Trivia: In episode 2x22 "Mother's Day" the scene at the start was filmed at Brooklyn's American Veterans Memorial Pier, at Bay Ridge Ave aka 69th St, in Bay Ridge (off Belt Pkwy). The location and design of this pier is distinctive and also sentimental (particularly to older Brooklynites).
Trivia: In episode 2x1 "Mercy" the last scene was filmed at Brooklyn's Valentino Pier (it's also known as Coffey Street Pier) in Red Hook. It had been renamed for a Brooklyn firefighter who heroically died in the line of duty, and the area's full name is Louis Valentino Jr Park and Pier. It's an appreciated green area in the middle of a lot of buildings. Bit of trivia to add, Valentino Pier is featured in a short video with Rosie Perez.
Answer: It's a common cop show plot device. If the police on the show always acted in the most logical way (i.e. not alerting a suspect to their presence) the show wouldn't have the opportunity for an exciting foot chase.