Ash clouds and probability

From the Daily Mail: New ash cloud could delay re-opening of London airports. We have this gem: "Critics said the agency used a scientific model based on 'probability' rather than fact to forecast the spread of the ash cloud." See the Telegraph as well.

What else are they supposed to do? The agency here -- the Met Office, which is the national weather service of the UK -- doesn't know what the ash cloud is going to do. If they waited to see what the cloud does, the planes would already be in the air. It would be too late.

Is wind chill misleading?

From Daniel Engber at Slate: Wind Chill Blows. Back when nobody read this blog, I wrote about how the heat index doesn't make sense to me, because I know what 95 degrees with "typical" humidity for my location feels like, and telling me it "feels like" 102 is misleading. (That's Fahrenheit, not Celsius; we're not literally boiling here in the summer.)

Something similar is true for wind chill. Both of these measures only take into effect two of the many variables that effect comfort -- temperature and either humidity or wind speed. They assume that these are the only two variables which actually vary -- clothing, amount of sunlight, weight, etc. are held constant. In reality, comfort is a function of many variables, and it's misleading to create an index that assumes it's just a function of two variables. People know that they should take more than the temperature into account, but I've seen quantitiatively unsophisicated people think of the wind chill as some perfect index of the weather.

But let's face it, a wind chill of zero sounds scarier than a temperature of sixteen. (Those are approximately the numbers I heard reported this morning in Philadelphia.) That means more people watch the news.

Weather and political polls

From a Philadelphia TV weather man, Glenn "Hurricane" Schwartz, upon observing that the low and high temperature for Philadelphia today (70 and 85, respectively) were the "normal" temperatures for the day:

"Exactly normal! It's not normal to be exactly normal!"

which, of course, is true. That's how distributions work.

The weather people on television try a lot less to explain why the weather did what it did than the political people; John and Zeno have talked about this "roller-coaster polling". I suspect this is because once the weather has happened we don't care why it happened that way, while the whole point of polls is to use them to forecast the upcoming election.

Lightning and lotteries

From a rerun of Friends:

Ross: Do you know what your odds are of winning the lottery? You have a better chance of being struck by lightning 42 times.
Chandler: Yes, but there's six of us, so we'd only have to get struck by lightning 7 times.
Joey: I like those odds!

Unsurprisingly, Chandler seems to know that probability doesn't work this way; Joey doesn't.

Also, Ross is wrong. It seems the record for getting struck by lightning is Roy Sullivan, seven times. So nobody's been hit 42 times, while plenty of people have won the lottery.

I don't know how to calculate the odds that someone gets hit 42 times by lightning in their life; the lifetime incidence of getting hit is three thousand to one, and if you figure that lightning strikes are a Poisson process with rate 1/3000 per lifetime, as this article states, then the probability that lightning hits one person seven times is something like one in (1/3000)⁷/7!, or one in about 10²⁸. (That's the probability that a Poisson with parameter 1/3000 takes the value exactly 7; I'm ignoring the normalizing factor of exp(1/3000) and the even-more-negligible probability that someone gets hit eight or more times.)

Since the number of people who have existed is much less than 10²⁸, the existence of a person who's been hit seven times is very strong evidence that that's not the right model. My hunch is that events of each person getting hit by lightning are a Poisson process, but with a separate parameter depends on the person. Roy Sullivan was a park ranger.

But the 1 in 3000 figure can't be trusted; the article also claims the annual risk of getting hit by lightning is one in 700,000. People don't live 700,000/3,000 (i. e. 233) years.

How many cold nights are there?

Nineteen degrees Fahrenheit (-7 Celsius) in Philadelphia this morning.

A photographer that I know went out and took some beautiful pictures of downtown Philadelphia overnight; then he complained that he picked the "coldest day of the year" to do it. Of course, he meant the coldest day of the year so far.

But I suspect it won't be the coldest day of the year, for the following reasons:

• the coldest weather in Philadelphia usually comes in late January. (But this is a bit specious; if this morning's low had been 5^oF (-15^o C) I wouldn't be saying that.)


• In the last eleven years, it has been 19 degrees or below on some day after January 3 every year. (There's more complete climate data available out there, but I don't feel like looking at it.)


• Forecasters are forecasting lows around 16 degrees F (-9 C) for tonight.

But that raises an interesting question: what can we expect the coldest temperature all year to be?

