What is the meaning of 'they estimated from above'?

Rashi: First they put oil (Shitah Mekubetzes - a Log), and oil remained in the morning. They successively used less until they found that half a Log suffices, even for the longest [winter] nights. They put this Shi'ur even for short [summer] nights, and discard what remains in the morning. Some say that they used thicker wicks in summer, so the half Log is consumed in the shorter night. (NOTE: It seems that they needed to experiment several times. After the first night, they could measure the remaining oil, and deduct that amount! Really, the Mitzvah is to burn until dawn; more burned until Hatavas ha'Neros. They could have made a precise calculation how much to deduct! Perhaps calculating is like poverty in the place of wealth. Or, even if they calculated the amount needed in Nisan (when Avodas ha'Mishkan began), they needed to experiment to know the amount needed for long nights of Teves, especially since oil burns slightly slower in cold weather. - PF)

Tosfos: The Yerushalmi says that they used thicker wicks in summer and thin wicks in winter. Initially, they used average wicks. Even though they could have measured with Chulin oil, 'there is no poverty in the place of wealth' applies.

Daf Al ha'Daf citing Ba'al Chidushei ha'Rim, Beis ha'Levi (on Torah, Derushim l'Chanukah), : This answers the Beis Yosef's question, why we say that the miracle of Chanukah was also on the first day. They used wicks an eighth as thick as normal [for winter], in order to fulfill the Mitzvah for all eight days with a tiny flame. Miraculously, the flame was big as with a normal wick, and still the oil lasted eight days.

Daf Al ha'Daf: The Me'iri writes that we inaugurate the Menorah via lighting it at night, and we put in enough oil to burn night and day. Our Gemara requires only enough to burn the entire night! Degel ha'Torah (p.234 1) answers that Klei Shares are Mekadesh only when they are full (Zevachim 88). We can say that when the Avodah inaugurates them, they must be full. This is why Rashi said that they initially used a Log, which is enough for a day and night. This answers how Nes Chanukah was also on the first day. They needed to inaugurate the Menorah with a full Log in each Ner; the flask should have sufficed only for half a Log in each.

NOTE: Half a Log suffices for the longest nights. This could be 12 hours In Eretz Yisrael or places of similar latitude, in late December. There are almost 14 hours from sunset to sunrise; if there are e.g. 40 minutes between sunset and Tzeis ha'Kochavim, and 72 minutes between dawn and sunrise, the night is about 12 hours. If so, a whole Log lasts 24 hours (night and day). Many say that we light the Menorah before sunset, but Shevet ha'Levi (11, Kuntres ha'Kodshim 2) says that according to one version in Pesachim, we may light it at night. (PF)

What is the meaning of 'they estimated from below'?

Rashi: First they used a quarter Log, and saw that it was not enough. They successively used more until they found that a half Log is the right Shi'ur.

Daf Al ha'Daf: Keren Orah asked, if so, they did not fulfill Mitzvas Hadlakah until they found the correct Shi'ur! Klei Chemdah (Emor p.210) asked, why did we not say that the two opinions argue about whether or not it is a Mitzvah to put in the Menorah enough oil to burn from night until morning? Regarding Nes Chanukah, Mizrachi (on SMaG) and the Beis Yosef (in one answer) say that it is only l'Chatchilah to light "me'Erev Ad Boker." Surely it is Me'akev! Rather, they mean that it is only l'Chatchilah to put in enough oil from the beginning to last until morning. B'Di'eved, one may start with a little, and if necessary, add more so it will last until morning. Mekor Chayim (670) said, lighting the Chanukiyah is the Mitzvah. If it did not have enough oil to last the required time, he was not Yotzei. Also lighting the Menorah is the Mitzvah. If it did not have enough oil at the time, and he added more later, why was he Yotzei? Mitzvas Ner Chanukah ends with lighting. If it extinguished, he need not relight it. Mitzvah ha'Menorah is the entire night. If it extinguished, he must relight it.