An Objective Determination of Probability of Fog Formation *

Home , Dew, Dew point depression, Wind direction

158 BULLETIN AMERICAN METEOROLOGICAL SOCIETY

LOUIS BERKOFSKY

Atmospheric Analysis Laboratory, Base Directorate for Geophysical Research, Air Force Cambridge Research Laboratories, 230 Albany St., Cambridge, Mass.

ABSTRACT

A method of approach to objective fog forecasting, based on the use of probability charts is suggested. Given two parameters, say wind speed and dew-point depression, at sunset as ordinate and abscissa of a chart, occurrences and non-occurrences of fog following sunset are plotted as functions of these parameters. Isolines of relative frequency are drawn, giving a probability chart. Two more such charts, using four additional parameters, are constructed, and a total probability of fog occurrence following sunset is computed as a linear function of the three individual probabilities. This result is used as a criterion for the forecast. Time of formation equations are developed, to be applied in the event a fog proves to be likely.

I. INTRODUCTION sive—during which period the fog was mainly non-frontal. Only those cases were considered in ANY objective methods of forecasting which precipitation was not occurring at time of fog deal primarily with determination of fog formation, in order to simplify the investiga- time of formation. The determination M tion.1 of likelihood of formation is largely subjective. If it is decided subjectively that a fog must be II. THE PARAMETERS forecast, the objective time graphs are then en- No so-called objective method is truly objective, tered. Such graphs must of necessity consider due to the fact that it is necessary for the investi- only cases of occurrence, and therefore should gator to select certain parameters. He selects apply quite well if the forecast of occurrence is those which, for statistical reasons, or by reason correct. of personal experience, show an inter relationship. It frequently occurs that, under apparently quite Obviously, if the exact nature of the relationship similar conditions, one situation may produce fog among the parameters were known, objective and the other not, at a given location. If, then, methods would be unnecessary. An attempt is the problem is to be handled statistically, consid- made to reduce the amount of subjectivity by con- eration must be given to the probability of fog sidering those parameters which show the best occurrence following a given "deadline" time, be- statistical relationship. fore any attempt is made to predict time of for- The following parameters were selected after a mation. Such an approach would be more com- careful survey of Stewart Field's data for 1943- pletely objective. 1947 inclusive: We have applied such a method to fog in the a. Wind Speed at Sunset.—It is generally study reported here, basing it on the work of Brier agreed that, if the wind is above a certain maxi- [1] and Mook [2], Fog, as defined here, will mum, or below a certain minimum speed, fog is mean a ceiling of 500 feet or less and/or a visibility not likely to form after sunset. At Stewart Field, of one mile or less. Cases of non-occurrence as the data indicated that most cases of fog occurred well as occurrence of fog are considered in order with wind not less than 3 miles per hour nor to arrive at numerical probabilities. Using these greater than 15 miles per hour at sunset. There figures, the actual forecasts can be stated as per- were, however, some cases where fog formed al- centages. However, they are called "yes" or "no" though the wind at sunset was calm. forecasts, depending upon whether the probability b. Dew-Point Depression at Sunset.—If other does or does not exceed 50%. conditions are favorable, a small dew point de- The study, prepared for Stewart Field, New pression at sunset is generally an indication of fog likelihood following sunset. At Stewart Field, York, included only months of maximum foggi- ness—July through November, 1943, 1947 inclu- 1 Miller, J. E. and Mantis, H. T., 1947: An Objective Method of Forecasting Visibility, New York University, * Adapted from a Master's thesis completed at New College of Engineering, Research Division, Department York University, 1948. of Meteorology, p. 3.

