A method and system is provided for detecting at risk structures due to mask overlay that occur during lithography processes. The method can be implemented in a computer infrastructure having computer executable code tangibly embodied on a computer readable storage medium having programming instructions. The programming instructions are operable to obtain a simulation of a metal layer and a via, and determine a probability that an arbitrary point (x, y) on the metal layer is covered by the via by calculating a statistical coverage area metric followed by mathematical approximations of a summing function.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A method implemented in a computer infrastructure having computer executable code tangibly embodied on a computer readable storage medium having programming instructions operable to: obtain a simulation of a metal layer and a via; and determine a probability that an arbitrary point (x, y) on the metal layer is covered by the via by calculating a statistical coverage area metric followed by a summing function; detecting at risk structures of a semiconductor device based on the determined probability, wherein determining the probability that the arbitrary point (x, y) on the metal layer is covered by the via comprises: determining that the metal layer is inside the via by calculating: P i   n  ( via   covers  [ x , y ] ) =  1 - P  ( O x ≤ - x &  O y ≤ - y ) =  1 - P  ( O x ≤ - x )  P  ( O y ≤ - y ) =  1 - 1 4  P  (  O x  ≥ x )  P  (  O y  ≥ y ) ( equation   1 ) wherein: Pin is representative of a probability that the via covers the metal layer, at the arbitrary point; determining that the metal layer is outside the via by calculating: P out  ( via   covers  [ x , y ] ) =  P  ( O x ≥ ( R v - x ) &  O y ≥ ( R v - y ) ) =  P  ( O x ≥ ( R v - x ) )  P  ( O y ≥ ( R v - y ) ) ( equation   2 ) wherein: Pout is representative of the probability that the via covers the metal layer, at the arbitrary point outside the nominal via shape; and Ox and Oy follows Gaussian distributions to calculate for the Pin and Pout; Rv represents the radius of the nominal via shape, wherein the arbitrary point (x, y) is defined by an x coordinate and a y coordinate in a cartesian coordinate system, wherein P is a probability, and Ox and Oy is an overlay in an x direction and a y direction, respectively. 2. The method of claim 1, further comprising computing a probabilistic area of contact made by a ring having radius “dr” at a distance “r” from a center of the via, wherein: dA P =  P  ( via   covers  [ r , r ] ) · M  ( r ) · A  ( r ) =  P  ( via   covers  [ r , r ] ) · M  ( r ) · 2  π   r   dr ( equation   3 ) wherein: M(r) is a fraction of the ring on which metal exists ε (0,1), where “0” represents no metal on the ring and “1” represents metal existing on an entire portion of the ring; providing equi-probability regions generated by sizing the via by a maximum overlay amount, wherein the equi-probability regions are represented by Co, C1 . . . Cn; and calculating a total probabilistic area of contact made by the via by integrating over r∞ by: A p = ∫ r = 0 ∞   A p = ∫ r = 0 ∞  P  ( via   covers  [ r , r ] )  Mr   2  π   r   r ; ( equation   4 ) and wherein the mathematical approximations of the summing function is provided by discretizing equation (4), and wherein [r, r] is a point, Mr2πrdr is an area of the ring covered by the metal, and ε is a member of a set of (0, 1). 3. The method of claim 2, wherein the discretizing equation (4) comprises providing an approximation by finding P and [r,r], and calculating for Mr2πrdr. 4. The method of claim 2, wherein the discretizing equation (4) comprises: representing shapes as always covered by the via; representing shapes as often covered by the via; representing shapes as sometimes covered by the via; for shapes always covered and often covered by the via calculate Pin using Equation (1); for shapes sometimes covered by the via calculate Pout using Equation (2); and for each shape, multiplying an area of the metal covering by its appropriate Pin or Pout, and adding each of the multiplied results. 5. The method of claim 1, wherein the summing function includes mathematical approximations, and the mathematical approximations include engineering approximations to detect the at risk structures of overlay error. 