<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 December 31, 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 December 31, 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
                                                             ------------

                      208 South Marienfeld, Midland, Texas
 ------------------------------------------------------------------------------
(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 December 31, 1998
     --------------------------------        --------------------------------
     Common Stock, $.33 1/3 par value                 5,405,794 shares



                                     -1-
<PAGE>   2



                           DAWSON GEOPHYSICAL COMPANY

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

                  Statements of Operations --
                        Three Months ended December 31, 
                        1998 and 1997                                                       3

                  Balance Sheets --
                        December 31, 1998 and September 30,
                        1997                                                                4

                  Statements of Cash Flows --
                        Three Months Ended December 31,
                        1998 and 1997                                                       5

                  Notes to Financial Statements                                             6

                  Management's Discussion and Analysis of
                        Financial Condition and Results of
                        Operations                                                          8
</TABLE>


Part II.          Other Information



















                                       -2-




<PAGE>   3







                          PART I. FINANCIAL INFORMATION



                           DAWSON GEOPHYSICAL COMPANY

                            STATEMENTS OF OPERATIONS
                                   (UNAUDITED)

<TABLE>
<CAPTION>
                                                                  Three Months Ended
                                                                      December 31
                                                          ---------------------------------
                                                               1998                 1997
                                                          ------------         ------------
<S>                                                       <C>                  <C>         
Operating revenues                                        $  8,018,000         $ 13,787,000

Operating costs:
    Operating expenses                                       6,433,000            9,093,000
    General and administrative                                 546,000              482,000
    Depreciation                                             2,711,000            2,111,000
                                                          ------------         ------------

                                                             9,690,000           11,686,000
                                                          ------------         ------------

Income (loss) from operations                               (1,672,000)           2,101,000
Other income (expense):
         Interest income                                       186,000              147,000
         Other                                                   6,000               15,000
         Interest expense                                         --               (125,000)
         Gain on disposal of assets                             11,000              145,000
                                                          ------------         ------------

Income (loss) before income tax                             (1,469,000)           2,283,000

Income tax benefit (expense):
         Current                                               615,000             (666,000)
         Deferred                                             (120,000)            (134,000)
                                                          ------------         ------------

                                                               495,000             (800,000)
                                                          ------------         ------------

Net income (loss)                                         $   (974,000)        $  1,483,000
                                                          ============         ============


Net income (loss) per common share                        $       (.18)        $        .31
                                                          ============         ============

Net income (loss) per common share-assuming
         dilution                                         $       (.18)        $        .31
                                                          ============         ============

Weighted average equivalent common
         shares outstanding                                  5,368,790            4,712,391
                                                          ============         ============


Weighted average equivalent common
         shares outstanding-assuming dilution                5,371,017            4,745,341
                                                          ============         ============
</TABLE>


See accompanying notes to the financial statements.






                                       -3-


<PAGE>   4

                           DAWSON GEOPHYSICAL COMPANY

                                 BALANCE SHEETS

<TABLE>
<CAPTION>

                                                             December 31, 1998    September 30,1998
                                                             -----------------    -----------------
                                                                (UNAUDITED)
<S>                                                          <C>                  <C>         
ASSETS
Current assets:
   Cash and cash equivalents                                   $  8,757,000         $  5,745,000
   Short-term investments                                         6,484,000            6,515,000
   Accounts receivable                                            8,093,000           11,821,000
     Income taxes receivable                                      1,701,000            1,050,000
   Prepaid expenses                                                 360,000              416,000
                                                               ------------         ------------

           Total current assets                                  25,395,000           25,547,000
                                                               ------------         ------------


Property, plant and equipment                                    73,500,000           73,584,000
   Less accumulated depreciation                                (30,307,000)         (27,672,000)
                                                               ------------         ------------

           Net property, plant and equipment                     43,193,000           45,912,000
                                                               ------------         ------------

                                                               $ 68,588,000         $ 71,459,000
                                                               ============         ============

LIABILITIES AND STOCKHOLDERS' EQUITY
Current liabilities:
   Accounts payable                                                 633,000            1,766,000
   Accrued liabilities:
     Payroll costs and other taxes                                  309,000              635,000
     Other                                                             --                810,000
                                                               ------------         ------------

           Total current liabilities                                942,000            3,211,000
                                                               ------------         ------------


Deferred income taxes                                             2,726,000            2,606,000

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, 5,405,794
     and 5,361,000 shares issued and
     outstanding respectively                                     1,802,000            1,787,000
   Additional paid-in capital                                    38,492,000           38,256,000
   Retained earnings                                             24,626,000           25,599,000
                                                               ------------         ------------

           Total stockholders' equity                            64,920,000           65,642,000
                                                               ------------         ------------

