TR-214 TECHNICAL REPORT FITTING A SET OF STRAIGHT LINES TO A DIGITAL BT PROFILE APRIL 1969 ; NAVAL OCEANOGRAPHIC OFFICE S WASHINGTON, D. C. 20390 noite oH Price 60 cents A Bisah RVAY Cai) A “‘three level tolerance algorithm” for fitting sets of straight lines to points digitized at regular depth intervals from a BT profile is discussed. The application of this algorithm to the reduction of other profile data similarly digitized is discussed. MBL/WHOI OO FOREWORD The rapid increase in the rate of growth of oceanographic data files has stimulated an awareness of the problems related to the management of large data masses and has motivated attempts to devise new techniques for resolving these problems effectively. Accordingly, the National Data Program for the Marine Environment is concerned with the overall problem as if exists now and as it is projected into the future. To deal with the immediate problem of pro- cessing existing data with reasonable economy in a reasonable time frame, the Marine Sciences and the Research and Development Departments of the Naval Oceanographic Office and the National Oceanographic Data Center are jointly developing an improved system for the management of ocean data files. This system, which has been referred to as a "live atlas", is conceived as a tool for providing the oceanographer a quick response computer feed-back data analysis capability, whereby questions formulated in response to a product display may be resolved immediately with the appropriate selection of subsequent displays. It is essential that the live atlas provide the most extensive, reliable and flexible data base that can be achieved within the limited reserves of quick access computer memory. Accordingly, considerable attention has been given to the problems of improving data quality and of formatting data for minimum storage requirements and maximum speed processing capabilities. This report documents an algorithm developed as a result of studies for improving upon the quality and manageability of the National Oceanographic Data Center Bathythermograph (BT) file. This algorithm is important, since it may be used to improve the quality and manageability of various profile forms, including expendable BT (XBT) and analog salinity-temperature-depth (STD) data. T. K. TREADWELL Captain, U. S. Navy Commander U.S. Naval Oceanographic Office O 0301 O0b9189 § q Oe oe fl 4 on; 7 - e e ae i : i ; Lm T ay I j v4 ‘Fi - iy % rt sy, t\ , Ws dish fi mt ‘b sna bi 7? \ Gols ing oh. 7 t i Th Gin a oe roy? vey th ie Tha, REY nek, SOR TWipevhig ad "oy Wah bear ey BID rey - au eae te. aie Og tier Be an En | hz % , oy i + iegalqn: ind Nas iy due i ’ a): a Wil tag ; ; iimeog Re eat r igh - an hey waay. waee a Ps. eG Daet \ y | : 3 2 hail : oe fF \ i WeAMy Seif y't mh An feet} ca * y ¥ ee \ he ) e i : it a t 1 . a ee Bee The peter By) aL ae! i, yee Tae Bagel hie eye: ound RC ey ¥ t { Se , fi at ya ~~, ty Gepeulvat ff ; ' i) hal wile AEs =y ar } ; + i AY x , ? \ i" ; 2 i : Fi ibn + : ; ‘ Pe RTA ce hl f ii DA SEN Bad val Src mn rani if j ‘ 4 ia, ey 3 al ; { H Saied tes s 2 A ‘ es; % A . aed LAN Heit 48 i, é = : Hepes ees t 4A) hea. i és) ’ rs , ye TABLE OF CONTENTS Page PNUTRODWETMONE Stier oh fi siiter saya el vere ente Scypctce: aE Ss te anew 1 PROBIEMIDERINIMIOINiceticis 2) ied. 2) Sercrps: Se. Rot ertoncis de silepte be: aye comets 1 Preserve Features of Profile Gradient. 2... 0.2000 ce eee 1 Eliminate Spurious Gradient Irregularities. ........2.0-0-+208- 4 Identify Straight Line Subsets in a Data Point Aggregate ........ 4 PROBIEMIS@UUMONGcete (i 2s ohare eens Joie peer: GraphicalSolutrombpemersircm -r <1 icl oi oucn-iaen iontciont-i ey i ental Meio 6 Tolerance a Function of Trace Slope... .. 2... 22.2202 eee 6 Trial and Error Regression Method. .........-. ne eee »- 8 Imtersectioniy. ots cmc. oe) ote) stray omelce cot pepromiodre Mey ceren cunein get wuelite 10. STAC IMAC|sb.d ON LONG OeCuDNosDUGEe ids 5V0 5 ro 1-080 doc O0 fen ceviehae 12 Widthiof theswiihing Stylus) lines y. 5 ccna sien enelgeme depen anteart 12 Measure of "Appropriateness of Fit" .... 2.25.2. 2.5226- oes 14 GOMMENITISFANDIGONGIUSIONS aceon cmreitenon ion tele meant: 17 Room for Analytical Studies... 