Exhibit 10.4

Clarifying Amendment

Restricted Stock Award

2007 Long-Term Incentive Plan

WHEREAS, Atwood Oceanics, Inc., a Texas corporation (the “Company”), has adopted and maintains the Atwood Oceanics, Inc. Amended and Restated 2007 Long-Term Incentive Plan, as amended (the “2007 LTIP”), under which awards may be granted to selected employees, officers and directors of the Company;

WHEREAS, pursuant to the 2007 LTIP, the Compensation Committee (the “Committee”) of the Board of Directors of the Company previously granted to Robert J. Saltiel (“Awardee”) that certain Restricted Stock Award Amended and Restated Agreement dated December 21, 2010 (the “Award”);

WHEREAS, the number of shares of the Company’s stock ultimately earned under the Award is subject to the achievement performance measures described in “Exhibit A” to the Award; and

WHEREAS, the Committee and Awardee wish to enter into this amendment to the Award to clarify certain terms of the performance measures as described in “Exhibit A” to the Award;

NOW, THEREFORE, the parties hereto agree as follows:

1. The term “Employment Date” as used in Exhibit A means December 14, 2009.

2. Unless the Award is vested sooner according to Section 2 or Section 7 of the Award, the Restriction Period expires and the Award shall vest on the date that is four (4) years after the date of grant of the Award, which expiration date is December 15, 2013.

3. The term “Cause” as used in Section 2 of the Award shall have the meaning assigned to such term in the Employment Agreement between the parties, dated December 8, 2009.

4. If Awardee’s employment with the Company is terminated prior to the expiration of the Restriction Period for reasons other than those specified in Section 2 of the Award (without Cause) or Section 3 of the Award (due to death, disability or Retirement), then the Award shall be forfeited in its entirety.

5. During the Restriction Period, if any dividends or other distributions with respect to the Company’s common stock are paid in other than shares of common stock, then (1) any such dividends or distributions in the form of cash shall be held in escrow by the Company and payment of such cash amounts shall be subject to the same restrictions as the shares included in the Award and (2) the number of shares subject to the Award shall be equitably adjusted, as determined by the Committee in its sole and reasonable discretion, in order to prevent enlargement or dilution of the Award as a result of any such dividend or distribution that is deemed extraordinary in nature, as determined by the Committee in its sole and reasonable discretion.

6. If, as a result of merger, acquisition or a similar corporate transaction, a member of the performance peers with respect to the stock price performance measure of Exhibit A ceases to be publicly traded within the first 365 days after the date of the Award, then the following alternative stock price performance payout ranking will apply:

If, as a result of merger, acquisition or a similar corporate transaction, a member of the performance peers with respect to the stock price performance measure of Exhibit A ceases to be publicly traded subsequent to the first 365 days after the date of the Award, then such member of the peer group shall remain in the peer group and the stock price performance of such member of the peer group shall be determined by assuming that its performance for the remainder of the performance period was equivalent to the arithmetic average percentage gain or loss (up or down) of the remaining members of the peer group (excluding the Company) over the remainder of the performance period.

7. The shares of Company common stock with respect to the Award that are subject to safety performance based on “TRIR” shall be determined on an annual basis as described in Exhibit A. However, any such shares annually determined to be awarded shall remain subject to forfeiture and shall not vest until the expiration of the Restriction Period. Similarly, any shares relating to safety performance that are not deemed to be awarded based on the annual determination remain subject to full vesting and award (i) in the event of a Change in Control prior to the end of the Restriction Period or (ii) subject to Section 3 of the Award in the event of death, disability or Retirement prior to the end of the Restriction Period.

8. In determining stock price performance, dividends are excluded.