<PAGE>   1
                                  UNITED STATES
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

                                    FORM 10-K

           [ X ] ANNUAL REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE
                         SECURITIES EXCHANGE ACT OF 1934

    FOR THE FISCAL YEAR ENDED SEPTEMBER 30, 1998   COMMISSION FILE NO. 0-10144

                           DAWSON GEOPHYSICAL COMPANY

       INCORPORATED IN THE STATE OF TEXAS                 75-0970548
                                                       (I.R.S. EMPLOYER
                                                      IDENTIFICATION NO.)

                 508 WEST WALL, SUITE 800, MIDLAND, TEXAS 79701
                          (PRINCIPAL EXECUTIVE OFFICE)
                         TELEPHONE NUMBER: 915-684-3000

                           --------------------------

        SECURITIES REGISTERED PURSUANT TO SECTION 12(b) OF THE ACT: NONE

           SECURITIES REGISTERED PURSUANT TO SECTION 12(g) OF THE ACT:

      TITLE OF EACH CLASS             NAME OF EACH EXCHANGE ON WHICH REGISTERED
      -------------------             -----------------------------------------
 COMMON STOCK, $.33 1/3 PAR VALUE                     NONE

         Indicate by check mark whether the Registrant (1) has filed all reports
required to be filed by Section 13 or 15(d) of the Securities Exchange Act of
1934 during the 12 preceding months, and (2) has been subject to such filing
requirements for the past 90 days. Yes X   No
                                      ----   ----
         Indicate by check mark if disclosure of delinquent filers pursuant to
Item 405 of Regulation S-K ( 229.405 of this chapter) is not contained herein,
and will not be contained, to the best of registrant's knowledge, in definitive
proxy or information statements incorporated by reference in Party III of this
Form 10-K or any amendment to this Form 10-K. [  ]

         The aggregate market value of the Common Stock of the Registrant based
upon the mean between the closing high and low price of the Common Stock as of
November 13, 1998 (as reported by NASDAQ), held by non-affiliates was
approximately $53,632,635 (See Item 12). On that date, there were 5,361,000
shares of Dawson Geophysical Company Common stock, $.33 1/3 par value,
outstanding.

                       DOCUMENTS INCORPORATED BY REFERENCE

         The information required by Items 1, 5, 6, 7 and 8 of Parts I and II
hereof is incorporated by reference to the Registrant's 1998 Annual Report filed
or to be filed with the Commission no later than 120 days after the end of the
fiscal year covered by this Form 10-K.

         The information required by Items 4, 10, 11 and 12 of Parts I and III
hereof is incorporated by reference to the Registrant's definitive proxy
statement filed or to be filed with the Commission no later than 120 days after
the end of the fiscal year covered by this Form 10-K.


<PAGE>   2




                                     PART I

ITEM 1.   BUSINESS

          There are no patents, trademarks, franchises or concessions held by
the Registrant. Software licenses held by the Registrant are considered ordinary
and replaceable. Although the Registrant has individual customers that comprise
more than 10% of its total annual revenues, the Registrant does not consider the
loss of any individual customer to have a material adverse effect on the
Registrant due to the demand for the Registrant's services and for the services
of the industry in which the Registrant competes. Competitors of the Registrant
consist primarily of subsidiary companies of large corporations. Services
provided by competitor companies other than provided by the Registrant may
include marine geophysics, speculative acquisition of seismic data, a library of
seismic data, or a combination of these services. The Registrant considers price
and quality of service to be its principal methods of competition. Indicative of
its level of commitment to the proprietary data of its customers, the Registrant
does not maintain a library of seismic data or participate in speculative
seismic data acquisition. Although the business of the Registrant is not
considered seasonal, it does depend on favorable weather.

          At September 30, 1998, the Company had 377 full-time employees. None
of the Company's employees are subject to a collective bargaining agreement. The
Company considers its relations with its employees to be good.

          Additional information required by this Item 1 is hereby incorporated
by reference to the Registrant's 1998 Annual Report (inside front cover, page 2
and page 20) filed or to be filed by the Registrant with the Securities and
Exchange Commission pursuant to Regulation 14A of the Securities and Exchange
Act of 1934 within 120 days after the end of the fiscal year covered by this
Form 10-K. (Exhibit 13 hereto.)

