<PAGE>   1
                                    FORM 10-Q

                       SECURITIES AND EXCHANGE COMMISSION
                              WASHINGTON D.C. 20549

                                   (Mark One)

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

                  For the quarterly period ended June 30, 1998

                                       OR

     ( ) TRANSITION REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES
                              EXCHANGE ACT OF 1934

        For the transition period from                 to
                                       ---------------    ---------------

     For Quarter Ended June 30, 1998         Commission File number 2-71058


                           DAWSON GEOPHYSICAL COMPANY
                          ----------------------------
             (Exact name of Registrant as specified in its Charter)

              TEXAS                                      75-0970548
-------------------------------              ---------------------------------
(State or other jurisdiction of              (IRS Employer Identification No.)
 incorporation or organization)

508 West Wall, Suite 800, Midland, Texas                  79701
----------------------------------------                ----------
(Address of principal executive offices)                (Zip Code)

        (Registrant's telephone number, including area code) 915/684-3000

                                      NONE
--------------------------------------------------------------------------------
 (Former Name, Former Address & Former Fiscal Year if changed since last report)

  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 preceding 12 months (or for such shorter period that the
registrant was required to file such reports), and (2) has been subject to such
filing requirements for the past 90 days. Yes X . No   .
                                             ---    ---

  Indicate the number of shares outstanding of each of the issuer's classes of
common stock, as of the latest practicable date.

                  CLASS                      Outstanding at June 30, 1998
     --------------------------------        ----------------------------
     Common Stock, $.33 1/3 par value              5,356,000 shares


                                       -1-
<PAGE>   2

                           DAWSON GEOPHYSICAL COMPANY

                                      INDEX

<TABLE>
<CAPTION>
                                                                            Page No.
                                                                            --------
<S>                                                                         <C>
Part I.   Financial Information:

          Statements of Operations --
                Three Months and Nine Months
                ended June 30, 1998 and 1997                                   3

          Balance Sheets --
                June 30, 1998 and September 30,
                1997                                                           4

          Statements of Cash Flows --
                Nine Months Ended June 30, 1998
                and 1997                                                       5

          Notes to Financial Statements                                        6

          Management's Discussion and Analysis of
                Financial Condition and Results of
                Operations                                                     8


Part II.  Other Information
</TABLE>



                                       -2-
<PAGE>   3

                          PART I. FINANCIAL INFORMATION

                           DAWSON GEOPHYSICAL COMPANY

                            STATEMENTS OF OPERATIONS
                                   (UNAUDITED)

<TABLE>
<CAPTION>
                                             Three Months Ended                 Nine Months Ended
                                                   June 30                            June 30
                                       ------------------------------      ------------------------------
                                           1998              1997              1998              1997
                                       ------------      ------------      ------------      ------------
<S>                                    <C>               <C>               <C>               <C>         
Operating revenues                     $ 18,647,000      $ 12,520,000      $ 45,991,000      $ 34,304,000

Operating costs:
    Operating expenses                   11,193,000         8,127,000        30,094,000        22,840,000
    General and administrative              478,000           386,000         1,443,000         1,052,000
    Depreciation                          2,461,000         1,685,000         6,816,000         5,456,000
                                       ------------      ------------      ------------      ------------

                                         14,132,000        10,198,000        38,353,000        29,348,000
                                       ------------      ------------      ------------      ------------

Income from operations                    4,515,000         2,322,000         7,638,000         4,956,000

Other income (expense):
      Interest income                       209,000            86,000           579,000           167,000
      Interest expense                           --          (111,000)         (125,000)         (341,000)
      Gain on disposal of assets             19,000             3,000           167,000           196,000
      Other income                            5,000             6,000            28,000            15,000
                                       ------------      ------------      ------------      ------------


Income before income tax                  4,748,000         2,306,000         8,287,000         4,993,000

Income tax expense:
      Current                            (1,386,000)         (622,000)       (2,333,000)       (1,222,000)
      Deferred                             (275,000)         (183,000)         (567,000)         (523,000)
                                       ------------      ------------      ------------      ------------
                                         (1,661,000)         (805,000)       (2,900,000)       (1,745,000)
                                       ------------      ------------      ------------      ------------