6. The method of claim 1, wherein the statistical coverage area metric is computed using a Gaussian distribution for an overlay of the via. 7. The method of claim 6, wherein the statistical coverage area metric is based on equi-probability contours of the via. 8. The method of claim 1, wherein the via can be any shape of choice. 9. The method of claim 8, wherein the shape of the via is a circular shape. 10. The method of claim 1, wherein the programming instructions of claim 1 are integrated into a rule checking system. 11. The method of claim 1, wherein the detecting at risk structures of the semiconductor device occurs during lithography based on the determined probability. 12. A system implemented in hardware, comprising an overlay aware optical rule checking module configured to determine a probability that an arbitrary point (x, y) on a metal layer is covered by a via by calculating a statistical coverage area followed by a summing factor and detecting at risk structures of a semiconductor device based on the determined probability, wherein determining the probability that the arbitrary point (x, y) on the metal layer is covered by the via comprises: determining that the metal layer is inside the via (Pin); determining that the metal layer is outside the via (Pout); providing equi-probability regions generated by sizing the via by a maximum overlay amount, wherein the equi-probability regions are represented by Co . . . Cn; and calculating a total probabilistic area of contact made by the via by an approximate probabilistic coverage area of Ap found by Ap=PCoACo+ . . . PCnACn, wherein the metal area covered by Co is represented as ACo and the metal area covered by Cn is ACn, wherein PCo, . . . PCn is calculated as either Pin or Pout by: P i   n  ( via   covers  [ x , y ] ) =  1 - P  ( O x ≤ - x &  O y ≤ - y ) =  1 - P  ( O x ≤ - x )  P  ( O y ≤ - y ) =  1 - 1 4  P  (  O x  ≥ x )  P  (  O y  ≥ y ) ( equation   1 ) wherein: Pin is representative of a probability that the via covers the metal layer, at an arbitrary point; and P out  ( via   covers  [ x , y ] ) =  P  ( O x ≥ ( R v - x ) &  O y ≥ ( R v - y ) ) =  P  ( O x ≥ ( R v - x ) )  P  ( O y ≥ ( R v - y ) ) ( equation   2 ) wherein: Pout is representative of the probability that the via does not cover the metal layer, at the arbitrary point, and multiplying each respective Pout and Pin by a covered area, and summing results, wherein the arbitrary point (x, y) is defined by an x coordinate and a y coordinate in a cartesian coordinate system, and wherein P is a probability, and Ox and Oy is an overlay in an x direction and a y direction, respectively. 13. The system of claim 12, further comprising computing a probabilistic area of contact made by a ring having radius “dr” at a distance “r” from a center of the via, wherein: dA P =  P  ( via   covers  [ r , r ] ) · M  ( r ) · A  ( r ) =  P  ( via   covers  [ r , r ] ) · M  ( r ) · 2  π   r   dr ( equation   3 ) wherein: M(r) is a fraction of the ring on which metal exists ε (0,1), where “0” represents no metal on the ring and “1” represents metal existing on an entire portion of the ring; providing equi-probability regions generated by sizing the via by a maximum overlay amount, wherein the equi-probability regions are represented by Co, C1 . . . Cn; and calculating a total probabilistic area of contact made by the via by integrating over r∞ by: A p = ∫ r = 0 ∞   A p = ∫ r = 0 ∞  P  ( via   covers  [ r , r ] )  Mr   2  π   r   r ; ( equation   4 ) and wherein the mathematical approximations of the summing function is provided by discretizing equation (4), and wherein [r, r] is a point, Mr2πrdr is an area of the ring covered by the metal, and ε is a member of a set of (0, 1). 14. The system of claim 12, wherein the summing factor includes a mathematical approximation. 15. The system of claim 14, wherein the mathematical approximation is a Gaussian distribution. 16. The system of claim 12, wherein the summing factor is provided by an exact distribution. 17. The system of claim 12, wherein the detecting at risk structures of the semiconductor device occurs during lithography based on the determined probability.