The normal low temperature in Philadelphia on any given night in late January is 25 degrees -- that's when the normal low is at its lowest -- but only an idiot would say that this means the average lowest temperature all year is 25 degrees. (In fact, even without looking at the records, I'd be shocked to learn that there has been a year in recorded history in which Philadelphia didn't go below 25 at some point.) One tempting thing to do is to make the following assumptions:

• the coldest day of the year will always fall between, say, January 5 and February 10, a period of 37 days. Call this number t;


• the low temperature on any day in that interval is normally distributed with mean 25 degrees and standard deviation σ


• thus the annual minimum temperature should be at the 1/(t+1) quantile of that normal distribution.

That turns out to be roughly 2 standard deviations below the mean. Unfortunately, I don't know what the standard deviation is!
And there's a much bigger problem here -- I've implicitly assumed that the annual minimum always falls in a certain cold period, and that the temperature on each day is independent of each other day! The second assumption is spectacularly bad. If it's colder than average today, it'll probably be colder than average tomorrow.

Also, I've assumed that the low temperatures on a given day of the year are normally distributed, which probably isn't true...

However, one could use a method like this to estimate, say, the mean number of days below, say, fifteen degrees in a given year. If we know the distribution then we can compute, say, that the probability of it being below fifteen degrees on the night of January 3 is ten percent; adding up these probabilities for every night of the year gives an expected mean number of cold nights; call this μ. But the variance is important here, as well. All I can say instinctively is that the variance is probably greater than that of a Poisson distribution with mean μ (which is what you'd usually use to model "rare events"), since if it's cold today it'll probably be cold tomorrow, and it probably doesn't tell us much about whether next week will be cold; in particular I suspect that within the winter season, there aren't pairs of days for which being cold is inversely correlated.

(Of course, this is all testable, but I don't care that much.)

Weather forecasting

One way I procrastinate and/or pass time is by reading the archives of long-running blogs. Today it's glenn mcdonald's furialog, because I was thinking of buying the My So-Called Life box set, but decided against it because it's $70; mcdonald used to write a music review column called The War Against Silence which I liked, the first installment of which was written five days after the last episode of that tragically short-lived television show aired. mcdonald was a fan of the show, and I was bored this morning, so I thought of his blog. (For the record, he's a programmer by trade but not by training, although this may be irrelevant.)

Anyway, he wrote on 13 January 2005, commenting on weather forecasting:

But what I would hold someone responsible for, if I thought it weren't a pervasive cultural flaw, is the destructive precision with which uncertain predictions are communicated. Weather is merely the most obvious daily public manifestation of a fundamental reluctance, or perhaps an inability, to say what we really know, rather than what we wish we knew.

Similarly, I've watched a bunch of Walter Lewin's introductory physics lectures in the last few days (I've mentioned this before) and he has drummed into my head that a measurement without uncertainty is useless. What is true for measurements is doubly true for predictions.

It would be interesting to see error bars on weather predictions, but we never will. One can get something like this, though, for the tracks of tropical storms at Weather Underground; they show the outputs of various computer models. If these models all show a storm going in the same direction, one can be fairly sure it will actually go in that direction; if the projected paths are spread out, one knows not to be so sure. I wonder if it is possible to determine the accuracy of more mundane weather predictions by, say, tuning into a large number of different TV channels, web sites, and so on which make independent forecasts. If it were possible to get archived weather forecasts I'd take a look; as it is I don't think I can find out what people on July 14, 2006 (say) thought the weather would be like on July 18, 2006, so the only way I could collect data would be to wait, or perhaps to call up some weather-forecasting entity and ask, and I don't care quite that much.

The fundamental reason we don't see error bars on weather forecasts, though, is because they are brought to us for the most part by profit-seeking entities, and such entities really don't want to advertise "we don't know what's going to happen!" -- even though everybody knows this. This is the "pervasive cultural flaw" that mcdonald refers to above.

Also, although I cannot produce data for this, I seem to perceive that at least certain weather forecasting outlets use only even numbers of (Fahrenheit) degrees in their forecasts, not odd numbers. If this is true, I suspect it means that they believe they can't be accurate to within one degree.

How much land is in the tropics?