Net income                             $  3,087,000      $  1,501,000      $  5,387,000      $  3,248,000
                                       ============      ============      ============      ============

Net income per common share            $        .58      $        .36      $       1.05      $        .78
                                       ============      ============      ============      ============

Net income per common share--
      assuming dilution                $        .57      $        .36      $       1.04      $        .77
                                       ============      ============      ============      ============
Weighted average equivalent shares
    outstanding                           5,353,582         4,199,247         5,136,652         4,177,432
                                       ============      ============      ============      ============

Weighted average equivalent shares
      outstanding--assuming
      dilution                            5,379,341         4,214,035         5,165,082         4,201,552
                                       ============      ============      ============      ============
</TABLE>

See accompanying notes to the financial statements.



                                       -3-
<PAGE>   4

                           DAWSON GEOPHYSICAL COMPANY

                                 BALANCE SHEETS

<TABLE>
<CAPTION>
                                                      June 30, 1998        September 30, 1997
                                                    -----------------      ------------------
                                                                   (UNAUDITED)
<S>                                                 <C>                    <C>
ASSETS
Current assets:
   Cash and cash equivalents                        $       2,473,000      $       4,774,000
   Marketable securities                                    6,527,000              3,968,000
   Accounts receivable                                     13,104,000              8,724,000
   Prepaid expenses                                           549,000                288,000
                                                    -----------------      -----------------

           Total current assets                            22,653,000             17,754,000
                                                    -----------------      -----------------


Property, plant and equipment                              80,036,000             63,267,000
   Less accumulated depreciation                          (33,996,000)           (27,460,000)
                                                    -----------------      -----------------

         Net property, plant and equipment                 46,040,000             35,807,000
                                                    -----------------      -----------------
                                                    $      68,693,000      $      53,561,000
                                                    =================      =================

LIABILITIES AND STOCKHOLDERS' EQUITY
Current liabilities:
   Current maturities of long-term debt             $              --      $       1,690,000
   Accounts payable                                         1,542,000              3,956,000
   Accrued liabilities:
          Payroll and other taxes                             380,000                566,000
          Other                                               497,000                494,000
                                                    -----------------      -----------------
           Total current liabilities                        2,419,000              6,706,000
                                                    -----------------      -----------------
Long-term debt, less current maturities                            --              7,893,000
                                                    -----------------      -----------------
Deferred income taxes                                       1,984,000              1,417,000

Stockholders' equity:
   Preferred stock - par value $1.00 per share;
       5,000,000 shares authorized, none
       outstanding                                                 --                     --
   Common stock - par value $.33 1/3 per share;
       10,000,000 shares authorized, 5,356,000
       and 4,199,250 shares issued and
       outstanding                                          1,785,000              1,400,000
   Additional paid-in capital                              38,215,000             17,174,000
   Retained earnings                                       24,290,000             18,971,000
                                                    -----------------      -----------------
           Total stockholders' equity                      64,290,000             37,545,000
                                                    -----------------      -----------------
                                                    $      68,693,000      $      53,561,000
                                                    =================      =================
</TABLE>

Contingencies (See Note 3)

See accompanying notes to the financial statements.



                                       -4-
<PAGE>   5

                           DAWSON GEOPHYSICAL COMPANY

                            STATEMENTS OF CASH FLOWS

                                   (UNAUDITED)

<TABLE>
<CAPTION>
                                                                    Nine Months Ended
                                                                         June 30
                                                             ------------------------------
                                                                 1998              1997
                                                             ------------      ------------
<S>                                                          <C>               <C>
Cash flows from operating activities:
         Net income                                          $  5,387,000      $  3,248,000

