The propagation of an rf asymmetric surface barrier discharge in atmospheric pressure air has been investigated. Measurements of the pulse-modulated 13.56 MHz voltage and current together with ICCD images of the plasma were recorded to study the visible plasma structure with respect to the rf pulses, time within the pulses and the rf waveforms. When exposing images over full rf pulses, which comprise over 150 oscillations of the applied voltage, clearly defined filamentary structures are observed indicating a strong memory effect. The discharge intensity decreases exponentially with distance from the electrode edge, and the average propagation length increases linearly with the applied voltage. Similar to some lower frequency asymmetric surface dielectric barrier discharges, two distinct breakdown events occur during one period of the voltage waveform. The number of filaments is found to be the same for both breakdown events, and collective effects are observed in both discharges.