2 2 6 2 ee eee et eee te ee 17 The "Three Level Tolerance Algorithm" ...... 2.242220 17 Compromise Regression Points... . +--+ ++ eee ee eee ees 18 Fitting Curvelinear Functions... . . 2.0.6 6 oye s e.e-ee so cg 18 Other Kinds of Profile Data Sets... 2-20 ee es wee wwe = 19 TABLE OF CONTENTS (CONTINUED) Page PROFILE REGRESSIOIN PROGRAM. (37-0. 2 - ce) =) cinco nem Cue 21 Input/Output Data Linkage sees oy ac Cee 21 Subroutines: 2... j6. ci. sc cantee Ieee woud a ouch cao ae Cerne “2 uh reale 22 Reduction of Data Memory Volume Requirements. .........-. 22 Production, Time: sss) es. 31 0% ineylieite) 91's) 6) ene: “ons ueqlommegtemremicl gen maeaae 23 APPENDIXKCA® oh6c3 ae ee lode lel eis tltetsco als « ele lo, (6) ch one mC mom A-1 FIGURES TF: Sample: BT Profiles 73 oo te Poe ee See is dee iene) feuns, (oe cree 2 2. ‘Lines Fit:to ao Profile’Data Set «5.7... Wo.) re eee ee 3 3; Extreme Gradient Error Range") 7... a we ee a ee es ee 5 4. Comparing Lines of Regression on Straight Line Sets. ..........-. 5 ‘52, Graphical*Solution Tolerance’). 4 ee nnn nn nee 7 Oo LEU duomoya ovolb O8oto Blo ON 0 6960.0 016 6 Rem or dee cre omen 11 7. The "Three Level Tolerance" Algorithm ........-2.c2ecccececee 13 8. Stylus Thickness Value, Regular Profile. ........2.c-e+ceecece 15 9. Stylus Thickness Value, Irregular Profile. ..........c-+e..2ee00-8 16 10%) Fitting Curvelineamunctions. sccm eects = eo ae eer eee 20 INTRODUCTION The Bathythermograph (BT) file is one of the more voluminous data files compiled by the National Oceanographic Data Center (NODC). It consists of temperature profile data visually digitized at five meter intervals from ozalid copies of bathythermograph slides (Figure 1). Although the size of an ozalid BT slide copy may vary, a representative scale is about two inches to 300 meters. Accordingly, a profile is digitized at 0.03 inch intervals on the BT graph scale. This fine resolution is ostensibly recommended for representing subtle temperature gradient irregularities. However, certain undesirable effects accompany such fine scale digitization intervals, namely: (1) the temperature gradient, which is calculated using differences between adjacent data points, is overly sensitive to slight variations caused by reader error and truncation, and is accordingly unreliable; (2) a considerable bulk of computer memory is required to store profile data points (as many as sixty data points for the standard scale BT profile). The computer algorithm described in this paper has been developed to reduce an NODC digital BT profile to a select set of regression lines (Figure 2). The objective of the algorithm is to represent a BT profile with economy and precision (negligible departure from the original BT data points). It has the feature that it provides reasonable estimates of temperature gradient between regression points (points of intersection between neighboring regression lines). This algorithm may, with appropriate modifications, be applied to the reduction of any kind of digital profile data set. For example, it may be used to reduce massive STD profile data sets to manageable proportions without compromising their fine scale descriptions of profile gradient; PROBLEM DEFINITION A. Preserve Features of Profile Gradient A BT profile trace may misrepresent not only real temperature values with depth, but real depth and temperature gradient values as well. The accuracy of a mechanical BT profile curve may be further compromised by mechanical and thermal inertia which causes hysteresis in the BT trace between sensor descent and ascent. Perhaps paradoxically, the hysteresis provides con- vincing evidence that the BT trace truthfully describes real irregularities in the temperature profile. Even in cases where hysteresis is pronounced, as a general rule, the humps and bumps registered in a trace during sensor descent match humps and bumps registered during sensor ascent (Figure 1(b)). It is a challenge to represent a BT trace digitally without losing the evidence of these subtle humps and bumps. FIGURE 1(a). A TYPICAL OZALID COPY BT PROFILE. NOTE THE SIZE AND SCALE OF THE GRAPH. <= FIGURE 1(b). HYSTERESIS IN A BT