         Adjustments to reconcile net income to net cash
             provided by operating activities:
         Depreciation                                           6,816,000         5,456,000
      Gain on disposal of assets                                 (167,000)         (196,000)
      Other                                                        99,000            (3,000)
      Deferred income tax expense                                 567,000           523,000
      Change in current assets and liabilities:
             Increase in accounts receivable                   (4,380,000)       (2,293,000)
             Increase in prepaid expenses                        (261,000)         (145,000)
             Decrease in income taxes receivable                       --           193,000
             Decrease in accounts payable                      (2,414,000)         (519,000)
             Decrease in accrued liabilities                     (183,000)         (113,000)
             Increase in federal and state income
                  taxes payable                                        --           174,000
                                                             ------------      ------------

Net cash provided by operating activities                       5,464,000         6,325,000
                                                             ------------      ------------

Cash flows from investing activities:
      Proceeds from disposal of assets                            264,000           288,000
      Capital expenditures                                    (17,322,000)       (2,340,000)
      Proceeds from sale and maturity of
                  marketable securities                        14,993,000           742,000
         Investment in marketable securities                  (17,531,000)       (3,377,000)
                                                             ------------      ------------

Net cash used in investing activities                         (19,596,000)       (4,687,000)
                                                             ------------      ------------

Cash flows from financing activities:
      Principal payments on debt                               (9,583,000)         (643,000)
      Issuance of common stock                                 21,371,000                --
      Proceeds from exercise of stock options                      43,000           157,000
                                                             ------------      ------------

Net cash provided (used) by financing activities               11,831,000          (486,000)
                                                             ------------      ------------

Net increase (decrease) in cash and cash equivalents           (2,301,000)        1,152,000

Cash and cash equivalents at beginning of period                4,774,000         1,493,000
                                                             ------------      ------------

Cash and cash equivalents at end of period                   $  2,473,000      $  2,645,000
                                                             ============      ============
</TABLE>

See accompanying notes to the financial statements.



                                       -5-
<PAGE>   6

                           DAWSON GEOPHYSICAL COMPANY

                          NOTES TO FINANCIAL STATEMENTS


1.       OPINION OF MANAGEMENT

         Although the information furnished is unaudited, in the opinion of
management of the Registrant, the accompanying financial statements reflect all
adjustments (consisting only of normal recurring accruals) necessary for a fair
presentation of the financial condition and results of operations for the
periods presented. The results of operations for the three months and the nine
months ended June 30, 1998, are not necessarily indicative of the results to be
expected for the fiscal year.

2.       NOTES PAYABLE

         In April 1997, the Company entered into a loan agreement, as amended
(the "Loan Agreement"), with a bank. The Loan Agreement consists of (1) a
revolving line of credit of $6,000,000 which matures on April 15, 1999, (2) a
term note in the aggregate principal amount of $6,000,000 bearing interest at
the bank's prime rate and which matures on March 25, 2003 and (3) a term note in
the aggregate principal amount of $5,000,000 bearing interest at the prime rate
as published in The Wall Street Journal and which matures on April 15, 2003. The
notes are secured by eligible accounts receivable and equipment purchased from
loan proceeds.

         On November 25, 1997, the Company repaid all outstanding principal and
interest on the two Term Promissory Notes which have no reborrowing capacity.
The Company has not utilized the revolving line of credit.

3.       CONTINGENCIES

         The Company is a defendant in two lawsuits pending in the 112th and
83rd District Courts of Pecos County, Texas 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, prejudgment 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


                                       -6-


<PAGE>   7

Notes to Financial Statements (continued)


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. 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 is 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.

         The Company is party to other legal actions arising in the ordinary
course of its business, none of which management believes will result in a
material adverse effect on the Company's financial position or results of
operation, as the Company believes it is adequately insured.

4.       PUBLIC OFFERING

         On November 21, 1997, the Company completed an offering of 1,150,000
shares. The proceeds of the offering were approximately $21,371,000 after
deducting costs payable by the Company.