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most cases occurred with a depression of less than thus calls for a "no fog" forecast. Calm cases 15 degrees Fahrenheit at sunset. were considered separately. c. Stability at Sunset.—The stability of the The quantities described in SECTION II were lower levels of the air was measured by consider- selected because they are quantities which are con- ing the difference in temperature at sunset be- sidered to have most effect on fog formation at tween Stewart and Bear Mountain. (Bear Moun- Stewart Field. These parameters, along with tain is about 18 miles SSE of Stewart Field, and wind direction, could be combined in various ways about 800 feet higher.) Fog rarely formed when other than in the objective method described be- the difference between Stewart Field's tempera- low. The combinations were chosen in the fol- ture and Bear Mountain's temperature was more lowing manner. First, those quantities generally than 7F°. (rSWF - TBMN < 7F° ; where SWF believed to be most effective in fog production = Stewart Field, BMN - Bear Mountain.) (wind speed and dew-point depression at sunset) d. Wind Shear at Sunset.—Since, in most were combined. Additional parameters (stability cases, the wind direction at sunset was the same and shear) were then introduced in an attempt to at both Stewart Field and at Bear Mountain, the improve the forecast of probability of formation. vertical shear of the horizontal wind was measured Finally, two more parameters (low cloud cover by taking the difference between observed winds and vertical dew-point gradient) were introduced.

at both places. This quantity (FBMN — ^SWF) There are actually fifteen ways in which the ele- ranged from +10 miles per hour to — 5 miles ments could have been combined, for a given per hour inclusive, in general. Observations quadrant. Trial and error attempts with several showed that fog occurred for all values of wind of the fifteen combinations led to the belief that shear within this range. those finally selected would yield best results. e. Vertical Dew-Point Gradient at Sunset.— All cases in the study were plotted as a function The difference between dew-point temperature at of wind speed and dew-point depression at sunset, Bear Mountain and at Stewart Field at sunset for each of Quadrants I, II, III. In the following, was calculated in order to obtain a measure of only Quadrant III will be described, but the pro- the vertical distribution of moisture. Most cases cedure is exactly the same for the others. In the of fog occurred when Bear Mountain's dew-point wind speed—dew-point depression chart for Quad- temperature was 10F° or less smaller than Stew- rant III, a line of separation could be drawn be- art Field's, or 2F° or more greater. tween a region of both occurrence and non-occur- f. Low Cloud Cover at Sunset.—Any cloudi- rence following sunset, and a region of non- ness below 10,000 feet was considered to have an effect on fog formation. Most cases formed with either very little low cloudiness or almost com- plete overcast at sunset, so that the likelihood of formation could not be determined merely on the basis of cloud cover.

III. COMBINING THE PARAMETERS In order to take into account the probability of fog formation with winds from different direc- tions, four quadrants were designated: Quadrant I—which includes all cases where the wind was N through ENE at sunset, Quadrant II—which includes all cases where the wind was E through SSE at sunset, Quadrant III—S through WSW at sunset, Quadrant IV—W through NNW at sunset. Quadrant IV was eliminated from further study when it was observed that fog occurred only 5 times out of 153 cases when the wind was from FIG. 1. Wind speed (mph) vs. dew-point depression this quadrant at sunset. This is due to adiabatic (°F) for Quadrant III (S through WSW) at sunset. warming of air flowing downslope from the Cats- Lines are isolines of probability of fog occurrence after kill Mountains. A Quadrant IV wind at sunset sunset.

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above statement holds to a good degree if we interpret dying down of turbulence to mean in- creasing fog probability and conversely. Thus, since the chart seems to follow the formula, we may assume that there is more justification in drawing for actual values rather than by large- scale smoothing. There does not seem to be any a priori reason for such smoothing. FIGURE 3 shows the distribution of isolines when all cases are again re-plotted as functions of vertical dew-point gradient and low cloud cover.

IV. A WORKING HYPOTHESIS FOR PROBABILITY OF FOG FORMATION At this point, given the data at sunset, a separate forecast of the probability of formation FIG. 2. Stability (°F) vs. wind shear (mph) for could be made from each of the charts in each Quadrant III (S through WSW) at sunset. Lines are quadrant. In order to combine all the charts, the isolines of probability of fog occurrence after sunset. assumption was made that the total probability is a linear function of the three separate proba- occurrence only. All cases on the side of the line bilities : away from the origin are non-occurrences; and, P = Ap + Bp + CPs + D, (1) therefore, if a plotted point falls on that side of c 1 2 the line, it is assigned a zero probability. The where Pc = computed probability of fog follow- region of both occurrence and non-occurrence was ing sunset, divided into arbitrary areas, and the relative fre- pi = probability of fog from FIGURE 1, quency of occurrence within each area was com- p2 = probability of fog from FIGURE 2, puted. Isolines of frequency were then drawn and (see FIG. 1) and these lines were considered as probability lines, and used as such for future pz = probability of fog from FIGURE 3. forecasts. The same cases were re-plotted as functions of stability and shear, and the same procedure as above was followed (see FIG. 2). Consider the expression for the Richardson Number:2