ITEM 2.   PROPERTIES

          The principal facilities of the Registrant are summarized in the table
below.

<TABLE>
<CAPTION>


                        Fee or                                         Building Area
Location                Leased       Purpose                            Square Feet
--------                ------       -------                           -------------
<S>                     <C>          <C>                               <C>
Midland, TX             Leased       Executive offices and                 18,400
                                     data processing

Midland, TX             Fee          Field office                          53,000
                                     Equipment fabrication
                                     Maintenance and repairs
</TABLE>


         The Registrant owns additional undeveloped real property used for
employee parking consisting of approximately 21,000 square feet in Midland,
Texas, in the vicinity of the headquarters office building.





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         The Registrant operates only in one industry segment and only in the
United States.

ITEM 3.  LEGAL PROCEEDINGS

         The Company is a defendant in two lawsuits pending in the 112th and
83rd District Courts of Pecos County, Texas (respectively, Cause No. 8812,
Ernestine Bernal, et al. vs. Javier Antonio Orona, et al.; and Cause No.
P5565-83-CV, Carla Jaquez, et al. vs. Javier Antonio Orona, et al.) relating to
a July 1995 accident involving a van owned by the Company which was used to
transport employees to various job sites and a non-Company owned vehicle. The
accident resulted in the deaths of four Company employees who were passengers in
such van. The Company is one of several named defendants in such suits. Other
named defendants include the estate of the deceased driver of such van, who was
an employee of the Company, the driver of such non-Company owned vehicle, who
was then an employee of the Company, the owner of such vehicle, and Ford Motor
Company, the manufacturer of the Company van involved in such accident. In
general, the claims against the Company include allegations of negligence, gross
negligence and/or intentional tort as a result of, among other things, the
Company's alleged failure to provide safe transportation for its employees and
to properly select, train and supervise the deceased driver of such van. The
plaintiffs in such suits are seeking actual damages from the defendants of $15.5
million, additional unspecified actual damages, pre-judgment and post-judgment
interest and costs of suit as well as exemplary and punitive damages in an
amount not to exceed four times the amount of actual damages. The Company
believes that it has meritorious defenses to the claims asserted against it in
such suits and it intends to continue to vigorously defend itself against such
claims. In addition, the Company believes that it has approximately $11 million
of liability insurance coverage to provide against an unfavorable outcome. Such
suits are currently in the discovery stage and the Company currently has pending
before the court a motion for summary judgment in Cause No. 8812 requesting that
the Company be dismissed from such suit based upon various legal theories. Such
motion has not yet been heard by the court. A trial date of February 22, 1999
has been set for these cases. A motion to consolidate such suits into a single
proceeding is currently pending before the courts. Due to the uncertainties
inherent in litigation, no assurance can be given as to the ultimate outcome of
such suits or the adequacy or availability of the Company's liability insurance
to cover the damages, if any, which may be assessed against the Company in such
suits. A judgment awarding plaintiffs an amount significantly exceeding the
Company's available insurance coverage could have a material adverse effect on
the Company's financial condition, results of operations and liquidity.

         In addition to the foregoing, from time to time the Company is a party
to various legal proceedings arising in the ordinary course of business.
Although the Company cannot predict the outcomes of any such legal proceedings,
the Company's management believes that the resolution of pending legal actions
will not have a material adverse effect on the Company's financial condition,
results of operations or liquidity.


ITEM 4.  SUBMISSION OF MATTERS TO A VOTE OF SECURITY HOLDERS

         No matter has been submitted during the fourth quarter of the 1998
fiscal year to a vote of security holders, through the solicitation of proxies
or otherwise. However, please refer to the Registrant's Proxy Statement dated
November 13, 1998, filed or to be filed with the Commission



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no later than 120 days after the end of the fiscal year covered by this Form
10-K, notifying as to the election of Directors, approval of the proposed Dawson
Geophysical Company 2000 Incentive Stock Plan and selection of KPMG Peat Marwick
LLP as independent certified public accountants of the Company (requiring an
affirmative vote of a majority of shares present or represented by proxy), at
the Annual Meeting to be held on January 12, 1999.

                                     PART II

ITEM 5.  MARKET FOR REGISTRANT'S COMMON EQUITY AND RELATED STOCKHOLDER MATTERS

         The information required by this Item 5 is hereby incorporated by
reference to the Registrant's 1998 Annual Report (inside back cover, "Common
Stock Information") referred to in Item 1 above.

