The myth of the "sunrise observation" was created by people who know "a little bit" about celestial navigation. The 157°-337° sunline line of position (LOP) was derived by an observation of the sun when the sun's azimuth was 67° true since the LOP is at right angles to the azimuth to the celestial body. When the sun rose in the vicinity of Howland on July 2, 1937 its azimuth was 67° true. Those with "a little bit" of knowledge fastened on this fact to claim that Noonan took a sunrise observation and used only it in planning the approach to Howland. (They apparently believe that Noonan then opened the door and dropped his sextant into the sea.) But what these people didn't understand is that the sun's azimuth stayed at 67° until 1847 Z, an hour after sunrise at Howland! This means that Noonan would have computed the same 157°-337° LOP from any sight taken during this one hour period. He would have taken several additional sights as he approached the LOP then more after the interception to ensure staying on it. See Single LOP Landfall procedure about this procedure.
Some also claim that Noonan improperly applied the refraction correction and the dip correction to his sextant observation of the "sunrise celestial calculation" which caused up to a 70 nm error. First, Noonan couldn't take a"sunrise" observation because the correction table he had in his Nautical Almanac and in H.O 208 only have corrections for angles above the horizontal that are greater that six degrees. Refraction becomes much greater very fast below this altitude. The refraction correction for 6 degrees is 8 minutes of arc but at zero altitude it is 34 minutes of arc. The reason that the refraction correction tables do not contain information for altitudes below 6° is because navigators have been trained for many years to avoid low altitude observations because of unknowable variations in the refraction at low altitudes. The 1920 edition of the the U.S. Navy navigational manual advises against taking observations below 10° and more recent editions caution against observations below 5°. See: American Practical Navigator, H.O. 9 (1920) A sunrise observed from 10,000 feet is actually at 1 degree and 37 minutes below horizontal. The refraction correction for such a negative observation is 50 minutes of arc but Noonan would not have known this because a refraction correction table covering negative altitudes was not available until 1951. Noonan would have had to wait about 25 minutes after the sun rose to allow it to climb above the 6 degrees covered by his refraction correction table leaving about a half hour period for taking sights. This means that the observation that resulted in the 157°-337° LOP was taken a minimum of 25 minutes after sunrise and up to an hour later than sunrise. See: Refraction correction to sextant observations
Second, a "dip correction" is not used with a bubble sextant like the one used by Noonan, only when using a marine sextant which utilizes the visible sea horizon as the horizontal reference.
By taking additional sights with his octant (sextant) after the sun was above six degrees Noonan would determine if he was short of the 157°-337° LOP that ran through Howland and how many more miles they had to travel to get to it. By the same observations he would known if they had overshot it and by how many miles. After they had intercepted the LOP, additional sights would confirm that they were staying on it and, if not, which way they should correct their heading to get back on to it. At sea level sunrise at Howland was at 1745 Z. At SEA LEVEL sunrise would occur one minute later for every 15 NM that the SEA LEVEL observer is west of Howland. ( Actually it's one minute for every 15 minutes of longitude west of Howland which is only 14.998 NM at the latitude of Howland.) For an airplane at 10,000 feet the sun will rise about 10 minutes earlier than for a person on the ground directly below the plane. You have probably seen this yourself while flying on an
airliner. You could still see the sun at your altitude but looking down you could see that the ground below you was in the dark. (As an aside, we pilots have the god-like power to make the sun rise in the west! I have demonstrated this a number of times. We sit in the airplane on the ramp and wait for the sun to set. Then we take off and climb and the sun then miraculously comes back up, rising in the west as we gain altitude.)
You are measuring the altitude of the body in degrees above horizontal to determine if you are on the LOP or how far off to the side you are. If your measured altitude is the same as you have calculated it would be at Howland for the same time of day then you are over Howland or on the LOP that runs through Howland. For every minute of arc that your measured altitude differs from what it would be if you were on Howland you are one nautical mile displaced from the LOP. If your measured altitude is greater, then you are closer to the sun and you have overshot the LOP. If your measured altitude is lower then you have not yet reached the LOP and need to continue further the number of nautical miles equal to the difference in minutes of arc of the altitude.
The landfall is plotted slightly differently than what is normally used to find a fix. The destination is used for the assumed position (AP) and the LOP is drawn through the AP on the azimuth of the LOP. You might plot several different LOPs on different azimuths corresponding to different arrival times. Then the sight is taken and compared to the precomputed altitude curve which shows that altitude that would have been measured at the destination at the same time of day so that you know instantly the distance between your position and the LOP. You do NOT plot the LOP for where you actually are as that is not important because using the destination as the AP means that the difference between the precomputed altitude at the destination and the altitude measured with your sextant is the distance you are away perpendicularly from the LOP that runs through the destination.
As the day goes on and the azimuth of the sun changes the LOP plotted through the destination will rotate as though on a pivot stuck on the destination. So a one degree change in the azimuth of the sun to 066° at 1847 Z (which lasted until 1923 Z including the time of the "must be on you" message) results in a 156°-336° LOP and would result in only a one nautical mile (NM) error in you position only if you were at least 60 NM out from the destination on the LOP. ( The sin of one degree is approximately 1/60th. so a one degree change results in a one mile change for each 60 miles flown. It is actually one part in 57.3 but navigators use the approximation of one part in 60.) A one mile error falls well within the margin of error of the sextant shot.
The azimuth of the sun stays at 066° until 1923 Z; 065° until 1945 Z; 064° until 2003 Z and was 063°at 2012 Z when they reported being on the LOP. So even at 2012 Z it would only produce a 4 NM error at 60 NM from Howland on the LOP and only 2 NM if 30 NM from the island. You would expect that this period would have allowed enough time for them to find Howland.
Eventually the azimuth does change enough so that you have to plot the new azimuth and plan your heading to track that new course. At this point you recalculate your landfall again. By 2230 Z, a time when they could have been approaching Gardner, the azimuth would be 040° and so they would have had to compute a new landfall and plan on following a 130°-310° LOP since the 157°-337° LOP no longer existed. This is why they couldn't have just followed the 157°-337° LOP to Gardner.
Even if it had been possible to take an observation at the time of sunrise the resulting growing dead reckoning uncertainty would not have allowed enough accuracy to find the island. See this illustration: