OH. (9)  7.29 x 102  6.77 x 102  5.28 x 102 (72.5%)  4.37 x 102 (64.6%)  6.25 x 101 (8.6%)  7.79 x 101 (11.5%) 5.85 x 101 (8.0%)  6.20 x 101 (9.2%)  R-1 (91)  1.79 x 104 9.29 x 103  1.19 x 104 (66.1%)  3.83 x 103 (41.2%)  3.17 x 103 (17.7%)  2.78 x 103 (30.0%)  7.39 x 102 (4.1%)  6.25 x 102 (6.7%) R-2 (91)  4.46 x 104  3.02 x 104  4.22 x 104 (94.7%)  2.63 x 104 (87.0%)  3.89 x 103 (8.7%)  2.86 x 103 (9.5%)  8.97 x 102 (2.0%) 1.29 x 103 (4.3%)  R-3 (105)  1.82 x 104  8.16 x 103  1.43 x 104 (78.8%)  4.66 x 103 (57.1%)  2.43 x 103 (13.4%)  1.96 x 103 (24.0%)  6.21 x 102 (3.4%)  5.55 x 102 (6.8%)  R-4 (144)  3.48 x 104  1.87 x 104  2.81 x 104 (80.8%)  1.09 x 104 (58.1%)  1.94 x 103 (5.6%)  2.17 x 103 (11.6%)  1.31 x 103 (3.8%)  1.51 x 103 (8.1%)  VA. (7)  1.14 x 103  7.92 x 102  9.26 x 102 (81.3%)  4.85 x 102 (61.3%)  7.01 x 101 (6.1%)  1.35 x 102 (17.0%)  4.94 x 101 (4.3%)  6.74 x 101 (8.5%)  ID. (16)  2.12 x 103  1.84 x 103  2.06 x 103 (97.2%)  1.63 x 103 (88.8%)  4.96 x 101 (2.3%)  1.25 x 102 (6.8%)  2.67 x 101 (1.3%)  3.87 x 101 (2.1%)  WA. (16)  1.91 x 103  2.44 x 103  1.69 x 103 (88.7%)  2.05 x 103 (84.8%)  1.55 x 102 (8.1%)  2.88 x 102 (11.8%)  3.51 x 101 (1.8%) 6.34 x 101 (2.6%)  WY. (16)  2.24 x 103  1.78 x 103  2.04 x 103 (91.2%) 1.54 x 103 (86.1%)  9.94 x 101 (4.4%)  1.07 x 102 (6.0%)  4.98 x 101 (2.2%)  7.08 x 101 (4.0%) Table 5: Summary of the ITM and TIREM Prediction Error Results at 450 m Terrain Extraction Interval Using Available-Case Data