ITEM 6.  SELECTED FINANCIAL DATA

         The information required by this Item 6 is hereby incorporated by
reference to the Registrant's 1998 Annual Report (page 1 "Financial Highlights")
referred to above in Item 1.

ITEM 7.  MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND RESULTS
         OF OPERATIONS

         The information required by this Item 7 is hereby incorporated by
reference to the Registrant's 1998 Annual Report (pages 12 to 14) referred to in
Item 1.

ITEM 8.  FINANCIAL STATEMENTS AND SUPPLEMENTARY DATA

         The report of independent public accountants appearing on page 15 and
the financial statements appearing on pages 16 through 24 of Registrant's 1998
Annual Report for the year ended September 30, 1998, referred to above in Item
1, are incorporated herein by reference.

ITEM 9.  CHANGES IN AND DISAGREEMENTS ON ACCOUNTING AND FINANCIAL DISCLOSURES

         None




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<PAGE>   5


                                    PART III

ITEM 10. DIRECTORS AND EXECUTIVE OFFICERS OF THE REGISTRANT

         The information required by this Item 10 with respect to Directors and
Executive Officers is hereby incorporated by reference to the Registrant's Proxy
Statement dated November 13, 1998 (page 2), filed or to be filed by the
Registrant with the Securities and Exchange Commission pursuant to Regulation
14A of the Securities and Exchange Act of 1934 within 120 days after the end of
the fiscal year covered by this Form 10-K.

ITEM 11. EXECUTIVE COMPENSATION

         The information required by this Item 11 is hereby incorporated by
reference to the Registrant's Proxy Statement (page 3) referred to above in Item
10.

ITEM 12. SECURITY OWNERSHIP OF CERTAIN BENEFICIAL OWNERS AND MANAGEMENT

         The information required by this Item 12 with respect to security
ownership of certain beneficial owners is hereby incorporated by reference to
the Registrant's Proxy Statement (page 6, "SECURITY OWNERSHIP OF CERTAIN
BENEFICIAL OWNERS AND MANAGEMENT") referred to above in Item 10.


ITEM 13. CERTAIN RELATIONSHIPS AND RELATED TRANSACTIONS

         None



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                                    PART IV

ITEM 14. EXHIBITS, FINANCIAL STATEMENT SCHEDULES AND REPORTS ON FORM 8-K

         (a)  1.  Financial Statements

         The following financial statements of the Registrant, included in pages
16 through 24 of the Registrant's 1998 Annual Report to Shareholders for the
year ended September 30, 1998, and the Independent Auditors' Report on page 15
of such report, are incorporated herein by reference:

                                   DESCRIPTION

                   Balance Sheets, September 30, 1998 and 1997

                            Statements of Operations
                               For the Years Ended
                        September 30, 1998, 1997 and 1996

                            Statements of Cash Flows
                               For the Years Ended
                        September 30, 1998, 1997 and 1996

                       Statements of Stockholders' Equity
                               For the Years Ended
                        September 30, 1998, 1997 and 1996

                          Notes to Financial Statements

                          Independent Auditors' Report

         (a)  2.  All schedules are omitted because they are not applicable, not
required or because the required information is included in the financial
statements or notes thereof.

         (a)  3.  Exhibits

         The exhibits and financial statement schedules filed as a part of this
report are listed below according to the number assigned to it in the exhibit
table of Item 601 of Regulation S-K:

         (3)  Restated Articles of Incorporation and Bylaws.

         (4)  Instruments defining the rights of security holders, including
              indentures.

         (9)  Voting Trust Agreement -- None; consequently, omitted.





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<PAGE>   7



         (10) Material Contracts.

         (11) Statement re: computation of per share earnings -- Not Applicable.

         (12) Statement re: Computation of ratios -- Not Applicable.

         (13) 1998 Annual Report.

         (18) Letter re: change in accounting principles -- Not Applicable.

         (19) Previously unfiled documents -- No documents have been executed or
              in effect during the reporting period which should have been
              filed; consequently, this exhibit has been omitted.

         (22) Subsidiaries of the Registrant -- There are no subsidiaries of the
              Registrant; consequently, this exhibit has been omitted.