In Tropics widen, fringe areas drier, experts say, from the Associated Press, I found the following quote:

Geographically, the tropical region is a wide swath around Earth's middle stretching from the Tropic of Cancer, just south of Miami, to the Tropic of Capricorn, which cuts Australia almost in half. It's about one-quarter of the globe and generally thought of as hot, steamy and damp, but it also has areas of brutal desert.

In fact, the tropics make up about two-fifths of the globe. To be more precise, I mean that about two-fifths of the total area of the surface of the earth is between the two tropics. The tropics are at latitudes +23.5 and -23.5 degrees (I'll use + and - for north and south here.) So it's easy to see where the "one-quarter" figure might come from -- the tropics span a total of 47 degrees of latitude, out of the full range of 180, and 47/180 is essentially 1/4.

But there's an interesting fact about the surface area of a sphere. Take a sphere of radius r. Cut it with two parallel planes of distance h apart. Then the area of your slice will be h/(2r) times the surface area of the sphere, or 2πrh, regardless of the way the sphere is cut; notice that this is also the surface area of a cylinder of height h and radius r. I saw this demonstrated once, when I first saw this fact in a calculus class, by cutting a spherical loaf of bread into slices of equal thickness; the slices get varying amounts of the interior of the bread but all get the same amount of crust.

The thickness of the slice in question, for the tropics, is (2 sin(23.5^o))r, where r is the radius of the earth. Thus this slice makes up sin(23.5^o) = 0.398 of the earth, which is just under two-fifths. I knew from the moment I read this fact that one-quarter was an underestimate, but I suspected that perhaps it was more accurate than, say, "one-third". But it's not.

(The actual point of the article is not that the tropics make up one-fourth, or two-fifths, or whatever fraction of the earth, but that they are "widening"; this uses an atmospheric definition of the tropics which is different from the astronomical one implied by the quote above.)

Yet more fun with phase lag

It's still warm here in Philadelphia. Highs are in the eighties this week, which is slightly above average for early September.

Walking home from campus yesterday, a little before 7 PM, the sun was low in the sky. I found myself digging for my sunglasses. And I remembered the circumstances under which I bought the sunglasses -- in late March and early April I was having the same problem. (I live west of campus, so I could actually have this problem twice a day, as I walked east in the mornings and west in the evenings.)

But wasn't it cold then, I thought?

But the position of the sun doesn't depend on the cold. The position on the horizon at which the sun rises and sets depends only on the distance of the current day from the summer solstice. Yesterday was the 75th day after the summer solstice (I'm using June 21 here); the 75th day before the solstice was April 7. The high temperature in Philadelphia that day was 41 degrees. (The average high then is 59.) (The declination, which is the celestial analogue of latitude, varies essentially sinuisoidally, but that doesn't mean the position of sunrise varies sinusoidally because there's another circle to deal with.)

In short: in terms of insolation right now is like early April, but in terms of temperature it's like early June. Maybe this sort of asymmetry explains why spring and fall feel different, since we're sensitive to both temperature and amount of light.

Of course, it's hard to know for sure, because you also have to take into account the derivatives of insolation and temperature; in the spring they're both increasing, and in the fall they're both decreasing. (A Jewish friend of mine once said that he doesn't think it's coincidence that Yom Kippur, when Jews are obliged to fast from sundown to sundown, falls near the autumn equinox; that makes the fast a few minutes less than it would be if it fell near the spring equinox.) And there's a big psychological factor, as well. Right now, an hour and ten minutes before my first class of the year, I feel exhilirated intellectually; in, say, May I'll feel exhausted. There's not much of a way to control for the cycles we humans have imposed on the year.

But what would it be like to live in a world where the season with the most sunlight was the cold season? I suspect we'll never know, because the only way to create that sort of world is to isolate people from the natural world for a very long time, and if you're willing to pay people to live in your cave for a while you can probably think of more worthwhile things to find out by using them as test subjects.

information-theoretic entropy in the weather

Right now, in Philadelphia, it's sixty-one degrees, with a light rain.

I have long maintained that this is "generic Philadelphia weather". By this I do not mean that it's always like this here. What I mean is that if I for some reason do not know what season it is, and I head outside and find it is sixty-one degrees with a light rain, this gives me very little information, because this sort of weather can happen any time of year. Another characteristic of such a day is that the high and low temperatures are very close together, say within ten degrees of each other; it's been between 60 and 64 since midnight and probably won't get out of the sixties (in either direction) all day.