PROFILE, CAUSED BY SENSOR MECHANICAL INERTIA. NOTE CORRESPONDENCE BETWEEN "HUMPS AND BUMPS" RECORDED DURING SENSOR DESCENT AND ASCENT. DEPTH (METERS) 50 fo) ro) I50 200 250 50 55 60 65 70 75 TEMPERATURE (°F) FIGURE 2, THE "SOLUTION PLANE" RECTILINEAR GRID 1S DEFINED TO APPROXIMATE THE TEMPERATURE/DEPTH SCALE OF THE STANDARD CURVELINEAR GRID BATHY THERMOGRAPH. THE BT PROFILE (INSET) WAS VISUALLY DIGITIZED AT FIVE METER OR FIFTEEN FOOT INTERVALS. DATA POINTS (DOTS) ARE PLOTTED ON THE SOLUTION PLANE. "REGRESSION POINTS" (ASTERISKS) WERE CALCULATED USING THE THREE LEVEL TOLERANCE ALGORITHM DESCRIBED IN THIS REPORT. B. Eliminate Spurious Gradient Irregularities Temperatures on a "standard scale bathythermograph" (Figure 1) are represented from 28° to 90° Fahrenheit in a graph interval of about 3.3 inches. Thus, an increment of 0.1° Fahrenheit is represented by 0.0053 inches on the standard scale BT slide. The author has concluded from measurements on this scale that a stylus trace thickness varies between 0.01 and 0.02 inches, cover- ing a temperature range of from 0.2° to 0.4° Fahrenheit. Assuming that at this graph scale the value read for a data point may vary within the thickness of a stylus trace, then two different readers may record values for the same depth on the same profile that differ by as much as 0.4° Fahrenheit on a vertical trace. Graphical methods (Figure 3) may be used to illustrate the extreme range of error that may be expected when temperature gradient is calculated using the differences between adjacent data points. Recalling that data points are digitized at 0.03 inch intervals on the depth scale, suppose that two different readers were to independently digitize opposite temperature extremes within a trace thickness at consecutive levels on a vertical trace. If the line thickness is 0.02 inches, the two readers will, in effect, digitize graphical temperature slopes that deviate plus or minus 33° from the true slope of the stylus trace, or 66° from each other. It is desirable to eliminate such spurious irregularities from the digital profile, if possible. It may be expected that an algorithm designed to eliminate spurious irregularities in a digitized profile data set will sacrifice real profile irregularities of short interval and small amplitude. It is required to minimize such sacrifice, and it is desirable to provide an estimate of the possible sacrifice that may have been sustained. C. Identify Straight Line Subsets in a Data Point Aggregate When two different readers digitize points at identical depths from a trace that is straight, the lines computed to fit the resulting data sets will tend to agree more closely, depth for depth, than corresponding points in the original data sets (Figure 4). Moreover, the slope of the straight line computed to fit either data set will more likely provide a better approximation of the slope of the trace than slopes caluclated from the differences between consecutive data points. Accordingly, it would seem desirable to replace the set of points digitized at closely spaced intervals from a straight line trace by vectors selected to represent the line of regression on the data points and the regression interval. Of course, it happens that most BT profiles cannot be represented by a straight line. However, a profile can be represented by sets of straight lines, each line selected to fit a series of data points that fall within a negligible k— 02" — fee eet 03" (a) ~ (b) FIGURE 3. READERS DIGITIZING BT POINTS CAN RECORD TEMPERATURES AT EXTREMES WITHIN THE THICKNESS OF A STYLUS TRACE TO PROVIDE AN ESTIMATE OF THE GRAPHICAL GRADIENT ANGLE THAT VARIES FROM THE TRUE GRADIENT ANGLE BY AS MUCH AS PLUS OR MINUS 33°. (a) FIGURE 4. TWO READERS DIGITIZE POINTS FALLING WITHIN THE THICKNESS OF THE SAME STRAIGHT STYLUS TRACE. Ly IS THE REGRESSION LINE ON THE FIRST DATA SET (a), and Lo IS THE REGRESSION LINE ON THE SECOND DATA SET (b). THE RESULTS ARE SUPERIMPOSED (c) TO ILLUSTRATE THAT THE REGRESSION LINES L} AND Ly WILL TEND TO AGREE MORE CLOSELY, DEPTH FOR DEPTH, THAN THE ORIGINAL DATA POINTS. distance of a straight line. It remains to determine what constitutes a "negligible distance" of a