5.       NET INCOME PER COMMON SHARE

<TABLE>
<CAPTION>
                                               Three Months Ended             Nine Months Ended
                                                    June 30                       June 30
                                           -------------------------     -------------------------
                                              1998           1997           1998           1997
                                           ----------     ----------     ----------     ----------
<S>                                        <C>            <C>            <C>            <C>       
Numerator:
    Net income and numerator for basic
      and diluted net income per
      common share-income available
      to common stockholders               $3,087,000     $1,501,000     $5,387,000     $3,248,000
                                           ----------     ----------     ----------     ----------

      Denominator:
      Denominator for basic net income
         per common share-weighted
         average common shares              5,353,582      4,199,247      5,136,652      4,177,432
      Effect of dilutive securities-
         employee stock options                25,759         14,788         28,430         24,120
                                           ----------     ----------     ----------     ----------
      Denominator for diluted net
         income per common share-
         adjusted weighted average
         common shares and assumed
         conversions                        5,379,341      4,214,035      5,165,082      4,201,552
                                           ----------     ----------     ----------     ----------

    Basic net income per common share      $      .58     $      .36     $     1.05     $      .78
                                           ==========     ==========     ==========     ==========

    Diluted net income per common
      share                                $      .57     $      .36     $     1.04     $      .77
                                           ==========     ==========     ==========     ==========
</TABLE>



                                       -7-
<PAGE>   8

                     MANAGEMENT'S DISCUSSION AND ANALYSIS OF
                  FINANCIAL CONDITION AND RESULTS OF OPERATIONS


OVERVIEW

The results of operations for the three months ended June 30, 1998 are
indicative of near-perfect operating conditions along with continued strong
demand for the Company's 3-D seismic services. Hot and dry weather, favorable
permitting and relatively large prospects contributed to minimal down time and
optimal operating results. As the Company continues to enjoy favorable weather
during the fourth quarter of fiscal 1998, it is experiencing typical transitions
as crews complete one prospect and move to another. In May 1998, a 3,100-channel
Input/Output System Two Remote Recording System (RSR) became productive,
replacing the Halliburton MDS-18X recording system. With this upgrade the
Company operates four Input/Output System Two conventional cable connected
telemetry systems and two RSR systems. Capital expenditures for fiscal 1998 of
$17,322,000 are comprised of the system upgrade, additional data channels for
the existing crews, automotive units, and peripheral equipment. In November
1997, the Company completed a secondary public offering of 1,150,000 common
shares with net proceeds of $21,371,000. In reviewing the Company's financial
statements it should be noted that quarterly fluctuations in the Company's
results of operations can occur due to weather, land use permitting and other
factors.

RESULTS OF OPERATIONS

The Company's operating revenues for the first nine months of 1998 totaled
$45,991,000 versus $34,304,000 for the same period of fiscal 1997, an increase
of 34.1%. For the three months ended June 30, 1998, operating revenues totaled
$18,647,000 versus $12,520,000 for the same period of fiscal 1997, an increase
of 48.9%. The increase for the nine-month period reflects the Company's added
production capacity in response to continued strong demand for 3-D seismic
services. In August 1997, the Company added its first RSR system to bring the
number of crews to six. To further complement capacity, the Company has
continually added channels to each of its crews. The increase in the quarter
ended June 30, 1998 versus the same period of fiscal 1997 reflects the
near-perfect conditions described above in the Overview.

Operating expenses for the nine months ended June 30, 1998 totaled $30,094,000,
an increase of $7,254,000, or 31.8%, over the same period of fiscal 1997. For
the quarter ended June 30, 1998, operating expenses increased $3,066,000, or
37.7%. Operating expenses increased primarily as a result of increased personnel
and other expenses associated with equipment acquisitions and technological
upgrades.

General and administrative expenses for the nine months ended June 30, 1998
totaled $1,443,000, an increase of $391,000 from the same period of fiscal 1997.
For the quarter ended June 30, 1998, general and administrative expenses totaled
$478,000, an increase of $92,000 from the same quarter of fiscal 1997. The
increase primarily reflects


                                       8
<PAGE>   9

additional personnel required to support expanding operations. Due to the
Company's significant growth in recent years, the office building occupied since
1960 became outgrown, and the functions performed in that building have been
moved to a facility that accommodates the Company's continued growth. General
and administrative expenses totaled 3.1% of operating revenues for the nine
months ended June 30, 1998 as well as for the same period of the prior year.