Looking at the data from the last year, June 14 was pretty close to this, although it was just overcast, not rainy; January 6 might look that way on paper, except I remember it clearly and it was actually a freakishly warm and sunny day. I wore shorts. It was January. We had the New Year's Day parade that day. November 8, 13, and 14 fit as well; also October 11 and a few other October days; September 1. (I remember the weather on September 1 quite well, because I moved that day. The rain was light for most of the day and got heavier about an hour after my movers were done getting everything in.) I'm deliberately being vague about what constitute a day like this.

Not surprisingly, this sort of weather is most common in the spring and fall (mostly because I care about temperature) but it is possible in the winter or summer as well. And this gets me wondering -- in general, what is the information content of the weather? If it's 100 degrees and sunny, there might be a 2% chance that it's July 24; a 0.5% chance it's September 1; an 0.01% chance that it's November 1; and a one-in-a-million chance it's January 15. This sort of weather is very localized towards a certain time of year. One could imagine calculating the Shannon entropy corresponding to this distribution; it would be a lot smaller than the entropy you'd get from a similar distribution if you conditioned on sixty degrees and light rain.

Of course, in this formulation, the whole idea is kind of silly -- when am I not going to know what the date is? But looking at the information-theoretic entropy of weather seems like a potentially useful way to quantify how extreme the seasons in some place might be; it's possible but not normal to get the same weather in winter and summer in Philadelphia, say; routine in San Francisco; unheard of in Minneapolis. (I am picking these places without looking at actual statistics, so I might be wrong.) Why one would want to quantify that, though, I'm not sure.

climate phase lag trickery

Why does it get hot? Because of the sun.

But in temperate climates in the northern hemisphere, the most light from the sun comes around the summer solstice -- June 21 -- and yet the hottest part of the year is the second half of July. In certain places with a lot of maritime influence, the heat comes even later; the canonical northern-hemisphere example is probably San Francisco, where September is the warmest month. Similarly, the hottest time of the day isn't noon (even correcting for daylight savings time; solar noon is generally around 1 PM clock time under daylight savings time), but perhaps two to four hours later.

This phase lag comes about because the atmosphere holds some heat; the temperature as a function of the day of the year could be modeled by a differential equation, where the rate of change of temperature is some decreasing function of the temperature, plus a sinusoidal input. Thus the output should be similar to the input, but phase-lagged.

Furthermore, I am starting to believe (without evidence) that the temperature in my apartment building is lagged by maybe a couple weeks relative to the temperature outside during the day; I believe with somewhat more evidence that the hottest time of day in my apartment is generally around eight or nine in the evening. And so I find myself fantasizing about a world where that phase lag existed to such an extent that it's warm in my apartment at night and cold during the day. (Without using the air conditioner, that is.)

A bit more ridiculously, what if I could exploit that sort of seasonal phase lag to the point where it was cold in the summer and hot in the winter? But I'd need some sort of ridiculously large heat reservoir, something bigger than the ocean. Still, it's an interesting fantasy. The hottest time outside is a month after the time with the most solar radiation; the hottest time inside might be a few weeks after that; what if I had some sort of series of nested boxes each of which could shift the hottest time of year a few more weeks? If I nested enough of those boxes...

of course, it would be cheaper to just run the heat.

Seriously, though, there is some new construction that attempts to use the insulating properties of certain building materials in this way, although not so much to reverse the seasons as to smooth out seasonal variation. One particularly interesting one I heard about a few months ago depended on the thermal properties of a certain type of wood, which underwent a phase change around seventy degrees Fahrenheit; thermally it's very similar to keeping an ice cube in your house, except it's a magical seventy-degree ice cube instead of the thirty-two degrees of a normal ice cube.

80-degree lows in Philadelphia

I find myself interested in the weather, both because my main means of transportation is walking, and because it's a complex system that enough other people are interested in that they try to predict it and which is a source of large data sets. (Surprisingly, I have no weather-forecasting ability. I figure I could learn how if I wanted to, but I'm not that curious, because I'd just be discouraged when I realized the professionals are better than me at it.) It hit 97 here in Philadelphia yesterday; more crazy-sounding is that Wednesday morning's low was 80 degrees. This has only happened 39 40 times in recorded Philadelphia weather history, i. e. since 1876.

I find myself wondering how the number of such days is distributed, but it's hard to come to any sort of conclusion. My instinct, though, is something like this: the number of clusters of 80-degree lows (that is, consecutive days with them) in a given summer is probably Poisson-distributed.