                                                               $ 68,588,000         $ 71,459,000
                                                               ============         ============
</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>
                                                                                 Three Months Ended
                                                                                    December 31
                                                                         ---------------------------------
                                                                              1998                 1997
                                                                         ------------         ------------
<S>                                                                      <C>                  <C>         
Cash flows from operating activities:
         Net income (loss)                                               $   (974,000)        $  1,483,000

         Adjustments to reconcile net income to net cash
             provided by (used in) operating activities:
         Depreciation                                                       2,711,000            2,111,000
         Gain on disposal of assets                                           (11,000)            (145,000)
         Non-cash interest income                                                --                (22,000)
         Deferred income taxes                                                120,000              134,000
         Other                                                                319,000               49,000
         Change in current assets and liabilities:
         Decrease (increase) in accounts receivable                         3,728,000           (3,886,000)
         Increase in income taxes receivable                                 (651,000)                --
         Decrease in prepaid expenses                                          56,000                2,000
         Decrease in accounts payable                                      (1,133,000)            (136,000)
         Decrease in accrued liabilities                                   (1,136,000)            (629,000)
         Increase in federal and state income taxes payable                      --                680,000
                                                                         ------------         ------------

Net cash provided by (used in) operating activities                         3,029,000             (359,000)
                                                                         ------------         ------------

Cash flows from investing activities:
         Proceeds from disposal of assets                                      11,000              209,000
         Capital expenditures                                                 (50,000)          (4,276,000)
         Proceeds from maturity of marketable securities                    1,000,000            1,000,000
         Investment in marketable securities                                 (978,000)          (9,992,000)
                                                                         ------------         ------------

Net cash used in investing activities                                         (17,000)         (13,059,000)
                                                                         ------------         ------------

Cash flows from financing activities:
         Issuance of common stock                                                --             21,371,000
         Principal payments on debt                                              --             (9,583,000)
         Proceeds from exercise of stock options                                 --                  7,000
                                                                         ------------         ------------

Net cash provided by financing activities                                        --             11,795,000
                                                                         ------------         ------------

Net increase (decrease) in cash and cash equivalents                        3,012,000           (1,623,000)

Cash and cash equivalents at beginning of period                            5,745,000            4,774,000
                                                                         ------------         ------------

Cash and cash equivalents at end of period                               $  8,757,000         $  3,151,000
                                                                         ============         ============
</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 December 31,
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 (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, 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, 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 claims asserted against it in
such suits and it intends to


                                       -6-

<PAGE>   7


Notes to Financial Statements (continued)


vigorously defend itself against such claims. In addition, the Company believes
that it has approximately $11 million of liability insurance coverage to
provided 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. 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.

         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

         The following table sets forth the computation of basic and diluted net
income per common share:
<TABLE>
<CAPTION>

                                                                 Three Months Ended
                                                                     December 31

                                                              1998                1997
                                                          -----------         -----------
<S>                                                       <C>                 <C>
Numerator:
  Net income (loss) and numerator for basic
    and diluted net income per
    common share-income available
    to common stockholders                                $  (974,000)        $ 1,483,000
                                                          -----------         -----------

Denominator:
  Denominator for basic net income
    per common share-weighted
    average common shares                                   5,368,790           4,712,391
  Effect of dilutive securities-
    employee stock options                                      2,227              32,950
                                                          -----------         -----------
  Denominator for diluted net
    income per common share-
    adjusted weighted average
    common shares and assumed
    conversions                                             5,371,017           4,745,341
                                                          -----------         -----------

Net income (loss) per common share                        $      (.18)        $       .31
                                                          ===========         ===========

Net income (loss) per common
  share--assuming dilution                                $      (.18)        $       .31
                                                          ===========         ===========
</TABLE>





                                       -7-


<PAGE>   8


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


The following discussion should be read in conjunction with the Company's
financial statements. In addition, in reviewing the Company's financial
statements it should be noted that the Company's revenues directly relate to oil
and gas exploration and production activity and fluctuations in the Company's
results of operations can occur due to weather, land use permitting and other
factors. Decreases in oil and gas activities have adversely affected the demand
for the Company's services and the Company's results of operations.

This report may contain certain forward-looking information regarding the
Company, including projections, estimates, forecasts, plans and objectives.
Although management believes that all such statements are based upon reasonable
assumptions, no assurance can be given that the actual results will not differ
materially from those contained in such forward-looking statements.


OVERVIEW

The revenues of the Company were severely impacted during the quarter ended
December 31, 1998 due to low crude oil prices. Demand for the Company's services
is expected to continue to be directly related to crude oil prices. The Company
has taken cost reduction measures in response to decreased demand for its
services. The Company reduced the number of operating crews from six to three
during the quarter.


RESULTS OF OPERATIONS

The Company's operating revenues for the first quarter of 1999 totaled
$8,018,000 versus $13,787,000 for the same period of fiscal 1998, a decrease of
41.8%. Demand for the Company's services was negatively impacted by low crude
oil prices. During the quarter ended December 31, 1998, the Company reduced the
number of operating crews from six to three.

