Let (R,m) be a local ring andIan ideal ofR. In this work we find conditions onIthat allow us to describe simple relations among depthR(It), depth grI(R), depth S(I) and depth S(I/I2). These relations are useful also from a practical point, of view since it is usually difficult to evaluate depth grI(R) and depth S(I/I2) even with the help of a computer. Furthermore we study the class of ideals that satisfy one of the required conditions and we show that ideals generated by quadratic sequences are in this class