data point from a straight line, and to exploit methods for dividing a profile data set into subsets each of which is appropriately identified with a straight line. PROBLEM SOLUTION A. Graphical Solution The problem of BT profile reduction, stated in the last section, may be regarded as a graphical problem of fitting lines to points ina plane. The metric of the plane is defined by the dimensions of the standard BT slide scale. It is assumed that a stylus trace on this plane will be of constant thickness independent of its rotation in the plane. And it is assumed appropriate to fit a straight line to points that align within the thickness of a stylus trace. It may be seen that the depth/temperature grid of the standard BT is not linear (Figure 2). A "solution plane" may be defined in which temperature and depth are related by a rectilinear grid in proportions that approximate the BT grid. The grid of the solution plane is adequate for representing stylus trace thickness as a constant independent of rotation. B. Tolerance a Function of Trace Slope Accordingly, a zonal section of the trace will vary in magnitude with the slope of the trace, and points falling within a trace thickness on the plane may be separated by a distance exceeding trace thickness on a zonal section (Figure 5(a)). If a line is computed for least squares fit with points on the trace, then the extreme zonal departure of a point from the straight line may be permitted to vary with the slope of the line. Referring to Figure 5(a), the permitted range of departure is +e, where e = d/(2sin @). (1) This is translated to units of depth and temperature as follows. Suppose that one inch on the plane of the BT slide equals k1 intervals on the depth scale and k2 intervals on the temperature scale. Then, referring to Figure 5(b), wiohcs (shee 812 eI) 4 sin@ = 1/ 1+(% A oat! (2) If the permitted range of departure on a vertical trace is expressed in units of temperature, §1, so that e = §T/(2 sin 8), then the equation (1) may be expressed (b) (a) FIGURE 5. (a) THE DISTANCE BETWEEN POINTS P; AND P2 ON A ZONAL SECTION OF THE STYLUS TRACE WILL EXCEED THE DISTANCE, d, BY THE FACTOR 1/sine 8. (b) THE GRAPHICAL GRADIENT ANGLE, 6, IS RELATED TO MEASURES OF TEMPERATURE AND DEPTH (SEE TEXT). en / ee ar 7 1+(5- 5-5) o C. Trial and Error Regression Method It is a simple task to fit a straight line to a point subset by the method of least squares. Assuming the intercept A and B the slope of the line of regression on (m - n + 1) points (zj, Tj), (j =m, m+1, .,n), then the solution for A and B is given in matrix notation A = (at @y! ate Hy), B wee Clos Nil, jf Gh Ssii i co 7 |, ane fl =| i (4) 1 241 al : Zm Zmt+l-°2n : Zn n A set of points may be tested for alignment within the thickness of a stylus trace by trial and error. 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ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATION . UNCLASSIFIED a ae aa 3. REPORT TITLE FITTING A SET OF STRAIGHT LINES TO A DIGITAL BT PROFILE 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Technical Report 5. AUTHOR(S) (First name, middle initial, last name) Walter E. Yergen 6- REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS April 1969 35 0 8a. CONTRACT OR GRANT NO. 94. ORIGINATOR’S REPORT NUMBER(S) . Prosect No. HF 05-243-301 TR-214 707-FC-DAR 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report) ~All distribution of this publication is controlled. All qualified DDC users shall submit requests for copies to the: Commander, U. S$. Naval Oceanographic Office, Washington, D. C. 20390 Attn: Code 40 - SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY U. S. Naval Oceanographic Office - ABSTRACT A "three level tolerance algorithm" for fitting sets of straight lines to points digitized at regular depth intervals from a BT profile is discussed. The application of this algorithm to the reduction of other profile data similarly digitized is discussed, DD (2"..1473 ‘PAse !) UNCLASSIFIED S/N 0101-807-6801 Security Classification UNCLASSIFIED Security Classification FITTING A SET OF STRIGHT LINES TO A DIGITAL BT | PROFILE DD *o%..1473 (sack) UNCLASSIFIED (PAGE: 2) Security Classification