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An Objective Determination of Probability of Fog Formation * 158 BULLETIN AMERICAN METEOROLOGICAL SOCIETY An Objective Determination of Probability of Fog Formation * 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. Unauthenticated | Downloaded 09/27/21 03:21 PM UTC VOL. 31, No. 5, MAY, 1950 159 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. Unauthenticated | Downloaded 09/27/21 03:21 PM UTC 160 BULLETIN AMERICAN METEOROLOGICAL SOCIETY 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 sep- arate 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.
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