  28. ???????????? Available Case and Augmented ITM Prediction Error Eigenvalue and Eigenvector Spectra for Phase 1 CO Mountains (200 m Terrain Profile Extraction Interval, nominal) itm prediction error covariance matrix eigenvalue spectrum, dataset:mt total var= 2.670713E+03 63.4 22.8 6.2 3.1 1.6 .8 .7 .5 .4 .2 .2 .1 1.69E+03 6.09E+02 1.67E+02 8.23E+01 4.38E+01 2.21E+01 1.74E+01 1.37E+01 9.61E+00 6.23E+00 4.07E+00 2.82E+00 itm prediction error covariance matrix eigenvectors, dataset:mt 1 2 3 4 5 6 7 8 9 10 11 12 1 ‑1.94E‑01 2.99E‑01 ‑3.53E‑01 7.45E‑01 2.84E‑01 ‑9.04E‑02 2.62E‑02 3.08E‑01 5.04E‑02 3.79E‑02 1.17E‑02 ‑7.22E‑02 2 ‑2.55E‑01 2.53E‑01 ‑3.71E‑01 ‑1.75E‑01 ‑5.62E‑01 ‑2.98E‑01 ‑5.00E‑01 4.98E‑02 ‑1.50E‑01 8.93E‑02 5.08E‑03 ‑1.21E‑01 3 ‑2.33E‑01 3.83E‑01 ‑3.36E‑01 ‑4.51E‑02 ‑1.73E‑01 3.92E‑01 4.33E‑01 ‑5.45E‑01 8.20E‑02 ‑3.75E‑02 ‑2.82E‑02 8.07E‑02 4 ‑2.88E‑01 2.01E‑01 5.02E‑02 ‑3.06E‑01 3.02E‑01 ‑4.27E‑01 3.26E‑01 4.29E‑02 ‑3.58E‑01 ‑2.09E‑01 4.73E‑01 ‑5.31E‑02 5 ‑2.81E‑01 2.15E‑01 1.29E‑02 ‑2.89E‑01 2.97E‑01 7.89E‑02 ‑1.97E‑01 2.15E‑01 1.52E‑01 2.22E‑01 ‑1.60E‑01 7.17E‑01 6 ‑2.79E‑01 1.72E‑01 6.25E‑02 ‑3.16E‑01 3.15E‑01 3.43E‑01 ‑2.11E‑01 1.77E‑01 2.32E‑01 ‑2.12E‑01 ‑2.02E‑01 ‑5.97E‑01 7 ‑2.64E‑01 1.17E‑01 3.88E‑01 3.66E‑02 ‑2.33E‑01 ‑4.16E‑01 4.07E‑01 7.80E‑02 1.86E‑01 1.75E‑01 ‑5.36E‑01 ‑1.13E‑01 8 ‑2.75E‑01 7.25E‑02 4.31E‑01 2.15E‑01 ‑3.15E‑01 8.42E‑02 ‑1.27E‑01 2.68E‑02 4.67E‑01 ‑2.78E‑01 5.02E‑01 1.26E‑01 9 ‑2.67E‑01 1.19E‑01 4.75E‑01 2.50E‑01 1.16E‑02 3.03E‑01 ‑2.13E‑01 ‑1.86E‑01 ‑6.22E‑01 2.53E‑01 ‑3.53E‑02 ‑4.33E‑02 10 ‑3.59E‑01 ‑4.64E‑01 ‑1.81E‑01 ‑8.88E‑02 ‑2.21E‑01 3.16E‑01 3.19E‑01 4.44E‑01 ‑6.71E‑02 3.44E‑01 2.00E‑01 ‑7.04E‑02 11 ‑3.68E‑01 ‑4.18E‑01 ‑1.21E‑01 1.21E‑01 ‑2.89E‑02 ‑1.91E‑02 ‑5.00E‑02 ‑5.13E‑02 ‑2.10E‑01 ‑6.66E‑01 ‑3.43E‑01 2.21E‑01 12 ‑3.49E‑01 ‑4.01E‑01 ‑9.19E‑02 5.70E‑02 3.02E‑01 ‑2.67E‑01 ‑2.00E‑01 ‑5.34E‑01 2.73E‑01 3.44E‑01 1.01E‑01 ‑1.24E‑01 augmented itm prediction error covariance matrix eigenvalue spectrum, dataset:mt total var= 2.743119E+03 61.0 23.4 7.7 3.2 1.8 .9 .6 .5 .3 .3 .2 .1 1.67E+03 6.42E+02 2.12E+02 8.76E+01 4.81E+01 2.42E+01 1.71E+01 1.42E+01 8.86E+00 7.47E+00 4.93E+00 2.80E+00 augmented itm prediction error covariance matrix eigenvectors, dataset:mt 1 2 3 4 5 6 7 8 9 10 11 12 1 ‑1.77E‑01 2.67E‑01 ‑3.96E‑01 ‑6.72E‑01 ‑4.20E‑01 1.32E‑01 ‑1.84E‑01 2.23E‑01 ‑4.65E‑03 ‑5.79E‑02 3.33E‑02 ‑8.57E‑02 2 ‑2.38E‑01 2.15E‑01 ‑4.05E‑01 9.62E‑02 5.78E‑01 3.60E‑01 3.86E‑01 2.74E‑01 1.60E‑01 ‑3.49E‑02 2.00E‑02 ‑9.50E‑02 3 ‑2.11E‑01 3.41E‑01 ‑4.29E‑01 5.14E‑02 1.84E‑01 ‑4.79E‑01 ‑1.10E‑01 ‑5.87E‑01 ‑1.16E‑01 1.86E‑02 ‑5.68E‑02 1.29E‑01 4 ‑2.99E‑01 2.04E‑01 6.62E‑02 3.38E‑01 ‑2.57E‑01 4.02E‑01 ‑2.76E‑01 ‑2.10E‑01 3.06E‑01 4.22E‑01 ‑3.03E‑01 ‑1.97E‑01 5 ‑2.91E‑01 2.15E‑01 2.87E‑02 3.29E‑01 ‑2.73E‑01 ‑2.26E‑02 6.66E‑02 3.55E‑01 ‑1.80E‑01 ‑1.42E‑01 ‑1.34E‑01 6.96E‑01 6 ‑2.87E‑01 1.74E‑01 5.09E‑02 3.55E‑01 ‑2.53E‑01 ‑3.12E‑01 1.17E‑01 2.24E‑01 ‑1.98E‑01 ‑7.66E‑02 4.09E‑01 ‑5.67E‑01 7 ‑2.79E‑01 1.50E‑01 3.49E‑01 ‑7.78E‑02 2.25E‑01 3.68E‑01 ‑3.70E‑01 ‑2.51E‑01 ‑9.06E‑02 ‑4.11E‑01 4.45E‑01 1.01E‑01 8 ‑2.77E‑01 1.27E‑01 3.99E‑01 ‑3.01E‑01 2.72E‑01 ‑4.80E‑02 6.84E‑02 9.36E‑02 ‑5.62E‑01 2.88E‑01 ‑3.87E‑01 ‑1.39E‑01 9 ‑2.86E‑01 1.60E‑01 4.14E‑01 ‑2.73E‑01 1.32E‑02 ‑3.51E‑01 2.90E‑01 ‑8.85E‑03 6.56E‑01 ‑6.06E‑02 ‑2.35E‑02 6.61E‑02 10 ‑3.55E‑01 ‑4.70E‑01 ‑1.38E‑01 6.07E‑02 2.21E‑01 ‑2.49E‑01 ‑5.21E‑01 2.97E‑01 1.50E‑01 ‑2.27E‑01 ‑2.72E‑01 ‑1.09E‑01 11 ‑3.62E‑01 ‑4.32E‑01 ‑9.71E‑02 ‑1.11E‑01 6.29E‑03 ‑7.43E‑03 6.40E‑02 ‑3.11E‑02 ‑2.44E‑02 5.87E‑01 4.93E‑01 2.59E‑01