Of course, counting these "clusters" is a silly thing to do, because it seems to imply that, say, the lows on August 16, 2002 and August 18, 2002 (both 80 degrees) are independent events, just because it didn't get up to eighty on August 17. (The low that day was 77.) If you look at the data, there have been 109 summers with no 80-degree lows, 20 summers with one cluster of them, one summer with two clusters, and two summers with three clusters; that last one makes me suspicious, but one of those is 2002 (July 30, August 16, August 18) and one is 1995 (July 15, 26, 29).

Second, how long is each individual cluster? There are 19 clusters of length 1, 7 of length 2, and 2 of length 3; I'm inclined to suggest a geometric distribution. Once there's an 80-degree low, the probability of having an 80-degree low on the next day is a constant, approximately 1/4. It seems reasonable that this probability would be less than 1/2 but not too much less; an 80-degree low in Philadelphia is very unusual, so the next day is expected to be a bit cooler. I've heard that the simplest weather-forecasting rule is that "tomorrow will be like today". I suspect that a slightly better rule is "tomorrow will be like today, but a little less so" -- that is, it will regress to the mean a bit. But to apply this rule one has to have some idea what the mean is. On a day with a high of 70 in Philadelphia in July I'd want this rule to predict the next day would be warmer; on the same day in November I'd want it to predict the next day would be colder.

So the number of 80-degree days in a given summer, I expect, is the sum of some number of geometrically distributed variables with p = 3/4, the number of such variables being given by a Poisson distribution with mean about .21 (there were 28 observed clusters in 132 years). Determining what this says in terms of actual probabilities is left as an exercise for the reader, mostly because I don't trust these numbers enough. It seems like a reasonable first stab at a model, though. It only has any chance of working for extreme temperatures, though; if I replaced 80 with 70 (the average low in Philly this time of year) then the model wouldn't work so well, because it depends on this clustering phenomenon. (The highest-recorded low temperature ever in Philadelphia is 82, so you can see that 80 is extreme.)

While I'm on the subject of weather, check out weatherbonk.com, which displays the weather, live, on a map, showing the temperature at various volunteer-run weather stations. I'm never sure how much to trust any of these stations, though, because you hear that sometimes they're in an asphalt-paved parking lot next to the place where the heat comes out for a central AC unit. I suspect it would be more interesting in a place like San Francisco where the microclimates are pronounced enough that you can see them even over that noise.

you can't predict the climate, either

Julie Rehmeyer, of MathTrek, writes about how predicting the weather -- or climate -- is basically impossible. The article begins:

Climate models may never produce predictions that agree with one another, even with dramatic improvements in their ability to imitate the physics and chemistry of the atmosphere and oceans. That's the conclusion of a report by James McWilliams, an applied mathematician and earth scientist at the University of California, Los Angeles. The mathematics of complex models guarantees that they will differ from one another, he argues. Therefore, says McWilliams, climate modelers need to change their approach to making predictions.

I had been under the impression that this was already known. It's known as "sensitive dependence on initial conditions"; if we know the weather to within a certain precision ε right now, then after one day we know the weather to within kε, after two days we know it to within k²ε, and so on, where k is some constant larger than 1 . More technically, the Lyapunov exponent of the weather is positive.

I've probably read several dozen versions of the story that is usually told about Edward Lorenz's toy model of the weather. (It would be interesting to see a web page that gives the various ways this particular story has been told, something like this page which gives over a hundred versions of the story of the young Gauss summing 1 + 2 + ... + 100.) The story, if you're not familiar with it, goes like this: Lorenz had a toy model of the weather in his computer, a system of differential equations. (I want to say it was a system of three equations, but I might be confusing it with the Lorenz attractor. Then again, they may actually be the same system.) It was the sixties, so computers were slow. Lorenz had his computer print out the position of the system in phase space at time 0, 1, 2, ...; one day he was looking at one of these printouts and saw a pattern he wanted to investigate. He fired up the computer again and typed in a line from the printout and told it to evolve the system from that point. The system evolved differently in the second run than the first; Lorenz thought it was a mistake, but eventually realized that the figures on the printout were rounded versions of the actual numbers in the computer, so he was introducing a small error by doing this, which was quickly amplified.

