A 52-year-old woman presents with a 3-day history of acute onset swelling, pain, and warmth in her left lower limb. She returned from a 12-hour flight 5 days ago and has been immobilized due to a recent knee arthroscopy 2 weeks prior. On examination, the left calf is swollen with a positive Homan's sign and pitting edema. Duplex ultrasonography confirms thrombosis in the left popliteal vein. She has no contraindications to anticoagulation. What is the most appropriate immediate management?