  29. Available Case and Augmented TIREM Prediction Error Eigenvalue and Eigenvector Spectra for Phase 1 CO Mountains (200 m Terrain Profile Extraction Interval, nominal)

  30. Available Case and Augmented ITM Prediction Error Eigenvalue and Eigenvector Spectra for Phase 1 CO Plains (200 m Terrain Profile Extraction Interval, nominal)

  31. Available Case and Augmented TIREM Prediction Error Eigenvalue and Eigenvector Spectra for Phase 1 CO Plains (200 m Terrain Profile Extraction Interval, nominal)

  32. Available Case and Augmented ITM Prediction Error Eigenvalue and Eigenvector Spectra for Phase 1 NE OHIO (200 m Terrain Profile Extraction Interval, nominal)

  33. Available Case and Augmented TIREM Prediction Error Eigenvalue and Eigenvector Spectra for Phase 1 NE OHIO (200 m Terrain Profile Extraction Interval, nominal)

  34. Discussion/Conclusions • Prediction Errors' Univariate Statistics Quoted In a Recent Study Are In General Agreement With Those Found In the Current Study For the Individual Datasets Which Overlap Both Studies • These Statistics Should Be Viewed with Considerable Caution: The Measurement Data within Many of the Individual Datasets are Correlated, thereby Contaminating the Prediction Errors' Univariate Statistics • Multivariate Statistical Techniques and/or More Data are Needed to Make Meaningful Statements (e.g., To Combine Data from Different Datasets, etc.) About Either Model's Predictive Qualities Relative to Measured Data

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