The MathTrek article is about climate, not weather, though, and it addresses this point (even anticipating my complaint about Lorenz!). Still, my instinct would have been -- even before reading this -- that the climate is a complex system just like the weather. The Lyapunov time -- the reciprocal of the Lyapunov exponent -- is much larger for climate than for weather. (I am quite confident saying that the average high temperature in Philadelphia next August will be about eighty-three degrees, and I am confident enough in this that when I take my air conditioner down when the summer ends, I will store it in my closet, instead of selling it. But I have a much worse idea what the weather will be on August 2, 2008.) Roughly speaking, climate is the average of weather, and averages change much less quickly than the things being averaged. I am confident that the Phillies will win about 29 of their remaining 55 games and just miss the playoffs, which is something I've gotten quite used to. (I'd be pleasantly surprised if they prove me wrong.) I have no idea whether they'll win tomorrow. (I would have said "I have no idea whether they'll win today," but they're up by four runs right now. It's only the fourth inning, though, so they have time to fall apart.)

The actual study is available here. Apparently the state of the art in climate and weather forecasting is to run a variety of different models on the same initial data; if they end up giving similar results then you can be fairly confident in the correctness of the forecast, while if they vary widely you know the forecast isn't so good. This is an experimental way of determining how sensitive the forecast is to the assumptions of the model. Although I'm not a meteorologist, it would be kind of interesting to see this on weather forecasts. I'm not sure how useful it would be for temperature; would I take a forecast high of "94, plus or minus 3" any differently than a forecast high of just "94"? Probably not. But for, say, snowfall estimates it could be incredibly useful. I don't care so much if they say there will be "two inches of snow". What I really want to know is if there's a chance of having some amount of snow that will seriously inconvenience me (say, over six inches). But I doubt you'll hear this on the TV news, because "we don't really know what the weather is going to be" kills the ratings -- even though "everyone knows" that the TV weather people don't really know what the weather is going to be. I've been known to actually use something like this "ensemble forecasting" myself -- I go to a bunch of different weather forecasts and see what they say. I'm not sure if it actually helps me, but it makes me feel better, usually because when I'm checking multiple weather web sites it means I'm procrastinating.

what's the heat index, anyway?

It's never really been clear to me what the "heat index" means.

Today in Philadelphia, at 4 pm, it's supposed to hit 95 degrees with 34% humidity; that corresponds, according to weather.com, to a heat index of 98. Tomorrow we expect 95 degrees and 43% relative humidity, for a heat index of 102.

What I've noticed is that the heat index here is almost always higher than the actual temperature. It'll get up to 95 on Tuesday, and someone will say "it `feels like' 102 degrees". No! It feels like 95. This is what 95 feels like around here. Sure, it might feel hotter than 95 degrees in the desert, but I've never been to the desert.

There are various formulas for the heat index. I suspect the second of the three given at Wikipedia is the "most accurate" in some sense, because it involves exponentiation of the reciprocal temperature, which is something which arises often in statistical mechanics; the other two are probably just polynomial approximations of it, which are easier to calculate, but ease of calculation is not so important. The various sources online seem a bit vague, though.

The following document from the National Weather Service states that the third approximation given at Wikipedia is, indeed, a polynomial approximation of the "true" heat index, which apparently depends on a fairly complicated biological model. In the end it probably just makes sense to resort to tables.

What surprises me is that none of the online formulas take into account wind. (The NWS document says that the wind is assumed to be a constant 5 knots in the model.) This seems silly to me. When I'm inside I can turn on a fan and be cooler than I would if the fan weren't on, because the fan dissipates the hot air around my body; shouldn't the same be true outside? (Michael Bluejay taught me this about fans.)

I think that the heat index is misleading, because it makes people think it's hotter than it actually is. I'd actually support replacing it with a number with an arbitrary scale, say zero to ten. If someone told me that tomorrow's going to be a nine on that scale, I'd know I don't want to go outside.

Also, weather.com has two forecasts that I look at regularly -- their ten-day forecast and their hour-by-hour forecast. The ten-day forecast says the high will be 97 today and tomorrow; the hourly forecast only goes up to 95. I suspect the reason is that at any given hour the expected temperature is indeed 95 degrees and the exact time that it will reach 97 is not known. Similarly, the forecast low Tuesday morning according to the ten-day forecast is 77, and the hourly only gets down to 78; the forecast low Wednesday morning is 76 but the hourly only gets down to 77.