Operating expenses for the quarter ended December 31, 1998 totaled $6,433,000, a
decrease of 29.3% over the same period of fiscal 1998. Operating expenses
decreased as a result of decreased demand for the Company's services. Additional
cost reduction measures, such as employee layoffs and salary reductions, were
implemented in January 1999.

General and administrative expenses for the quarter ended December 31, 1998
totaled $546,000, an increase of 13.3% over the same period of fiscal 1998. The
increase reflects personnel and other expenses added throughout fiscal 1998 that
were required to support expanding operations. Cost reduction measures were
implemented in January 1999 to reduce general and administrative expenses. Rent
expense is also reflected in the quarter ended December 31, 1998 as the office
building owned and occupied since 1960 became outgrown, and the functions
performed in that building were moved to a more accommodating, leased facility
in July 1998.

Depreciation for the quarter ended December 31, 1998 totaled $2,711,000, an
increase of 28.4% from the same quarter of fiscal 1998. Depreciation continued
to increase as a result of capital expansion during fiscal 1998. See the
discussion below in "Liquidity and Capital Resources."

                                       8

<PAGE>   9



Total operating costs for the first quarter of fiscal 1999 totaled $9,690,000, a
decrease of 17.1% from the first quarter of fiscal 1998 due to the factors
described above. The 41.8% decrease of revenues as compared to the 17.1%
decrease of total operating costs reflects the high proportion of relatively
fixed total operating costs (including personnel costs of active crews and
depreciation costs).


LIQUIDITY AND CAPITAL RESOURCES

Cash Flows

Net cash provided by operating activities of $3,029,000 in the quarter ended
December 31, 1998 reflects decreases in accounts receivable and current
liabilities which occurred due to reduced revenues and cost cutting measures.
The increase in income taxes receivable in fiscal 1999 is a result of the
Company's recognition of a tax benefit generated by the net loss for the quarter
ended December 31, 1998.

Net cash used in investing activities decreased to $17,000 from $13,059,000
resulting from significantly decreased capital expenditures in fiscal 1999 as
compared to fiscal 1998; and, in fiscal 1998, the Company invested proceeds from
the November 1997 public offering in U.S. Treasury instruments.

The cash flows provided by financing activities for the quarter ended December
31, 1997 represent the difference between the offering proceeds and the 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. Depreciation has increased as a new crew has been placed into
service each year for the past several years. Capital expenditures for fiscal
1999 are not anticipated to be material. The Company will maintain equipment in
and out of service in anticipation of increased future demand of the Company's
services.

Capital Resources

The Company believes that its capital resources including its short-term
investments, 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.

Litigation

The Company is a defendant in two lawsuits relating to a July 1995 accident
involving a van owned by the Company in which four Company employees died. The
Company believes that it has meritorious defenses to the claims asserted against
it in such suits. Further, while the plaintiffs seek damages in excess of the
Company's liability insurance policies, the Company believes that its liability
insurance should provide adequate coverage of the damages, if any, which may be
assessed against the Company in such litigation. 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 any such damages. A judgment 


                                       9

<PAGE>   10



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.

Year 2000

The Company utilizes software and technologies throughout its operations that
may be vulnerable to the date change in the year 2000. Identification,
assessment, and in some cases, replacement of equipment that may be affected by
the year 2000 is underway. Software controlled by the Company, including its
propriety seismic processing package, has been tested successfully. Replacements
and upgrades have not been accelerated by the year 2000 issue and do not
represent costs in addition to normal operating expenditures. The Company has
begun communications with its significant suppliers to determine if those
parties have appropriate plans to remedy year 200 issues when their systems
interface with the Company's systems or may otherwise impact the operations of
the Company. 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. To date, the Company has spent approximately $20,000 primarily in the
assessment of and testing for year 2000 compliance. Assessment will continue
throughout fiscal 1999, with an additional estimated cost of $30,000. Although
the Company is not aware of any material operational issues, there can be no
assurance that there will not be a delay in, or increased costs associated with,
the implementation of the necessary systems and changes to address the year
2000. A potential source of risk includes, but is not limited to, the inability
of principal suppliers to be year 2000 compliant, which could result in an
interruption of the Company's services. The Company currently does not have a
formal contingency plan. If unforeseen problems are encountered that relate to
the year 2000, possible solutions will be evaluated and the most efficient will
be enacted.



                                       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
                                       Chief Financial Officer



DATE:       February 4, 1999
     -----------------------



                                      -11-



<PAGE>   12



                               Index to Exhibits
<TABLE>
<CAPTION>
Exhibit No.                Description
-----------                -----------
<S>                       <C>  
  27                       Financial Data Schedule

</TABLE>