We introduce nonlinear magnification fields as an abstract representation of nonlinear magnification, providing methods for converting transformation routines to magnification fields and vice-versa. This new representation provides ease of manipulation and power of expression. By removing the restrictions of explicit foci and allowing precise specification of magnification values, we can achieve magnification effects which were not previously possible. Of particular interest are techniques we introduce for expressing complex and subtle magnification effects through magnification brushing, and allowing intrinsic properties of the data being visualized to create data-driven magnifications.