Depreciation for the nine months ended June 30, 1998 totaled $6,816,000, an
increase of $1,360,000 from the same period of fiscal 1997. For the quarter
ended June 30, 1998, depreciation increased $776,000, or 46.1%. Depreciation
increased as a result of the capital expansion discussed below in "Liquidity and
Capital Resources."

Total operating costs for the first nine months of fiscal 1998 totaled
$38,353,000, an increase of 30.7%, from the same period of fiscal 1997 due to
the factors described above. For the quarter ended June 30, 1998, operating
costs increased 38.6% from the same period of the prior year. These increases
are consistent with the high proportion of relatively fixed total operating
costs (including personnel costs and depreciation) in conjunction with the
addition of the Company's sixth crew in August 1997. Income from operations of
$4,515,000 represents 24.2% of revenues for the three months ended June 30, 1998
as compared to a ratio of 18.5% for the same period of the prior year. For the
nine-month periods of fiscal years 1998 and 1997, the ratios of income from
operations to revenues are 16.6% and 14.4% respectively.

LIQUIDITY AND CAPITAL RESOURCES

Cash Flows

Net cash provided by operating activities for the nine months ended June 30,
1998 of $5,464,000 as compared to $6,325,000 for the same period of the prior
year is a decrease of $861,000. The net decrease is due to fluctuations in
working capital components as net income and depreciation increased for the
first nine months of fiscal 1998 as compared to the same period of fiscal 1997.
The increase in accounts receivable reflects increased activity, and accounts
receivable are considered collectible.

Net cash used in investing activities increased to $19,596,000 from $4,687,000
resulting from investment of offering proceeds and increased capital
expenditures in the first nine months of fiscal 1998 as compared to the same
period of fiscal 1997.

The cash flows provided by financing activities for the fiscal 1998 represent
the net of the offering proceeds reduced by the retirement of debt.

Capital Expenditures

The Company continually strives to supply market demand with technologically
advanced 3-D data acquisition recording systems and leading edge data processing
capabilities. In April of 1998 the Company announced the purchase of a
3,100-channel


                                        9
<PAGE>   10

Input/Output System Two Remote Recording System to replace the Halliburton
MDS-18X recording system. The approximate cost of the new system is $5,000,000.
Capital expenditures to date of $17,322,000 include the new system as well as
additions and replacements of cables and geophones, vehicles and other
peripheral equipment. For the remainder of fiscal 1998, the Company anticipates
the purchase of seismic data processing hardware and software, ancillary seismic
data acquisition instrumentation, and leasehold improvements for the move of its
corporate office. Depreciation has increased as a new crew has been placed into
service each year for the past several years.

Capital Resources

The Company believes that its capital resources including its holdings of
marketable securities, the availability of bank borrowings, and cash flow from
operations are adequate to meet its current operational needs and finance
capital needs as determined by market demand and technological developments.

Year 2000

The Company utilizes software and technologies throughout its operations that
may be affected by the date change in the year 2000 (Year 2000 Issue). An
assessment of the systems that will be affected by the Year 2000 Issue is
underway. The Company does not believe the costs related the Year 2000 Issue
will materially impact its results of operations. However, there can be no
guarantee that the systems of other companies, on which the Company's systems
rely, will be timely converted or that a failure to convert by another company
or a conversion that is incompatible with the Company's systems would not have a
material adverse effect on the Company.


                                       10
<PAGE>   11

                                    SIGNATURE




                    Pursuant to the requirements 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.


                                        DAWSON GEOPHYSICAL COMPANY
                                        (REGISTRANT)



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



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



DATE:         August 10, 1998
     ---------------------------------


<PAGE>   12

                               INDEX TO EXHIBITS

<TABLE>
<CAPTION>
     EXHIBIT
     NUMBER                             DESCRIPTION
    --------                            -----------            
    <S>                       <C>
       27                     Financial Data Schedule
       27.1                   Restated Financial Data Schedule
</TABLE>