I missed seeing who sent the email to the satelite monitoring girl. Who told her to check out the coordinates on the two different days?
Vincent Kapoor (originally in the book as Venkat Kapoor), the head of Mars operations at NASA, sent the email to NASA Satellite Communications Engineer Mindy Park, asking her to pull the satellite images of the HAB's GPS coordinates. It is signed by him.
It does not ask her to compare the old and new images, but as a Sat Comm Engineer or Image Analyst, comparing a new satellite image to an older one for reference on any changes between the two would be standard procedure. Due to how widely reported the original picture would be, she could also have noticed that something was off on the new picture. This would explain her puzzled expression prior to flipping to the Sol 18 image.
This is covered in the source novel. Despite being initially blocked from taking any picture of the Ares 3 site, Venkat Kapoor (renamed Vincent Kapoor for the film) ordered some telemetry pictures. He was hoping to assess what assets would be available for future missions.
A flicker on her screen announced another set of images were ready for dispatch. She checked the name on the work order. Venkat Kapoor.
Posting the data directly to internal servers, she composed an email to Dr. Kapoor. As she entered the latitude and longitude of the image, she recognized the numbers.
“31.2°N, 28.5°W… Acidalia Planitia… Ares 3?”
The implication is that he managed to persuade his superior to authorise the pictures on the strength of using sympathy for Watney's death to leverage a better budget for a future Mars mission.
“Ok, consider this,” he said. “Sympathy for Watney’s family is really high. Ares 6 could bring the body back. We don’t say that’s the purpose of the mission, but we make it clear that would be part of it. If we framed it that way, we’d get more support in Congress. But not if we wait a year. In a year, people won’t care any more.”
Teddy rubbed his chin. “Hmm…”.
The same scene appears in the film script, but 'Vincent' is a little more forceful now:
VINCENT: Right now, the world is on our side. Sympathy for the Watney family. ARES 6 can bring his body home. Now, we don't say that's the purpose of the mission, but we make it clear that would be a part of it. If we frame it that way, more support from congress, but not if we wait a year. We wait a year, nobody gives a shit.