de> dz y R =

where 6 = potential temperature, 1 dO = stability, and 6 dz du = shear in x-direction. dz We see that, as stability increases and shear remains constant or decreases, turbulence dies down; as stability decreases and shear remains constant or increases, turbulence increases. As is seen FIG. 3. Vertical dew-point gradient (°F) vs. low cloud from the isolines of the stability-shear chart, the cover (tenths) for Quadrant III (S through WSW) at 2 Haurwitz, B., 1941: Dynamic Meteorology, 1st Ed., sunset. Lines are isolines of probability of fog occur- McGraw-Hill Book Co., Inc., p. 200. rence after sunset.

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Various combinations of parameters were cor- related with time of formation, and those correla- tions used in which the standard error of estimate was smallest. In the case of Quadrant III, the following results were obtained:

Clear: t = 0.91 (T - Td) + .14 7 + .58, (3)

<*>

where T — Td — dew-point depression at sunset in F°, V = wind speed at sunset, in miles per hour, and t = time of formation, in hours FIG. 4. Probability computed from least squares equa- after sunset. tion vs. observed frequency in each range, for Quadrant III. VI. A SAMPLE FORECAST A regression equation of the above type was The data at sunset for September 8, 1948, were : fitted, and the constants determined by least Wind speed 6 mph squares.3 PQ was given the value 0 for a non- Dew-point depression 12F° occurrence, 1 for an occurrence, and pi, p2, p3 Stability - 7°F were in units of percentage. The following equa- Shear — 2 mph tion was obtained for Quadrant III: Low cloud cover .7 Vertical dew-point gradient — 1F° Pc = .029/>I + .026/>2 + .616/>3 - .005. (2)

It is apparent that for certain values of the p's, FIGURE 1 gives pi = .09; FIGURE 2 gives p2 = .30; FIGURE 3 gives />3 = .75. Therefore, Pc = .467, the above equation will yield values of Pe less than 0. Computed probabilities were therefore and FIGURE 4 gives P — .96, so that the forecast compared with observed frequencies, and a curve is "yes." To determine time of formation, we use equation (4), giving t— 12.23 hours. Actually, of P vs. Pc was fitted by eye (see FIG. 4). This is a correction curve which enables one to adjust a fog did form 12 hours after sunset. any computed probability value (Pc) to a function (P) which is constrained to lie between 0 VII. VERIFICATION and 100. The verification was based on data for the

V. A WORKING HYPOTHESIS FOR DETERMINA- period August 1-September 22, 1942, inclusive, TION OF TIME OF FORMATION and July 1-November 30, 1948, inclusive—a total of 206 cases. The results are listed in the follow- If the final probability P is > 50%, a fog is ing table, for all cases: forecast to occur. It is at this point that the question of time of formation arises. It was as- "Fog" forecast and observed 13 sumed that a linear relationship exists among the "No fog" forecast, "no fog" observed 153 time of formation, in hours after sunset, and the "Fog" forecast, not observed 9 parameters wind speed, dew point depression, sta- "Fog" observed, not forecast 31 bility, wind shear, vertical dew point gradient, all Total 206 measured at sunset. For purposes of time of formation determination, it was found that best The overall accuracy is 80%, while the accuracy results were obtained if each quadrant was sub- for correct "fog" forecasts is only 30%. Inas- divided into clear (0-.5 inclusive low cloud cover much as this method was not compared with sub- at sunset), cloudy (.6-7 inclusive low cloud cover jective forecasts at Stewart Field, it is not known at sunset), overcast (.<8-1.0 inclusive low cloud whether the results obtained by the objective cover at sunset). method are at least as good as would have been obtained subjectively. If this were the case, the 3 See, for example: Freeman, H. A., 1942, Industrial Statistics, John Wiley and Sons, Inc., p. 118-119. use of the method would be justified. However,

Unauthenticated | Downloaded 09/27/21 03:21 PM UTC 162 BULLETIN AMERICAN METEOROLOGICAL SOCIETY the 30% result indicates a need for improvement short of expectations. For this reason, the study of the method. is offered, not as evidence of overwhelming suc- cess, but rather as a suggestion for further study VIII. COMMENTS in fog forecasting by objective means. The above study was based on a method pre- viously used for rainfall forecasting and for thun- IX. ACKNOWLEDGMENTS derstorm forecasting, and was adapted here for The author wishes to express his thanks to Mr. fog forecasting. That such a method is of value Jerome Spar, of New York University, for nu- is shown by the fact that one is able to incor- merous helpful suggestions, and to Mrs. Berkofsky porate a number of parameters into the study. for performing most of the calculations and pre- Although only seven parameters were used in the present study, it is possible, by slight variations of paring the charts. the method, to include even more. The important REFERENCES question remains: which are the most important [1] Brier, G. W., 1946: A Study of Quantitative Pre- parameters? The best parameter for one location cipitation Forecasting in the TVA Basin, U. S. may not be the best for another. Indeed, it is Department of Commerce, Weather Bureau Re- search Paper No. 26. difficult to determine whether one has selected the [2] Mook, C. P., 1948: An Objective Method of Fore- best possible parameters to begin with. This is casting Thunderstorms for Washington, D. C., in May. JJ. S. Department of Commerce, Weather undoubtedly the reason why the present study fell Bureau (unpublished).

Program, 107th National Meeting, Salt Lake City, June 20-22

(In connection with meeting of the American Association for the Advancement of Science, June 19-24)

Meeting headquarters will be located in the lobby of 3. Air Temperature Modification by Vertical Transport. the Union Building on the University of Utah campus, —R. G. Fleagle, University of Washington, Seattle, where Registration and Information Desks will be open Washington. 15 minutes. daily from 8:30 A.M. to 5:00 P.M. from June 19th 4. Atmospheric Sea Salt in a Tropical Storm.—A. H. through 24th. Fee for registration is one dollar; wives Woodcock, Woods Hole Oceanographic Institution, without charge; transportation and food extra. Chair- Massachusetts. 15 minutes. man, AMS Local Committee on Arrangements: O. Rex 5. The Possibilities of Forecasting the Intensity of Warner, U. S. Weather Bureau Regional Office, Salt Eddy Diffusion Coefficients in the Atmosphere.— Lake City, Utah. H. H. Lettau, Air Force Cambridge Research Lab- Monday, June 19, 1950 oratories, Mass. 20 minutes. AAAS Sessions: 6. Air Flow in the Los Angeles Basin and its Relation 9:00 A.M. Trip to Bingham Copper Mine and to Atmospheric Pollution.—J. E. Edinger, Meteorol- Great Salt Lake. ogy Department, University of California, Los An- 2:00 P.M. Organ Recital and Lecture, Salt Lake geles. 20 minutes. Tabernacle. 7. (Title to be Announced.)—A. J. Hamming, Los 8:30 P.M. Special Tabernacle Organ Recital, with Angeles Air Pollution Control District, Los Angeles, lecture on "Sound in Organ Pipes," by Alexander California. 20 minutes. Schreiner, Tabernacle organist. Tuesday, June 20th, 1:00 P.M. Tuesday, June 20th, 9:00 A.M. Field Trip to American Smelting and Refining Company, M. D. Thomas, Chairman Department of Agricultural Research 1. The Problems of the Transport of Hay-Fever Pollen . Tuesday, June 20th, 4:00 to 6:00 P.M. and the Measurement of Their Concentration in the General Reception at the home of the President of the Atmosphere.—A. N. Dingle, Ohio State University, University of Utah, Dr. Albert Ray Olpin and Mrs. Columbus, Ohio. 15 minutes. Olpin, at their home, 1259 East South Temple Street. 2. Frequency and Magnitude of Inversions at Seattle.— P. E. Church, University of Washington, Seattle, Tuesday, June 20th, 7:00 P.M. Washington. 20 minutes. Canyon picnic at Storm Mountain Picnic Area ( Continued i page 167)

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