         (23) Published report regarding matters submitted to vote of security
              holders -- None; consequently, omitted.

         (24) Consent of experts and counsel -- Not applicable.

         (25) Power of Attorney -- There are no signatures contained within this
              report pursuant to a power of attorney; consequently, this exhibit
              has been omitted.

         (b)  Reports on Form 8-K

              The Registrant has not filed any reports on Form 8-K during the
              last quarter of the year ended September 30, 1998.

         (28) Additional Exhibits -- None.

         (29) Information from reports furnished to state insurance regulatory
              authorities -- None.





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<PAGE>   8



                                  EXHIBIT INDEX


<TABLE>
<CAPTION>

NUMBER            EXHIBIT                                                               PAGE
------            -------                                                               ----
<S>               <C>                                                                   <C>
(1)               *
(2)               *
(3)               Articles of Incorporation and Bylaws                                  **
(4)               Instruments defining the rights of security                           **
                  holders, including indentures
(5)               *
(6)               *
(7)               *
(8)               *
(9)               Voting Trust Agreement                                                Omit
(10)              Material Contracts                                                    **
(11)              Statement re: computation of per share earnings                       Omit
(12)              Statement re: computation of ratios                                   Omit
(13)              1998 Annual Report to Stockholders                                    E-1
(14)              *
(15)              *
(16)              *
(17)              *
(18)              Letter re: change in accounting principles                            Omit
(19)              Previously unfiled documents                                          Omit
(20)              *
(21)              *
(22)              Subsidiaries of the Registrant                                        Omit
(23)              Published report regarding matters submitted                          Omit
                  to vote of security holders
(24)              Consent of experts
(25)              Power of Attorney                                                     Omit
(26)              *
(27)              *
(28)              Additional Exhibits                                                   Omit
(29)              Information from reports furnished to state                           Omit
                  insurance regulatory authorities
</TABLE>

 *This exhibit is not required to be filed in accordance with Item 601 of
Regulation S-K

**Incorporated by reference to Registrant's Form 10-Q, dated June 30, 1997
(Commission File No. 0-10144) and Registrant's Form S-1, dated October 21, 1997
(Registrant No. 333-38393).



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<PAGE>   9



                                   SIGNATURES

Pursuant to the requirements of Section 13 or 15(d) of the Securities Exchange
Act of 1934, the Registrant has duly caused this report to be signed on its
behalf by the undersigned, thereunto duly authorized, in the City of Midland,
and the State of Texas, on the 13th day of November, 1998.

                                        DAWSON GEOPHYSICAL COMPANY

                                        By: /s/ L. Decker Dawson
                                            ------------------------------------
                                            L. Decker Dawson, President

Pursuant to the requirements of the Securities Exchange Act of 1934, this report
has been signed below by the following persons in the capacities and on the date
indicated.

<TABLE>
<CAPTION>



SIGNATURE                                           TITLE                                DATE
---------                                           -----                                ----
<S>                                                 <C>                                  <C>
/s/ L. Decker Dawson                                President, Principal                 11-13-98
------------------------------------                Executive and Director
L. Decker Dawson                                    

/s/ Floyd B. Graham                                 Executive Vice President             11-13-98
------------------------------------                and Director
Floyd B. Graham                                     

/s/ Howell W. Pardue                                Executive Vice President             11-13-98
------------------------------------                and Director
Howell W. Pardue                                    

/s/ Calvin J. Clements                              Director                             11-13-98
------------------------------------
Calvin J. Clements

/s/ Matthew P. Murphy                               Director                             11-13-98
------------------------------------
Matthew P. Murphy

/s/ Tim C. Thompson                                 Director                             11-13-98
------------------------------------
Tim C. Thompson

/s/ Paula W. Henry                                  Secretary                            11-13-98
------------------------------------
Paula W. Henry

/s/ Christina W. Hagan                              Vice President and                   11-13-98
------------------------------------                Chief Financial Officer
Christina W. Hagan                                  

</TABLE>


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                               INDEX TO EXHIBITS

<TABLE>
<CAPTION>

EXHIBIT
NUMBER                   DESCRIPTION
-------                  -----------
<S>                      <C>

  13                     Excerpts of the Annual Report

  27                     Financial Data Schedule

  27.1                   Restated Financial Data Schedule
</TABLE>